Forallx Adelaide

forall𝓍ADELAIDE forall𝓍 ADELAIDE Antony Eagle Adelaide University Tim Button University College London P.D. Magnus University at Albany, State University of New York © 2005–2025 by Antony Eagle, Tim Button and P.D. Magnus. Some rights reserved. This version of forall𝓍ADELAIDE is current as of 19 December 2025. This book is a derivative work created by Antony Eagle, based upon Tim Button’s 2016 Cam‑ bridge version of P.D. Magnus’s forall𝓍(version 1.29). There are pervasive substantive changes in content, theoretical approach, coverage, and appearance. (For one thing, it’s more than twice as long.) You can find the latest release version of forall𝓍ADELAIDE at: The current version of forall𝓍Cambridge is available at github.com/OpenLogicProject/ forallx-cam, which has now also diverged noticeably from the 2016 version. Magnus’ original is available at fecundity.com/logic. This book, like its predecessors, is released under a Creative Commons license (Attribution 4.0). This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Typesetting was carried out entirely in X E L A T EX using the Memoir class. The body text is set in Palatino Linotype; math is set in STIX Two Math; sans serif in Avenir Next; and monospaced text in Iosevka Fixed Light. Natural deduction proofs are typeset using the fitch.sty package by Peter Selinger, revived by Richard Zach. Kaurna miyurna, Kaurna yarta, ngai tampinthi. Made on Kaurna land. Its sovereignty was never ceded. Contents 1 Key Notions 1 1 Arguments 2 2 Valid Arguments 5 3 Other Logical Notions 17 2 The Language of Sentential Logic 23 4 First Steps to Symbolisation 24 5 Connectives 34 6 Sentences of Sentential 47 7 Use and Mention 54 3 Truth Tables 60 8 Truth‑Functional Connectives 61 9 Complete Truth Tables 72 10 Semantic Concepts 77 11 Entailment and Validity 82 12 Truth Table Shortcuts 90 13 Partial Truth Tables 95 14 Expressiveness of Sentential 102 4 The Language of Quantified Logic 107 15 Building Blocks of Quantifier 108 iii iv CONTENTS 16 Sentences with One Quantifier 119 17 Multiple Generality 133 18 Identity 146 19 Definite Descriptions 155 20 Sentences of Quantifier 166 5 Interpretations 172 21 Extensionality 173 22 Truth in Quantifier 195 23 Semantic Concepts 210 24 Demonstrating Consistency and Invalidity 215 25 Reasoning about All Interpretations 221 6 Natural Deduction for Sentential 226 26 Proof and Reasoning 227 27 The Idea of Natural Deduction 232 28 Basic Rules for Sentential: Rules without Subproofs 238 29 Basic Rules for Sentential: Rules with Subproofs 250 30 Some Philosophical Issues about Conditionals, Meaning, and Negation 275 31 Proof‑Theoretic Concepts 281 32 Proof Strategies 291 33 Derived Rules for Sentential 293 34 Alternative Proof Systems for Sentential 304 7 Natural Deduction for Quantifier 309 35 Basic Rules for Quantifier 310 36 Derived Rules for Quantifier 332 37 Rules for Identity 336 38 Proof‑Theoretic Concepts and Semantic Concepts 343 39 Next Steps 346 Appendices 355 A Alternative Terminology and Notation 355 B Quick Reference 360 C Index of defined terms 365 Acknowledgements, etc. 369 List of Figures 1 How the sections depend on one another. ....................... viii 4.1 A phrase structure tree for Example 18. ......................... 26 4.2 A phrase structure tree for Example 19. ......................... 26 4.3 Paraphrase of Example 26 showing its subsentential structure, as in Example 27. . . 29 4.4 Different sentential structures for ‘Not A and B’ shown in schematic syntactic trees. 29 6.1 Formation tree for ‘¬(𝑃∧¬(¬𝑄∨𝑅))’. .......................... 50 21.1 An interpretation represented diagrammatically. .................... 182 21.2 A representation of ‘Every Rose Has Its Thorn’. ..................... 183 21.3 Representing the argument from §15. .......................... 184 21.4 A simple but ambiguous Euler diagram. ......................... 184 21.5 ‘Chat Systems’, xkcd.com/1810/.............................. 185 21.6 A simple graph. ....................................... 186 21.7 A graph with ‘loops’. ................................... 186 21.8 A more complicated graph. ................................ 187 21.9 Multiple techniques used to depict a complex interpretation. ............. 187 21.10 A graph of a reflexive relation on {1,3,4}......................... 188 21.11 A graph of the transitive relation ‘older than’ on some University of Adelaide buildings.189 21.12 An extract of Qantas’ route network. ........................... 189 21.13 ‘next to’: a symmetric but intransitive relation. ..................... 190 21.14 >(black arrows) and ≥(orange dotted arrows) on the domain {0,1,2}.. . . . . . . 191 21.15 Black arrows indicate both ⊂and ⊆, dotted arrows indicate ⊆only, on the domain ℘{𝑎,𝑏}={{𝑎,𝑏},{𝑎},{𝑏},∅}................................. 193 29.1 Proof of ¬(𝐴∨𝐵)∴(¬𝐴∧¬𝐵) .............................. 270 29.2 Proof that (𝐴∨𝐵),¬(𝐴∧𝐶),¬(𝐵∧¬𝐷)∴(¬𝐶∨𝐷)................... 271 29.3 A complicated proof .................................... 272 33.1 Disjunctive syllogism is derivable in the standard proof system. . . . . . . . . . . . 295 v vi LIST OF FIGURES 33.2 Modus tollens is derivable in the standard proof system. ................ 296 33.3 Tertium non datur is derivable in the standard proof system. .............. 298 34.1 Schematic ∨E proof. .................................... 306 34.2 Schematic proof using DS to emulate ∨E. ........................ 306 39.1 A red‑yellow spectrum. .................................. 352 How to Use This Book This book was designed for use in conjunction with logic courses at the University of Ad‑ elaide/Adelaide University. But it is suitable for self‑study as well. I have included a number of features to assist your learning. ›forall𝓍is divided into seven chapters, each further divided into sections and subsections. The sections are continuously numbered. –Chapter 1gives an overview of how I understand the project of formal logic; –Chapters 2–3cover sentential or truth‑functional logic; –Chapters 4–5cover quantified or predicate logic; –and Chapters 6–7cover the formal proof systems for our logical languages. ›The book contains many cross‑references to other sections. So a reference to ‘§6.2’ indic‑ ates that you should consult section 6, subsection 2 – you will find this on page 47. Cross‑ references and entries in the table of contents are hyperlinked – this includes numbered example sentences, such as this reference here: 71. ›Figure 1shows how the sections depend on one another. For example, the arrows coming from §21 in the diagram show that understanding that section requires familiarity with §11 and §20, and also any sections on which they depend. ›Each section in the book concludes with a box labelled ‘Key Ideas in §𝑛’. These are not a summary of the section, but contain some indication of what I regard as the main ideas that you should be taking away from your reading of the section. ›Logical ideas and notation are pretty ubiquitous in philosophy, and there are a lot of differ‑ ent systems. We cannot cover all the alternatives, but some indication of other terminology and notation is contained in Appendix A. ›A quick reference to many of the aspects of the logical systems I introduce can be found in Appendix B. vii 6 KEY NOTIONS 2. The conclusion might not follow from, or be a consequence of, the premises – even if the premises were true, they would not support the conclusion. To determine whether or not the premises of an argument are true is often a very important matter. But that is normally a task best left to experts in the field: as it might be, historians, scientists, or whomever. In our role as logicians, we are more concerned with arguments in general. So we are (usually) more concerned with the second way in which arguments can go wrong. 2.2 Conclusive Arguments As logicians, we want to be able to determine when the conclusion of an argument follows from the premises. One way to put this is as follows. We want to know whether, if all the premises were true, the conclusion would also have to be true. This motivates a definition: An argument is CONCLUSIVE if, and only if, the truth of the premises guarantees the truth of the conclusion. In other words: an argument is conclusive if, and only if: it is not possible for the premises of the argument to be true while the conclusion is false. Consider another argument: You are reading this book. This is a logic book. So: You are a logic student. This is not a terrible argument. Both of the premises are true. And most people who read this book are logic students. Yet, it is possible for someone besides a logic student to read this book. If your housemate picked up the book and thumbed through it, they would not immediately become a logic student. So the premises of this argument, even though they are true, do not guarantee the truth of the conclusion. This is not a conclusive argument. Or consider this pair of arguments: Mary has two brothers; So: There are three siblings in her fam‑ ily. Mary has two brothers; So: There are at least three siblings in her family. The argument on the left might be pretty good: that premise provides some reason to accept the conclusion, especially if you imagine a real conversation in which someone makes this case. It would be weird to use a premise about Mary’s brothers in an argument for a conclusion about her siblings unless Mary had no sisters. Weird, but not impossible. Strictly speaking, if Mary had a sister the speaker did not mention, it would be possible for the premise to be true while the conclusion is false. So the left argument is not conclusive, interpreted strictly. The argument on the right shows that the premise does make a compelling case for a related but more hedged conclusion. You might think ‘that second argument is a bit nit‑picking’. But that is exactly what makes it so watertight. It doesn’t depend on what would make for a ‘normal’ §2. VALID ARGUMENTS 7 conversation, or whether you are making the same assumptions as the speaker, etc. No matter what, the truth of the premises secures the truth of the conclusion: it is conclusive. The crucial thing about a conclusive argument is that it is impossible, in a very strict sense, for the premises to be true whilst the conclusion is false. Consider this example: Oranges are either fruits or musical instruments. Oranges are not fruits. So: Oranges are musical instruments. The conclusion of this argument is ridiculous. Nevertheless, it follows from the premises. If both premises were true, then the conclusion would just have to be true. So the argument is conclusive. Why is this argument conclusive? The most important factor for us in considering what makes an argument conclusive is to examine the argument’s STRUCTURE – the grammatical forms of the premises and conclusion. An argument will be conclusive if its structure guarantees that its premises support the conclusion. In the present case, one premise says that oranges are in one of two categories; the other premise says that oranges are not in the first category. We conclude that they are in the second category. The premises and conclusion are about oranges. But it is plausible to think that any argument with this same sort of structure must be conclusive, whether we are talking about oranges, or cars – or anything really. Conclusiveness of an argument is insensitive to the truth or falsity of the premises. An argument can be conclusive while nevertheless going wrong in the first of the ways we identified in §2.1. Consider this argument The earth has twenty‑eight moons; So: The earth has an even number of moons. The argument is conclusive, as there is no possible way in which the earth could have twenty‑ eight moons while having an odd number of moons. But the premise is so obviously false that this argument could never be used to persuade anyone. The premise supports the conclusion, but that support is moot given the evident falsity of the premise. 2.3 Reasons to Believe A conclusive argument, in the logician’s sense, links the premises to the conclusion. It turns the reasons you have for accepting to the premises into reasons to accept its conclusion. But a conclusive argument need not provide you with a reason to believe the conclusion. One way this can happen is when you don’t accept any of the premises in the first place. When the premises support the conclusion, that might just mean that they would be excellent reasons to accept the conclusion – if only they were true! So: we are interested in whether or not a conclusion follows from some premises. Don’t, though, say that the premises infer the conclusion. Entailment is a relation between premises and conclu‑ sions; inference is something we do. So if you want to mention inference when the conclusion follows from the premises, you could say that one may infer the conclusion from the premises. But even this may be doubted. Often, when you believe the premises, a conclusive argument provides you with a reason to believe the conclusion. In that case, it might be appropriate for you to infer the conclusion from the premises. 8 KEY NOTIONS But sometimes a conclusive argument shows that some premises support a conclusion you can‑ not accept. Suppose, for example, that you know the conclusion to be false. The fact that the argument is conclusive and has a false conclusion tells you that the premises cannot all be true. (Consider the argument from the previous section with the false conclusion ‘Oranges are musical instruments’: the second premise is as absurd as the conclusion.) In general, when an argument is conclusive ›the truth of all the premises guarantees the truth of the conclusion; and equally ›the falsity of the conclusion guarantees the falsity of at least one of the premises. In this sort of situation, you might find that the argument gives you a better reason to abandon any belief in one of the premises than to accept the conclusion. A conclusive argument shows there is some reason to believe its conclusion, if you accept its premises; it doesn’t mean there aren’t better reasons to reject its premises, if you reject its conclusion. Consider this sort of ex‑ ample:1 If I look in the cupboard, I’ll find some muesli. I am looking in the cupboard. So: I find some muesli. Someone might believe both premises, and not accept the conclusion, because on looking in the cupboard, they do not see the muesli. (Someone else finished it off earlier.) It would be silly for this person to ‘follow the argument where it leads’. Rather, they should use the fact that these premises entail a conclusion they now know to be false, having just looked, to reject one of the premises. The obvious candidate is the first premise. So this person should probably stop believing that they’ll find muesli if they look in the cupboard, and start believing instead that they are out of muesli or suchlike. Some cases are less straightforward. Consider this argument: It is immoral to cause avoidable suffering. Eating meat causes avoidable suffering. So: Eating meat is immoral. Many people will find the premises plausible, and the conclusion therefore compelling. This is part of the case for vegetarianism. But not everyone finds the conclusion acceptable; such people end up finding that one or both of the premises should be rejected. Interestingly, people may discover they still find the premises attractive even when they recognise a conclusion follows that they cannot bring themselves to accept. Here they find that there is some reason to accept the premises (perhaps they seem true at first glance), and some reason to reject them (they have, at second glance, consequences the person cannot accept). I don’t want to adjudicate the merits of this argument here. I only want to emphasise that even offering a conclusive argument to someone, with premises that they currently accept, is not enough to make them come to believe the conclusion. 1This sort of example is discussed by Gilbert Harman, Change in View, MIT Press, esp. ch. 2. §2. VALID ARGUMENTS 9 Sometimes people think that logic will provide a powerful tool to persuade and convince others of their point of view. They sometimes want to study logic as if it is some dark art enabling them to subdue beliefs that contradict their own. This is not really a very nice thing to want to do – to force a belief on someone, whether they want to believe it or not – and so it is not particularly regrettable that logic doesn’t help you to do it. Logic can show which claims follow from which others, and which contradict one another. It can help us elaborate the content of some claim, or delineate the commitments a certain belief would incur. But logic does not tell you what to believe, even when you have a conclusive argument. The question, what ought I to believe? is one of the deepest in the area of philosophy known as EPISTEMOLOGY, the theory of knowledge. Logic is not able to answer that question all by itself. Even if logic tells you that there is a conclusive argument from premise 𝒜to conclusion ℬ, logic can’t tell you whether you ought to believe both, or reject both. However, logic will tell you something important, even if it is only a limited part of the answer to the question of rational belief. It will tell you that, when you know an argument to be conclusive, you cannot both accept its premises while rejecting its conclusion – at least, not while being ideally rational. Thus conceived, logic is not even a science of reasoning, because it does not tell you what to think. Logic can tell you which packages of claims (premises and conclusions) can and cannot be simultaneously true together. But logic can’t tell you, or anyone else, which packages of claims to accept. (We return to the topic of logic and reasoning again in §§26.1,33.6.) 2.4 Conclusiveness for Special Reasons An argument can be conclusive for reasons unrelated to its structure. Take this example about my pet fox Juanita: Juanita is a vixen. So: Juanita is a fox. It is impossible for the premise to be true and the conclusion false. So the argument is conclusive. But this is not due to the structure of the argument. Here is an inconclusive argument with seem‑ ingly the same structure or form. The new argument is the result of replacing the word ‘vixen’ in the first argument with the word ‘cathedral’, but keeping the overall grammatical structure the same: Juanita is a cathedral. So: Juanita is a fox. This might suggest that the conclusiveness of the first argument is keyed to the meaning of the words ‘vixen’ and ‘fox’. But, whether or not that is right, it is not simply the FORM of the argument that makes it conclusive. It is instructive to compare the first argument with this modification, where we replace ‘vixen’ with the near‑synonym ‘female fox’: Juanita is a female fox. So: Juanita is a fox. This also seems to be conclusive. But now we might suspect the occurrence of the word ‘fox’ in both premise and conclusion is not mere coincidence, but an essential part of the explanation as to why this is conclusive. 10 KEY NOTIONS Equally, consider the argument: The sculpture is green all over. So: The sculpture is not red all over. Again, because nothing can be both green all over and red all over, the truth of the premise would guarantee the truth of the conclusion. So the argument is conclusive. But here is an inconclusive argument with the same form: The sculpture is green all over. So: The sculpture is not shiny all over. The argument is inconclusive, since it is possible to be green all over and shiny all over. (I might paint my nails with an elegant shiny green varnish.) Plausibly, the conclusiveness of this argu‑ ment is keyed to the way that colours (or colour‑words) interact. But, whether or not that is right, it is not simply the form of the argument that makes it conclusive. An argument can be conclusive due to its structure, and also be conclusive for other reasons. Arguably, this might be going on in the argument discussed at the end of §2.2, with the premise ‘Oranges are not fruits’. Some people might think this premise has to be false, because of what oranges are. (Many will say that being a fruit is an essential part of what it is to be an orange.) But if the premise ‘Oranges are not fruits’ has to be false, it is not possible for the premises to be true. So it is not possible for premises to be true while the conclusion is false. Hence the argument is conclusive – both because it has a good structure, but also because it has a premise that cannot be true.2 2.5 Validity Logicians try to steer clear of controversial matters like whether there is a definition of an or‑ ange that requires it to be a fruit, or whether there is a ‘connection in meaning’ between being green and not being red. It is often difficult to figure such things out from the armchair (a logi‑ cian’s preferred habitat), and there may be widespread disagreement even among subject matter experts. So logicians do not study conclusive arguments in general, but rather concentrate on those con‑ clusive arguments which have a good structure or form.3This is why the logic we are studying is sometimes called FORMAL LOGIC. We introduce a special term for the class of arguments logicians are especially interested in: 2When an argument has an impossible premise, any argument with that premise will be conclusive no matter what the conclusion is! So this is a weird kind of case of conclusiveness. But nothing much really turns on it, and it is simpler to simply count it as conclusive than to try and separate out such ‘degenerate’ cases of conclusive arguments. See also §3.3. 3It can be very hard to tell whether an invalid argument is conclusive or inconclusive. Consider the argument ‘The sea is full of water; so the sea is full of H2O’. This is conclusive, since water just is the same stuff as H2O. The sea cannot be full of that water stuff without being full of that exact same stuff, namely, H2O stuff. But it took a lot of chemistry and ingenious experiments to figure out that water is H2O. So it was not at all obvious that this argument was conclusive. On the other hand, it is generally very clear when an argument is conclusive due to its structure – you can just see the structure when the argument is presented to you. §2. VALID ARGUMENTS 11 An argument is VALID if, and only if, it is conclusive due to its structure; other‑ wise it is INVALID. The notion of the structure of a sentence, or an argument, is an intuitive one. I make the notion more precise in §4.1. Relying on our intuitive grasp of the notion for now, however, we can see the argument about ogres on the right has the same form as the argument on the left about oranges (slightly tweaked from our earlier presentation in §2.2 to make its structure clearer). It is easy to see that both of these arguments are conclusive and valid: Either Oranges are fruits or oranges are musical instruments. It is not the case that Oranges are fruits. So: Oranges are musical instruments. Either Ogres are fearsome or ogres are mythical. It is not the case that Ogres are fear‑ some. So: Ogres are mythical. The shared structure of these two arguments is something like this: Either 𝒜or ℬ. It is not the case that 𝒜. So: ℬ. Any argument with this structure will be conclusive in virtue of structure, and hence valid. It does not matter, really, what sentences we put in place of ‘𝒜’ and ‘ℬ’. (Within limits: you can’t put a question or an exclamation and get a valid argument – see §3.1.) This highlights that valid arguments do not need to have true premises or even true conclusions. We can put a true sentence in place of 𝒜and a false sentence in place of ℬ, and both premises and the conclusion will be false. The argument is still valid. Conversely, having true premises and a true conclusion is not enough to make an argument valid. Consider this example: London is in England. Beijing is in China. So: Paris is in France. The premises and conclusion of this argument are, as a matter of fact, all true. But the argument is invalid. If Paris were to declare independence from the rest of France, then the conclusion would be false, even though both of the premises would remain true. Thus, it is possible for the premises of this argument to be true and the conclusion false. The argument is therefore inconclusive, and hence invalid. Return briefly to another example we discussed earlier: Juanita is a female fox. So: Juanita is a fox. This seems to have something like this structure: 12 KEY NOTIONS 𝑎is an ℱ 𝒢. So: 𝑎is a 𝒢. For most adjectives ℱ, this structure yields a conclusive argument when you replace the schem‑ atic letters by English words. E.g., Bob is a tall man; So: Bob is a man. But not all: some adjectives like ‘fake’ or ‘alleged’ do not yield conclusive arguments when sub‑ stituted for ℱ: ‘This is a fake gun; so this is a gun’ is not a conclusive argument. We will return in §15 to the logical structure of examples like these, and to expressions like ‘fake gun’ in §16.6. 2.6 Soundness The important thing to remember is that validity is not about the actual truth or falsity of the sentences in the argument. It is about whether the structure of the argument ensures that the premises support the conclusion. Nonetheless, we shall say that an argument is SOUND if, and only if, it is both valid and all of its premises are true. So every sound argument is valid and conclusive. But not every valid argument is sound, and not every conclusive argument is sound. It is often possible to see that an argument is valid even when one has no idea whether it is sound. Consider this extreme example (after Lewis Carroll’s Jabberwocky): ’Twas brillig, and the slithy toves did gyre and gimble in the wabe. So: The slithy toves did gyre and gimble in the wabe. This argument is valid, simply because of its structure (it has a premise conjoining two claims by ‘and’, and a conclusion which is one of those claims). But is it sound? That would depend on figuring out what all those nonsense words mean! 2.7 Inductive Assessment of Arguments Many good arguments are inconclusive and invalid. Consider this one: In January 1997, it rained in London. In January 1998, it rained in London. In January 1999, it rained in London. In January 2000, it rained in London. So: It will rain next January in London. This argument generalises from observations about several cases to a conclusion about a future case. Though it is invalid, that doesn’t mean it is a bad argument. The premises appear to provide some support for the conclusion, though it falls short of being conclusive. INDUCTION describes a form of reasoning from evidence to hypotheses about that evidence. In the above example, we draw a conclusion that the pattern we’ve seen in the evidence to date will continue, at least in the short term. Another example: when we reason from a sample of eligible §2. VALID ARGUMENTS 13 voters to a hypothesis about how the whole population will vote, we are reasoning inductively. INDUCTIVE LOGIC is the attempt to generalise deductive logic to evaluate arguments in line with the canons of good inductive reasoning. The proponents of inductive logic think there might be a generalisation of deductive logic which enables us to evaluate arguments in a more fine‑ grained way than the options we’ve canvassed so far (i.e., just ‘valid’, and ‘invalid and conclusive’ and ‘invalid and inconclusive’). A significant part of the project of inductive logic is the attempt to classify inconclusive arguments in terms of whether their premises provide good inductive evidence in favour of their conclusions.4 In the example, the premises are of the form ‘In January 𝑛, it rains’, and the conclusion is of the form ‘Next January, it rains’. This argument is inconclusive, as nothing in the premises by them‑ selves necessitates anything about a future unobserved case. It might well be that the premises provide decent support for that conclusion; weather might be hard to predict, but broad climate patterns such as this one we do expect to continue, at least over short time scales. We might think that even more instances of the pattern, yet further premises of the same sort, would make the support of the conclusion even stronger. This principle – that the existence of an ongoing pattern is increasingly supported the more instances we adduce – might be part of our toolkit when evaluating arguments inductively. But, no matter how many premises of this form we add, the argument will remain inconclusive. Even if it has rained in London in every January thus far, it remains possible that London will stay dry next January. The same goes for the example of extrapolating from sample data. In that case, the inductive argument has a general conclusion drawn from a narrower body of data. (That is, ‘the population has a certain distribution’ is made plausible by the particular data ‘this sample has a certain distribution’.) The larger the sample, the better the inductive support it provides, until we survey the whole population.5 The point of all this is that most arguments which are inductively very strong are not (deduct‑ ively) valid. Arguments which represent very good examples of inductive reasoning generally are not watertight. Unlikely though it might be, it is possible for their conclusion to be false, even when all of their premises are true. In this book, our interest is simply in sorting the (deductively) valid arguments from the invalid ones. The project of inductive logic, of sorting the invalid ar‑ guments further into the inductively good and poor ones, we shall set aside entirely from here on. 2.8 Making Conclusive Arguments Valid Some arguments which are conclusive but invalid can be turned into valid arguments. So con‑ sider again the argument ‘The sculpture is green all over; therefore it is not red all over’. We can make a valid argument from this by adding a premise: The sculpture is green all over. If the sculpture is green all over, then it is not red all over. So: The sculpture is not red all over. 4The project of inductive logic was quickly seen to need resources going far beyond those of formal logic, and now in‑ ductive logics have more to do with probability theory than with logic strictly speaking. Branden Fitelson provides a useful overview in his ‘Inductive Logic’, pp. 384–93 in Jessica Pfeifer and Sahotra Sarkar, eds., Philosophy of Science: an encyclopedia, Routledge 2005. 5Even in that case, the argument will be inconclusive, because we will need the additional premise this sample is the whole population to rule out the possibility that there are other individuals not yet captured in our data. 14 KEY NOTIONS This new argument has a premise which makes explicit a fact about green and red that was merely implicit in the original argument. Since the original argument was conclusive – since the fact about green and red is true just in virtue of the meaning of the words ‘green’ and ‘red’ (and ‘not’) – the new argument remains conclusive. (We can’t undermine conclusiveness by adding further premises.) But the new argument is valid, because the additional premise we have added yields an argument with a structure that guarantees the truth of the conclusion, given the truth of the premises. The original argument is sometimes thought to be merely an abbreviation of the expanded valid argument. An argument with an unstated premise, such that it can be seen to be valid when the premise is made explicit, is called an ENTHYMEME.6Many inconclusive arguments can be treated as enthymematic, if the unstated premise is obvious enough: The Nasty party platform includes imprisoning people for chewing gum; So: The Nasty party will not form the next government. The unstated premise is something like ‘If a party platform includes imprisoning people for chew‑ ing gum, then that party will win too few votes to form the next government’. The unstated premise may or may not be true. But if it is added, the argument is made valid. Any conclusive argument you are likely to come across will either already be valid, or can be transformed into a valid argument by making some assumption on which it implicitly relies into an explicit premise. Not every inconclusive argument should be treated as an enthymeme. In particular, many strong inductive arguments can be made weaker when they are treated as enthymematic. Consider: In January 2015, it was hot in Adelaide. In January 2020, it was hot in Adelaide. So: In January 2025, it will be hot in Adelaide. This argument is inconclusive. It can be made valid by adding the unstated premise ‘Every Janu‑ ary, it is hot in Adelaide’. But that unstated premise is extremely strong – we do not have suf‑ ficient evidence to conclude that it will be hot in Adelaide in January for eternity. So while the premises we have been given explicitly are good reason to think Adelaide will continue to have a hot January for the foreseeable future, we do not have good enough reason to think that every January will be hot. Treating the argument as an enthymeme makes it valid, but also makes it less persuasive, since the unstated premise on which it relies is not one many people will share.7 6The term is from ancient Greek; the concept was given its first philosophical treatment by Aristotle in his Rhetoric. He gives this example, among others: ‘He is ill, since he has fever’. 7If we had claim that the unstated premise was rather ‘If it has been hot in January in recent representative years, it will be hot in January for the near future’, we would have had a valid argument and one that has a plausible unstated premise – though the unstated premise seems to be plausible only because the unamended inductive argument was fine to begin with! §2. VALID ARGUMENTS 15 Key Ideas in §2 ›An argument is conclusive if, and only if, the truth of the premises guar‑ antees the truth of the conclusion. ›An argument is valid if, and only if, the form of the premises and con‑ clusion alone ensures that it is conclusive. Not every conclusive argu‑ ment is valid (though they can be made valid by addition of appropriate premises). ›An argument can be good and persuade us of its conclusion even if it is not conclusive; and we can fail to be persuaded of the conclusion of a conclusive argument, since one might come to reject its premises. Practice exercises A. What is a conclusive argument? What, in addition to being conclusive, is required for an argu‑ ment to be valid? What, in addition to being valid, is required for an argument to be sound? B. Which of the following arguments are valid? Which are invalid but conclusive? Which are inconclusive? Comment on any difficulties or points of interest. 1. 1. Socrates is a man. 2. All men are carrots. So: Therefore, Socrates is a carrot. 2. 1. Abe Lincoln was either born in Illinois or he was once president. 2. Abe Lincoln was never president. So: Abe Lincoln was born in Illinois. 3. 1. Abe Lincoln was the president of the United States. So: Abe Lincoln was a citizen of the United States. 4. 1. If I pull the trigger, Abe Lincoln will die. 2. I do not pull the trigger. So: Abe Lincoln will not die. 5. 1. Abe Lincoln was either from France or from Luxembourg. 2. Abe Lincoln was not from Luxembourg. So: Abe Lincoln was from France. 6. 1. If the world ends today, then I will not need to get up tomorrow morning. 2. I will need to get up tomorrow morning. So: The world will not end today. 22 KEY NOTIONS C. For each of the following: Is it necessarily true, necessarily false, or contingent? 1. Caesar crossed the Rubicon. 2. Someone once crossed the Rubicon. 3. No one has ever crossed the Rubicon. 4. If Caesar crossed the Rubicon, then someone has. 5. Even though Caesar crossed the Rubicon, no one has ever crossed the Rubicon. 6. If anyone has ever crossed the Rubicon, it was Caesar. D. Look back at the sentences 7–10 in this section (about giraffes, gorillas and Martians in the wild animal park), and consider each of the following: 1. 8,9, and 10 2. 7,9, and 10 3. 7,8, and 10 4. 7,8, and 9 Which are jointly consistent? Which are jointly inconsistent? E. Are these sentences jointly consistent? 1. There are three people leaving the party: Atheer, Brigitte, and James. 2. Brigitte is wearing Atheer’s hat. 3. Each of the people is wearing a hat. 4. No person is wearing their own hat. 5. Atheer is wearing Brigitte’s hat. F. Could there be: 1. A conclusive argument, the conclusion of which is necessarily false? 2. An inconclusive argument, the conclusion of which is necessarily true? 3. Jointly consistent sentences, one of which is necessarily false? 4. Jointly inconsistent sentences, one of which is necessarily true? In each case: if so, give an example; if not, explain why not. G. Some feature of a set 𝐴is MONOTONIC if any set including all the members of 𝐴also has the feature. (So having at least 3 members is monotonic, since adding more members clearly preserves that feature.) Explain why inconsistency of a set of sentences is monotonic. Chapter 2 The Language of Sentential Logic 4 First Steps to Symbolisation 4.1 Argument Structure Consider this argument: It is raining outside. If it is raining outside, then Jenny is miserable. So: Jenny is miserable. and another argument: Jenny is an anarcho‑syndicalist. If Jenny is an anarcho‑syndicalist, then Dipan is an avid reader of Tolstoy. So: Dipan is an avid reader of Tolstoy. Both arguments are valid, and there is a straightforward sense in which we can say that they share a common structure. We might express the structure thus, when we let letters stand for phrases in the original argument: A If A, then C So: C This is an excellent argument STRUCTURE. Surely any argument with this structure will be valid. And this is not the only good argument structure. Consider an argument like: Jenny is either happy or sad. Jenny is not happy. So: Jenny is sad. Again, this is a valid argument. The structure here is something like: 24 §4. FIRST STEPS TO SYMBOLISATION 25 Aor B not: A So: B A superb structure! You will recall that this was the structure we saw in the original arguments which introduced the idea of validity in §2.5. And here is a final example: It’s not the case that Jim both studied hard and acted in lots of plays. Jim studied hard So: Jim did not act in lots of plays. This valid argument has a structure which we might represent thus: not both: A and B A So: not: B The examples illustrate the idea of validity – conclusiveness in virtue of structure. The validity of the arguments just considered has nothing very much to do with the meanings of English expressions like ‘Jenny is miserable’, ‘Dipan is an avid reader of Tolstoy’, or ‘Jim acted in lots of plays’. If it has to do with meanings at all, it is with the meanings of phrases like ‘and’, ‘or’, ‘not,’ and ‘if …, then …’. 4.2 Sentence Trees and Canonical Clauses Since arguments are made of sentences, the logical structure of an argument will be related to the grammatical structure of the sentences within it. We’ve already seen in §3.1 that arguments are made up of declarative sentences. One standard way of analysing the grammatical structure of a declarative sentence is to break it into constituent phrases. Such approaches are known as PHRASE STRUCTURE GRAMMARS. These structures are usefully depicted in a hierarchical SYNTACTIC TREE. Consider the example: (18) Tariq is a man. This sentence (phrase of type S) can be divided into two main parts: the NOUN PHRASE ‘Tariq’ (type NP), and the VERB PHRASE ‘is a man’ (type VP). In this example, the verb phrase itself divides into the verb ‘is’ and the DETERMINER PHRASE ‘a man’ (type DP). We will return to the internal structure of sentences in §15. This phrase structure is depicted in the tree in Figure 4.1. Let’s look at a more complicated example. Consider (19) Alice will ace the test, or Bob will. First we note that this sentence is elliptical, the verb phrase ‘ace the test’ being omitted after ‘Bob will’ as it is understood to be supplied by the first clause of the sentence.1The full tree, with the 1A closely related example is discussed by Paul Elbourne (2011) Meaning, Oxford University Press, pp. 74–6. He argues that this kind of elision is crucial evidence for phrase structure grammars, for only phrases can be omitted in this way: witness the ungrammatical ‘Alice will ace the test and Bob will the’, where arbitrary words are omitted that do not form a grammatical phrase. This is evidence that English syntax is sensitive to the phrasal structure of sentences, and does not treat them as merely a string of words. 26 THE LANGUAGE OF SENTENTIAL LOGIC S VP DP man a is NP Tariq Figure 4.1: A phrase structure tree for Example 18. S S VP VP ace the test will NP Bob Coord or S VP VP DP test the ace will NP Alice Figure 4.2: A phrase structure tree for Example 19. elided phrase supplied (marked by having the VP label in a box at the lower right), is depicted in Figure 4.2. In this example, the whole sentence is a compound of two clauses which are sentences in their own right, or SUBSENTENCES, connected by ‘or’. When analysing the structure of compound sentences, a useful notion is that of a CANONICAL CLAUSE.2These are the simplest units of English sentences that can constitute a sentence by them‑ selves. Here are some examples: (20) They knew the victim. (21) She has read your article. (22) Jenny is happy. A canonical clause has internal structure too, but its parts are not themselves sentences. It is composed of a GRAMMATICAL SUBJECT, typically but not always a noun phrase (‘Jenny’, ‘She’), and a PREDICATE, always a verb phrase (‘knew the victim’, ‘is happy’). 2A fuller treatment of canonical clauses can be found in Rodney Huddleston and Geoffrey Pullum (2005) A Student’s Introduction to English Grammar, Cambridge University Press, pp. 24–5. §4. FIRST STEPS TO SYMBOLISATION 27 The following, in contrast with the previous examples, are noncanonical clauses: (23) They did not know the victim. (24) She has read your article and she vehemently disagrees with it. (25) She says that Jenny is happy. These give us some characteristics of canonical clauses: ›They are positive, as in 20, rather than negative, as indicated by the underlined ‘not’ in 23. ›Canonical clauses are simple, and not coordinated with any other clause, as in 21. In 24, the COORDINATOR ‘and’ links two clauses that would be canonical on their own into a longer COMPOUND sentence. ›The underlined clause in 25 is a subordinate clause, part of a more complex clause. Canon‑ ical clauses are main clauses, as in 22. 4.3 Identifying Argument Structure Our topic is not English grammar. But the idea of a canonical clause is illuminating for the logic we are examining in this part of the course. Sentential logic is, more or less, the logic that helps us understand and analyse arguments whose conclusiveness depends on their constituent canonical clauses and how they are joined together by SENTENCE CONNECTIVES. The sentence connectives we focus on are a grammatically diverse group, including Coordinators including ‘and’, ‘or’, ‘if and only if ’ (also ‘but’) Adjunct heads including ‘if … then …’, ‘only if’ (also ‘unless’, ‘because’, ‘since’, ‘hence’ – though we treat these last as signalling premises and conclusions of arguments) Negatives including ‘not’, ‘‑n’t’, ‘it is not the case that’ (‘never’). And there are others that we won’t deal with (‘always’, ‘might’). How can we identify the logical structure of an argument? The logical structure of an argument is the form one arrives at by ‘abstracting away’ the details of the words in an argument, except for a special set of words – the STRUCTURAL WORDS. This connects with what we have just been saying, because in many of the examples from §4.1, the arguments can be analysed as involving canonical clauses which are linked by the structural words ‘and’, ‘or’, ‘not’, ‘if … then …’ and ‘… if and only if …’. So our efforts to identify the logical structure of arguments often coincide with linguist’s efforts to identify the canonical clauses that constitute the sentences in those arguments – or, at least, constitute some plausible paraphrase or reformulation of those sentences. We can see this if we return to this earlier argument: Jenny is either happy or sad. Jenny is not happy. So: Jenny is sad. We paraphrase the compound sentence ‘Jenny is either happy or sad’ into a compound of two canonical clauses, joined by the coordinator ‘or’: 28 THE LANGUAGE OF SENTENTIAL LOGIC ›Jenny is happy or Jenny is sad. And we paraphrase the noncanonical clause ‘Jenny is not happy’ as the negative clause ›It is not the case that: Jenny is happy. Identifying the coordinator ‘or’ and the negative clause ‘it is not the case that’ as structural words, and replacing the canonical clause ‘Jenny is happy’ with the placeholder letter ‘A’, and the canon‑ ical clause ‘Jenny is sad’ with the placeholder letter ‘B’, we arrive again at the argument structure we previously identified: Aor B not: A So: B In this and the other examples above, we removed all details of the arguments except for the special words ‘and’, ‘or’, ‘not’ and ‘if …, then …’, and replaced the other clauses in the argument (or its near paraphrase) by placeholder letters ‘A’, ‘B’, etc. (We were careful to replace the same clause by the same placeholder every time it appeared.) Another example: (26) Butter isn’t healthy, but it is delicious. We begin by paraphrasing. We note that the pronoun ‘it’ is actually referring to the previously mentioned subject ‘butter’, and we move the negative ‘isn’t’ into a position that reveals the ca‑ nonical clause ‘butter is healthy’. We also note that the effect of ‘but’ is roughly the same as our structural word ‘and’ (though it suggests a contrast that ‘and’ does not, ‘but’ expresses the roughly equivalent idea that each of the claims it connects are true) we obtain this more stilted paraphrase: (27) It is not the case that butter is healthy and butter is delicious. This paraphrase has the syntactic tree in Figure 4.3. Note that we have not further broken down the canonical clauses into subject‑predicate form, because our structural words, the sentence connectives, do not occur within any canonical clause and hence are not ‘visible’ to our analysis at the level of sentences. Finally, we replace the canonical clauses with placeholder sentences, and we reach the structure ‘Not A and B’. The schematic paraphrase ‘Not A and B’ is potentially ambiguous, because it is not obvious whether the ‘not’ applies just to the A‑clause, or to the whole ‘A and B’ clause. We could introduce parentheses to eliminate this ambiguity, distinguishing ‘Not (A and B)’ from ‘Not A and B’. This is the approach we will take in our formal language Sentential (see page 39). Alternatively, we can use the hierarchical nature of syntactic trees to see see that there are two different structures possible for ‘Not A and B’, as depicted in the schematic syntactic trees in Figure 4.4. Sharp‑eyed readers will have noticed that our list of special words doesn’t precisely line up with the canonical clauses we introduced in §4.2. Consider the noncanonical clause ‘She said that §4. FIRST STEPS TO SYMBOLISATION 29 S S butter is delicious and S S butter is healthy Not Figure 4.3: Paraphrase of Example 26 showing its subsentential structure, as in Example 27. S S B and S S A Not S S S B and S A Not Figure 4.4: Different sentential structures for ‘Not A and B’ shown in schematic syntactic trees. Jenny is sad’, in which the canonical clause ‘Jenny is sad’ is subordinate within an indirect speech report. Because ‘She said that’ is not on our special list of words, we cannot analyse this sentence as composed from a canonical clause and some structural expression. So for the purposes of sentential logic, we will treat ‘She said that Jenny is sad’ as if it were canonical, even though it is not from the point of view of English grammar. From the point of view of our structural words, this sentence doesn’t have any further structure that we can identify. Such QUASI‑CANONICAL CLAUSES – clauses that do not feature any of our list of structural words – will also be replaced by placeholder letters in our analysis. This raises another question. What makes the words on our list special? Logicians tend to take a pragmatic attitude to this question. They say: nothing! If you had chosen different words, you would have come up with a different structure. In terms of the syntactic trees we drew above, there is a hierarchy of levels when breaking down the top‑level sentence into constituent phrases, and different choices of structural words correspond to different choices concerning the level at which to stop our analysis. This pragmatic attitude is represented by the words of the logician Alfred Tarski, introducing something like the modern theory of validity: Underlying our whole construction is the division of all terms of the language dis‑ cussed into logical [i.e., structural] and extra‑logical. This division is certainly not quite arbitrary. … On the other hand, no objective grounds are known to me which permit us to draw a sharp boundary between the two groups of terms. It seems to 30 THE LANGUAGE OF SENTENTIAL LOGIC be possible to include among logical terms some which are usually regarded by logicians as extra‑logical without running into consequences which stand in sharp contrast to ordinary usage. In the extreme case we could regard all terms of the language as logical.3 While not quite taking Tarski’s ‘extreme’ position, linguists typically have a very expansive view of structural words, so that the class of canonical clauses is rather small, and there is a lot of structure to be identified in natural language. For example, the presence of modal auxiliaries (like ‘will’ or ‘must’, as in ‘James must eat’), verb inflections other than the present tense (e.g., ‘they took snuff’), and subordination (as in ‘Etta knew that peaches were abundant’), in addition to items on our list of structural words, suffices to make a clause noncanonical. And there are logics which do take these to be structural words: modal logics, tense logics, and epistemic logics treat these as structural words and provide recipes for the structural analysis of arguments that abstract away features other than these words. But, as Tarski emphaises, there is no fundamental principle that divides words once and for all into structural words and other words. So logicians are more interested in quasi‑canonical clauses, given some fixed chosen list of structural words, when using logic to model or represent natural languages. The pragmatic approach is to look at a particular argument or text one wishes to analyse, and choose as structural words those words which seem to be structuring the author’s discussion! This may be more of an art than a science. 4.4 Formal Languages In our approach, we will start simply, and focus on the list of truth‑functional sentence connect‑ ives, principally ‘and’, ‘or’, ‘not’, ‘if … then …’ and ‘… if and only if …’. There are practical reasons why these words are useful ones to focus on initially, because many arguments can be represented fruitfully by taking them to be structured by these expressions. Let me expand. In logic, a FORMAL LANGUAGE is a language which is particularly suited to representing the struc‑ ture or form of its sentences. Such languages have a precisely defined SYNTAX, that allows us to specify without difficulty the grammatical sentences of the language, and they also have a pre‑ cise SEMANTICS which both tells us what meanings are in those languages, and assigns meanings to the sentences and their constituents. Formal languages are not the same as natural languages, like English. The languages we will be looking at are much more limited in their expressive power than English, and not subject to ambiguity or imprecision in the way that English is. But a formal language can be very useful in representing or modelling features of natural language. In particular, they are very good at capturing the structure of natural language sentences and arguments, because we can design them specifically to represent a particular level of structural analysis. The semantics we give for such languages reflects that this is their primary use, as you will see. The languages assign fixed meanings only to those expressions which can be used to represent structure, but (unlike English) not every expression is treated as structural. In this chapter we will begin developing a formal language which will allow us to represent many sentences of English, and arguments involving those sentences. The language will have 3Alfred Tarski (1936) ‘On the Concept of Logical Consequence’, reprinted at pp. 409–20 in his Logic, Semantics, Metamathematics, Hackett 1983 – the quotation appears on pp. 418–19. §4. FIRST STEPS TO SYMBOLISATION 31 a very small basic vocabulary, since we are designing it to represent the sentential structure of the examples with which we began. So it will have expressions corresponding to the English sentence connectives ‘and’, ‘or’, ‘not’ and ‘if …, then …’ and ‘if and only if’. The language rep‑ resents these English connectives by its own class of dedicated sentence connectives which allow simple sentences of the language to be combined into more complex sentences. These words are a good class to focus on in developing a formal language, because languages with expressions analogous to these feature a good balance between being useful and being well behaved and easy to study. The language we will develop is called Sentential, and the study of that language and its features is called sentential logic. (It also has other names: see Appendix A.) Once we have our formal language Sentential, we will be able to go on (in chapter 3) to show that it has a very nice feature: it is able to represent the structure of a large class of valid natural language arguments. So while logic can’t help us with every good argument, and it can’t even help us with every conclusive argument, it can help us to understand arguments that are valid, or conclusive due to their structure, when that structure involves ‘and’, ‘or’, ‘not’, ‘if’ and various other expressions. We will see later in this book that one could take additional expressions in natural language to be involved is determining the structure of a sentence or argument. The language we develop in chapter 4is one that is suited to represent what we get when we take names like ‘Juanita’, predic‑ ates like ‘is a vixen’ or ‘is a fox’, and quantifier expressions like ‘every’ and ‘some’ to be structural – see also §16. And as we have mentioned, there are still other formal logical languages, unfortu‑ nately beyond the scope of this book, which take still other words or grammatical categories as structural: logics which take modal words like ‘necessarily’ or ‘possibly’ as structural, and logics which take temporal adverbs like ‘always’ and ‘now’ as structural.4What we notice here is that using logic to model natural language arguments inevitably involves a compromise. If we have lots of structural words, we can show many conclusive arguments to be valid, but our logic is complex and involves many different sentence connectives. If we take relatively few words as structural, we can represent fewer conclusive arguments as valid, but our formal language is a lot easier to work with. We will start now by putting those tantalising observations about richer logics out of your mind! We’ll focus to begin with on the language Sentential, and on how we can use it to model argu‑ ments in English featuring some particular sentence connectives as structural words, including those we’ve just highlighted above. This collection of structural words has struck many logi‑ cians over the years as providing a good balance between simplicity and strength, ideal for an introduction to logic. 4.5 Atomic Sentences In §4.1, we started isolating the form of an argument by replacing quasi‑canonical clauses (those including none of our list of structural words) within the argument with individual placeholder 4Some arguments will be valid in richer logical frameworks, but not according to the more austere framework for logical structure provided by Sentential. For example ‘Sylvester is always active; so Sylvester is active now’ is conclusive. From the point of view of temporal logic, the argument has this form ‘Always A; so now A’, which is valid in normal temporal logics. But it is not valid according to Sentential, because the premise ‘Sylvester is always active’ has no internal structural words that occur on the list of sentence connectives that Sentential represents. 38 THE LANGUAGE OF SENTENTIAL LOGIC 5.3 Conjunction Consider these sentences: (37) Adam is athletic. (38) Barbara is athletic. (39) Adam is athletic, and Barbara is also athletic. We will need separate atomic sentences of Sentential to symbolise sentences 37 and 38; perhaps 𝐴: Adam is athletic. 𝐵: Barbara is athletic. Sentence 37 can now be symbolised as ‘𝐴’, and sentence 38 can be symbolised as ‘𝐵’. Sentence 39 roughly says ‘A and B’. We need another symbol, to deal with ‘and’. We will use ‘∧’. Thus we will symbolise it as ‘(𝐴∧𝐵)’. This connective is called CONJUNCTION. We also say that ‘𝐴’ and ‘𝐵’ are the two CONJUNCTS of the conjunction ‘(𝐴∧𝐵)’. Notice that we make no attempt to symbolise the word ‘also’ in sentence 39. Words like ‘both’ and ‘also’ function to draw our attention to the fact that two things are being conjoined. Maybe they affect the emphasis of a sentence. But we will not (and cannot) symbolise such things in Sentential. Some more examples will bring out this point: (40) Barbara is athletic and energetic. (41) Barbara and Adam are both athletic. (42) Although Barbara is energetic, she is not athletic. (43) Adam is athletic, but Barbara is more athletic than him. Sentence 40 is obviously a conjunction. The sentence says two things (about Barbara). In English, it is permissible to refer to Barbara only once. It might be tempting to think that we need to symbolise sentence 40 with something along the lines of ‘𝐵and energetic’. This would be a mistake. Once we symbolise part of a sentence as ‘𝐵’, any further structure is lost. ‘𝐵’ is an atomic sentence of Sentential. Conversely, ‘energetic’ is not an English sentence at all. What we are aiming for is something like ‘𝐵and Barbara is energetic’. So we need to add another sentence letter to the symbolisation key. Let ‘𝐸’ symbolise ‘Barbara is energetic’. Now the entire sentence can be symbolised as ‘(𝐵∧𝐸)’. Sentence 41 says one thing about two different subjects. It says of both Barbara and Adam that they are athletic, and in English we use the word ‘athletic’ only once. The sentence can be para‑ phrased as ‘Barbara is athletic, and Adam is athletic’. We can symbolise this in Sentential as ‘(𝐵∧𝐴)’, using the same symbolisation key that we have been using. Sentence 42 is slightly more complicated. The word ‘although’ sets up a contrast between the first part of the sentence and the second part. Nevertheless, the sentence tells us both that Barbara is energetic and that she is not athletic. In order to make each of the conjuncts an atomic sentence, we need to replace ‘she’ with ‘Barbara’. So we can paraphrase sentence 42 as, ‘Both Barbara is energetic, and Barbara is not athletic’. The second conjunct contains a negation, so we paraphrase further: ‘Both Barbara is energetic and it is not the case that Barbara is athletic’. And now we can §5. CONNECTIVES 39 symbolise this with the Sentential sentence ‘(𝐸∧¬𝐵)’. Note that we have lost all sorts of nuance in this symbolisation. There is a distinct difference in tone between sentence 42 and ‘Both Barbara is energetic and it is not the case that Barbara is athletic’. Sentential does not (and cannot) preserve these nuances. Sentence 43 raises similar issues. There is a contrastive structure. The speaker who asserts it means something by that ‘but’ — something to the effect of there being a contrast between those two features. But their brief utterance doesn’t tell exactly which contrast is intended. These contrasts are not something that Sentential is designed to deal with. So we can paraphrase the sentence as ‘Both Adam is athletic, and Barbara is more athletic than Adam’. (Notice that we once again replace the pronoun ‘him’ with ‘Adam’.) How should we deal with the second conjunct? We already have the sentence letter ‘𝐴’, which is being used to symbolise ‘Adam is athletic’, and the sentence ‘𝐵’ which is being used to symbolise ‘Barbara is athletic’; but neither of these concerns their relative ‘athleticity’. So, to to symbolise the entire sentence, we need a new sentence letter. Let the Sentential sentence ‘𝑅’ symbolise the English sentence ‘Barbara is more athletic than Adam’. Now we can symbolise sentence 43 by ‘(𝐴∧𝑅)’. A sentence can be symbolised as (𝒜∧ℬ)if it can be paraphrased in English as ‘Both …, and …’, or as ‘…, but …’, or as ‘although …, …’. You might be wondering why I am putting parentheses around the conjunctions. The reason for this is to avoid potential ambiguity. This can be brought out by considering how negation might interact with conjunction. Consider: (44) It’s not the case that you will get both soup and salad. (45) You will not get soup but you will get salad. Sentence 44 can be paraphrased as ‘It is not the case that: both you will get soup and you will get salad’. Using this symbolisation key: 𝑆1: You will get soup. 𝑆2: You will get salad. We would symbolise ‘both you will get soup and you will get salad’ as ‘(𝑆1∧𝑆2)’. To symbolise sentence 44, then, we simply negate the whole sentence, thus: ‘¬(𝑆1∧𝑆2)’. Sentence 45 is a conjunction: you will not get soup, and you will get salad. ‘You will not get soup’ is symbolised by ‘¬𝑆1’. So to symbolise sentence 45 itself, we offer ‘(¬𝑆1∧𝑆2)’. These English sentences are very different, and their symbolisations differ accordingly. In one of them, the entire conjunction is negated. In the other, just one conjunct is negated. Parentheses help us to avoid ambiguity by clear distinguishing these two cases. Once again, however, English does feature this sort of ambiguity. Suppose instead of 44, we’d just said (46) You won’t get soup with salad. 40 THE LANGUAGE OF SENTENTIAL LOGIC The sentence 46 is arguably ambiguous; one of its readings says the same thing as 44, the other says the same thing as 45. Parentheses enable Sentential to avoid precisely this ambiguity. The introduction of parentheses prompts us to define some other useful concepts. If a sentence of Sentential has the overall form (𝒜∧ℬ), we say that its main connective is conjunction – even if other connectives occur within 𝒜or ℬ. Likewise, if the sentence has the form ¬𝒜, its main connective is negation. We define the scope of an occurrence of a connective in a sentence as the subsentence which has that connective as its main connective. So the scope of ‘¬’ in ‘¬(𝑆1∧𝑆2)’ is the whole sentence (because negation is the main connective), while the scope of ‘¬’ in ‘(¬𝑆1∧ 𝑆2)’ is just the subsentence ‘¬𝑆1’ – the main connective of the whole sentence is conjunction. Parentheses help us keep track of things the scope of our connectives, and which connective in a sentence is the main connective. I say more about the notions of a main connective and the scope of a connective in §6.3. 5.4 Disjunction Consider these sentences: (47) Either Denison will play golf with me, or he will watch movies. (48) Either Denison or Ellery will play golf with me. For these sentences we can use this symbolisation key: 𝐷: Denison will play golf with me. 𝐸: Ellery will play golf with me. 𝑀: Denison will watch movies. However, we shall again need to introduce a new symbol. Sentence 47 is symbolised by ‘(𝐷∨ 𝑀)’. The connective is called DISJUNCTION. We also say that ‘𝐷’ and ‘𝑀’ are the DISJUNCTS of the disjunction ‘(𝐷∨𝑀)’. Sentence 48 is only slightly more complicated. There are two subjects, but the English sentence only gives the verb once. However, we can paraphrase sentence 48 as ‘Either Denison will play golf with me, or Ellery will play golf with me’. Now we can obviously symbolise it by ‘(𝐷∨𝐸)’ again. A sentence can be symbolised as (𝒜∨ℬ)if it can be paraphrased in English as ‘Either …, or …’. Each of the disjuncts must be a sentence. Sometimes in English, the word ‘or’ excludes the possibility that both disjuncts are true. This is called an EXCLUSIVE OR. An exclusive ‘or’ is clearly intended when it says, on a restaurant menu, ‘Entrees come with either soup or salad’: you may have soup; you may have salad; but, if you want both soup and salad, then you have to pay extra. At other times, the word ‘or’ allows for the possibility that both disjuncts might be true. This is probably the case with sentence 48, above. I might play golf with Denison, with Ellery, or with both Denison and Ellery. Sentence 48 merely says that I will play with at least one of them. This is called an INCLUSIVE OR. The Sentential symbol ‘∨’ always symbolises an inclusive ‘or’. §5. CONNECTIVES 41 It might help to see negation interact with disjunction. Consider: (49) Either you will not have soup, or you will not have salad. (50) You will have neither soup nor salad. (51) You get either soup or salad, but not both. Using the same symbolisation key as before, sentence 49 can be paraphrased in this way: ‘Either it is not the case that you get soup, or it is not the case that you get salad’. To symbolise this in Sentential, we need both disjunction and negation. ‘It is not the case that you get soup’ is sym‑ bolised by ‘¬𝑆1’. ‘It is not the case that you get salad’ is symbolised by ‘¬𝑆2’. So sentence 49 itself is symbolised by ‘(¬𝑆1∨¬𝑆2)’. Sentence 50 also requires negation. It can be paraphrased as, ‘It is not the case that either you get soup or you get salad’. Since this negates the entire disjunction, we symbolise sentence 50 with ‘¬(𝑆1∨𝑆2)’. Sentence 51 is an exclusive ‘or’. We can break the sentence into two parts. The first part says that you get one or the other. We symbolise this as ‘(𝑆1∨𝑆2)’. The second part says that you do not get both. We can paraphrase this as: ‘It is not the case both that you get soup and that you get salad’. Using both negation and conjunction, we symbolise this with ‘¬(𝑆1∧𝑆2)’. Now we just need to put the two parts together. As we saw above, ‘but’ can usually be symbolised with ‘∧’. Sentence 51 can thus be symbolised as ‘((𝑆1∨𝑆2)∧¬(𝑆1∧𝑆2))’. This last example shows something important. Although the Sentential symbol ‘∨’ always symbolises inclusive ‘or’, we can symbolise an exclusive ‘or’ in Sentential. We just have to use a few of our other symbols as well. 5.5 Conditional Consider these sentences: (52) If Jean is in Paris, then Jean is in France. (53) Jean is in France only if Jean is in Paris. Let’s use the following symbolisation key: 𝑃: Jean is in Paris. 𝐹: Jean is in France. Sentence 52 is roughly of this form: ‘if P, then F’. We will use the symbol ‘→’ to symbolise this ‘if …, then …’ structure. So we symbolise sentence 52 by ‘(𝑃→𝐹)’. The connective is called THE CONDITIONAL. Here, ‘𝑃’ is called the ANTECEDENT of the conditional ‘(𝑃 →𝐹)’, and ‘𝐹’ is called the CONSEQUENT. Sentence 53 is also a conditional. Since the word ‘if’ appears in the second half of the sentence, it might be tempting to symbolise this in the same way as sentence 52. That would be a mistake. My knowledge of geography tells me that sentence 52 is unproblematically true: there is no way for Jean to be in Paris that doesn’t involve Jean being in France. But sentence 53 is not so straightforward: were Jean in Dijon, Marseilles, or Toulouse, Jean would be in France without being in Paris, thereby rendering sentence 53 false. Since geography alone dictates the truth of 42 THE LANGUAGE OF SENTENTIAL LOGIC sentence 52, whereas travel plans (say) are needed to know the truth of sentence 53, they must mean different things. In fact, sentence 53 can be paraphrased as ‘If Jean is in France, then Jean is in Paris’. So we can symbolise it by ‘(𝐹 →𝑃)’. A sentence can be symbolised as (𝒜 → ℬ)if it can be paraphrased in English as ‘If A, then B’ or ‘A only if B’ or ‘B if A’. In fact, many English expressions can be represented using the conditional. Consider: (54) For Jean to be in Paris, it is necessary that Jean be in France. (55) It is a necessary condition on Jean’s being in Paris that she be in France. (56) For Jean to be in France, it is sufficient that Jean be in Paris. (57) It is a sufficient condition on Jean’s being in France that she be in Paris. If we think really hard, all four of these sentences mean the same as ‘If Jean is in Paris, then Jean is in France’. So they can all be symbolised by ‘𝑃→𝐹’. It is important to bear in mind that the connective ‘→’ tells us only that, if the antecedent is true, then the consequent is true. It says nothing about a causal connection between two events (for example). In fact, we seem to lose a huge amount when we use ‘→’ to symbolise English conditionals. We shall return to this in §§8.6 and 11.5. 5.6 Biconditional Consider these sentences: (58) Shergar is a horse only if it he is a mammal. (59) Shergar is a horse if he is a mammal. (60) Shergar is a horse if and only if he is a mammal. We shall use the following symbolisation key: 𝐻: Shergar is a horse. 𝑀: Shergar is a mammal. Sentence 58, for reasons discussed above, can be symbolised by ‘𝐻 →𝑀’. Sentence 59 is importantly different. It can be paraphrased as, ‘If Shergar is a mammal then Shergar is a horse’. So it can be symbolised by ‘𝑀 →𝐻’. Sentence 60 says something stronger than either 58 or 59. It can be paraphrased as ‘Shergar is a horse if he is a mammal, and Shergar is a horse only if Shergar is a mammal’. This is just the conjunction of sentences 58 and 59. So we can symbolise it as ‘(𝐻 →𝑀)∧(𝑀 →𝐻)’. We call this a BICONDITIONAL, because it entails the conditional in both directions. §5. CONNECTIVES 43 We could treat every biconditional this way. So, just as we do not need a new Sentential symbol to deal with exclusive ‘or’, we do not really need a new Sentential symbol to deal with bicondi‑ tionals. However, we will use ‘↔’ to symbolise the biconditional. So we can symbolise sentence 60 with the Sentential sentence ‘𝐻 ↔𝑀’. The expression ‘if and only if’ occurs a lot in philosophy and logic. For brevity, we can abbreviate it with the snappier word ‘IFF’. I shall follow this practice. So ‘if’ with only one ‘f’ is the English conditional. But ‘iff’ with two ‘f’s is the English biconditional. Armed with this we can say: A sentence can be symbolised as (𝒜 ↔ ℬ)if it can be paraphrased in English as ‘A iff B’; that is, as ‘A if and only if B’. Other expressions in English which can be used to mean ‘iff’ include ‘exactly if’ and ‘exactly when’, or even ‘just in case’. So if we say ‘You run out of time exactly when the buzzer sounds’, we mean: ‘if the buzzer sounds, then you are out of time; and also if you are out of time, then the buzzer sounds’. A word of caution. Ordinary speakers of English often use ‘if …, then …’ when they really mean to use something more like ‘… if and only if …’. Perhaps your parents told you, when you were a child: ‘if you don’t eat your vegetables, you won’t get any dessert’. Suppose you ate your vegetables, but that your parents refused to give you any dessert, on the grounds that they were only committed to the conditional (roughly ‘if you get dessert, then you will have eaten your vegetables’), rather than the biconditional (roughly, ‘you get dessert iff you eat your vegetables’). Well, a tantrum would rightly ensue. So, be aware of this when interpreting people; but in your own writing, make sure you use the biconditional iff you mean to. 5.7 Unless We have now introduced all of the connectives of Sentential. We can use them together to symbolise many kinds of sentences, but not every kind. It is a matter of judgment whether a given English connective can be symbolised in Sentential. One rather tricky case is the English‑ language connective ‘unless’: (61) Unless you wear a jacket, you will catch cold. (62) You will catch cold unless you wear a jacket. These two sentences are clearly equivalent. To symbolise them, we shall use the symbolisation key: 𝐽: You will wear a jacket. 𝐷: You will catch a cold. How should we try to symbolise these in Sentential? Note that 62 seems to say: ’either you will catch cold, or if you don’t catch cold, it will be because you wear a jacket’. That would have this symbolisation in Sentential: ‘(𝐷∨(¬𝐷 → 𝐽)’. This turns out to be just a long‑winded way of saying ‘(¬𝐷→𝐽)’, i.e., if you don’t catch cold, then you will have worn a jacket. 44 THE LANGUAGE OF SENTENTIAL LOGIC Equally, however, both sentences mean that if you do not wear a jacket, then you will catch cold. With this in mind, we might symbolise them as ‘¬𝐽 →𝐷’. Equally, both sentences mean that either you will wear a jacket or you will catch a cold. With this in mind, we might symbolise them as ‘𝐽∨𝐷’. All three are correct symbolisations. Indeed, in chapter 3we shall see that all three symbolisations are equivalent in Sentential. If a sentence can be paraphrased as ‘Unless A, B,’ then it can be symbolised as 𝒜∨ℬ, or ¬𝒜→ℬ, or ¬ℬ→𝒜. Again, though, there is a little complication. ‘Unless’ can be symbolised as a conditional; but as I said above, people often use the conditional (on its own) when they mean to use the bicondi‑ tional. Equally, ‘unless’ can be symbolised as a disjunction; but there are two kinds of disjunction (exclusive and inclusive). So it will not surprise you to discover that ordinary speakers of English often use ‘unless’ to mean something more like the biconditional, or like exclusive disjunction. Suppose I say: ‘I shall go running unless it rains’. I probably mean something like ‘I shall go running iff it does not rain’ (i.e., the biconditional), or ‘either I shall go running or it will rain, but not both’ (i.e., exclusive disjunction). Again: be aware of this when interpreting what other people have said, but be precise in your writing, unless you want to be deliberately ambiguous. We should not take ‘unless’ to always have the stronger biconditional form. Consider this ex‑ ample: (63) We’ll capture the castle, unless the Duke tries to stop us. This certainly says that if we fail to capture the castle, it will have been because of that pesky Duke. But what if the Duke tries but fails to stop us? We might in that case capture the castle even though he tried to stop us. While 63 is still true, it is not the case that ‘if the Duke tries to stop us, we won’t capture the castle’ is true. Key Ideas in §5 ›Sentential features five connectives: ‘∧’ (‘and’), ‘∨’ (‘or’), ‘¬’ (‘not’), ‘→’ (‘if …, then …’) and ‘↔’ (‘if and only if’ or ‘iff’). ›These connectives, alone and in combination, can be used to symbolise many English constructions, even some which do not feature the English counterparts of the connectives – as when we approached the symbolisa‑ tion of sentences involving ‘unless’. ›Figuring out how to symbolise a given natural language sentence might not be straightforward, however, as in the cases of ‘A if B’ and ‘A only if B’, which have quite different symbolisations into Sentential. ‘A unless B’ is perhaps even trickier, being sometimes used ambiguously by English speakers and being able to be symbolised in many good ways. ›What matters in symbolisation is that the ‘spirit’ of the argument is pre‑ served and modelled appropriately, not that every nuance of meaning is preserved. §5. CONNECTIVES 45 Practice exercises A. Using the symbolisation key given, symbolise each English sentence in Sentential. 𝑀: Those creatures are men in suits. 𝐶: Those creatures are chimpanzees. 𝐺: Those creatures are gorillas. 1. Those creatures are not men in suits. 2. Those creatures are men in suits, or they are not. 3. Those creatures are either gorillas or chimpanzees. 4. Those creatures are neither gorillas nor chimpanzees. 5. If those creatures are chimpanzees, then they are neither gorillas nor men in suits. 6. Unless those creatures are men in suits, they are either chimpanzees or they are gorillas. B. Using the symbolisation key given, symbolise each English sentence in Sentential. 𝐴: Mister Ace was murdered. 𝐵: The butler did it. 𝐶: The cook did it. 𝐷: The Duchess is lying. 𝐸: Mister Edge was murdered. 𝐹: The murder weapon was a frying pan. 1. Either Mister Ace or Mister Edge was murdered. 2. If Mister Ace was murdered, then the cook did it. 3. If Mister Edge was murdered, then the cook did not do it. 4. Either the butler did it, or the Duchess is lying. 5. The cook did it only if the Duchess is lying. 6. If the murder weapon was a frying pan, then the culprit must have been the cook. 7. If the murder weapon was not a frying pan, then the culprit was either the cook or the butler. 8. Mister Ace was murdered if and only if Mister Edge was not murdered. 9. The Duchess is lying, unless it was Mister Edge who was murdered. 10. If Mister Ace was murdered, he was done in with a frying pan. 11. Since the cook did it, the butler did not. 12. Of course the Duchess is lying! C. Using the symbolisation key given, symbolise each English sentence in Sentential. 𝐸1: Ava is an electrician. 𝐸2: Harrison is an electrician. 𝐹1: Ava is a firefighter. 𝐹2: Harrison is a firefighter. 𝑆1: Ava is satisfied with her career. 𝑆2: Harrison is satisfied with his career. 46 THE LANGUAGE OF SENTENTIAL LOGIC 1. Ava and Harrison are both electricians. 2. If Ava is a firefighter, then she is satisfied with her career. 3. Ava is a firefighter, unless she is an electrician. 4. Harrison is an unsatisfied electrician. 5. Neither Ava nor Harrison is an electrician. 6. Both Ava and Harrison are electricians, but neither of them find it satisfying. 7. Harrison is satisfied only if he is a firefighter. 8. If Ava is not an electrician, then neither is Harrison, but if she is, then he is too. 9. Ava is satisfied with her career if and only if Harrison is not satisfied with his. 10. If Harrison is both an electrician and a firefighter, he must be satisfied with his work. 11. It cannot be that Harrison is both an electrician and a firefighter. 12. Harrison and Ava are both firefighters if and only if neither of them is an electrician. D. Give a symbolisation key and symbolise the following English sentences in Sentential. 1. Alice and Bob are both spies. 2. If either Alice or Bob is a spy, then the code has been broken. 3. If neither Alice nor Bob is a spy, then the code remains unbroken. 4. The German embassy will be in an uproar, unless someone has broken the code. 5. Either the code has been broken or it has not, but the German embassy will be in an uproar regardless. 6. Either Alice or Bob is a spy, but not both. E. Give a symbolisation key and symbolise the following English sentences in Sentential. 1. If there is food to be found in the pridelands, then Rafiki will talk about squashed bananas. 2. Rafiki will talk about squashed bananas unless Simba is alive. 3. Rafiki will either talk about squashed bananas or he won’t, but there is food to be found in the pridelands regardless. 4. Scar will remain as king if and only if there is food to be found in the pridelands. 5. If Simba is alive, then Scar will not remain as king. F. For each argument, write a symbolisation key and symbolise all of the sentences of the argu‑ ment in Sentential. 1. If Dorothy plays the piano in the morning, then Roger wakes up cranky. Dorothy plays piano in the morning unless she is distracted. So if Roger does not wake up cranky, then Dorothy must be distracted. 2. It will either rain or snow on Tuesday. If it rains, Neville will be sad. If it snows, Neville will be cold. Therefore, Neville will either be sad or cold on Tuesday. 3. If Zoog remembered to do his chores, then things are clean but not neat. If he forgot, then things are neat but not clean. Therefore, things are either neat or clean; but not both. G. We symbolised an exclusive ‘or’ using ‘∨’, ‘∧’, and ‘¬’. How could you symbolise an exclusive ‘or’ using only two connectives? Is there any way to symbolise an exclusive ‘or’ using only one connective? 6 Sentences of Sentential The sentence ‘either apples are red, or berries are blue’ is a sentence of English, and the sentence ‘(𝐴∨𝐵)’ is a sentence of Sentential. Although we can identify sentences of English when we encounter them, we do not have a formal definition of ‘sentence of English’. But in this chapter, we shall offer a complete definition of what counts as a sentence of Sentential. This is one respect in which a formal language like Sentential is more precise than a natural language like English. Of course, Sentential was designed to be much simpler than English. 6.1 Expressions We have seen that there are three kinds of symbols in Sentential: Atomic sentences 𝐴,𝐵,𝐶,…,𝑍 with subscripts, as needed 𝐴1,𝐵1,𝑍1,𝐴2,𝐴25,𝐽375,… Connectives ¬,∧,∨,→,↔ Parentheses ( , ) We define an EXPRESSION of Sentential as any finite nonempty string of symbols of Sentential. Take any of the symbols of Sentential and write them down, in any order, and you have an expression of Sentential. Expressions are sometimes called ‘strings’, because they are just a string of symbols from the approved list above. No restriction is placed on expressions apart from having to contain at least one character, and not going on infinitely long. 6.2 Sentences We want to know when an expression of Sentential amounts to a sentence. Many expressions of Sentential will be uninterpretable nonsense. ‘)𝐴17𝐽𝑄𝐹¬𝐾))∧)()’ is a perfectly good expression, but doesn’t look likely to end up a correctly formed sentence of our language. Accordingly, we 47 7 Use and Mention In this chapter, I have talked a lot about sentences. So I need to pause to explain an important, and very general, point. 7.1 Quotation Conventions Consider these two sentences: ›Malcolm Turnbull is the Prime Minister. ›The expression ‘Malcolm Turnbull’ is composed of two upper case letters and thirteen lower case letters. When we want to talk about this ex‑Prime Minister, we USE his name. When we want to talk about his name, we MENTION that name. And in English, we normally do so by putting it in quotation marks. There is a general point here. When we want to talk about things in the world, we just use words. When we want to talk about words, we typically have to mention those words.1We need to indicate that we are mentioning them, rather than using them. To do this, some convention is needed. We can surround the expression in matched left and right quotation marks, or display them centrally in the page (say). So this sentence: ›‘Malcolm Turnbull’ is the Prime Minister. says that some expression is the Prime Minister. And that’s false. The man is the Prime Minister; his name isn’t. Conversely, this sentence: ›Malcolm Turnbull is composed of two upper case letters and thirteen lower case letters. 1More generally, when we want to talk about something we use its name. So when we want to talk about an expres‑ sion, we use the name of the expression – which is just the expression enclosed in quotation marks. Mentioning an expression is using a name of that expression. 54 §7. USE AND MENTION 55 also says something false: Malcolm Turnbull is a man, made of meat rather than letters. One final example: ›‘ ‘Malcolm Turnbull’ ’ is the name of ‘Malcolm Turnbull’. On the left‑hand‑side, here, we have the name of a name (it consists of an expression in quotation marks, and that embedded expression itself contains quotation marks). On the right hand side, we have a name (of an expression). Perhaps this kind of sentence only occurs in logic textbooks, but it is true. Those are just general rules for quotation, and you should observe them carefully in all your work! To be clear, the quotation‑marks here do not indicate indirect speech. They indicate that you are moving from talking about an object, to talking about the name of that object. 7.2 Object Language and Metalanguage These general quotation conventions are of particular importance for us. After all, we are de‑ scribing a formal language here, Sentential, and so we are often mentioning expressions from Sentential. When we talk about a language, the language that we are talking about is called the OBJECT LAN‑ GUAGE. The language that we use to talk about the object language is called the METALANGUAGE. For the most part, the object language in this chapter has been the formal language that we have been developing: Sentential. The metalanguage is English. Not conversational English exactly, but English supplemented with some additional vocabulary which helps us to get along. Now, I have used italic upper case letters for atomic sentences of Sentential: 𝐴,𝐵,𝐶,𝑍,𝐴1,𝐵4,𝐴25,𝐽375,… These are sentences of the object language (Sentential). They are not sentences of English. So I must not say, for example: ›𝐷is an atomic sentence of Sentential. Obviously, I am trying to come out with an English sentence that says something about the ob‑ ject language (Sentential). But ‘𝐷’ is a sentence of Sentential, and no part of English. So the preceding is gibberish, just like: ›Schnee ist weiß is a German sentence. What we surely meant to say, in this case, is: ›‘Schnee ist weiß’ is a German sentence. Equally, what we meant to say above is just: 56 THE LANGUAGE OF SENTENTIAL LOGIC ›‘𝐷’ is an atomic sentence of Sentential. The general point is that, whenever we want to talk in English about some specific expression of Sentential, we need to indicate that we are mentioning the expression, rather than using it. We can either deploy quotation marks, or we can adopt some similar convention, such as placing it centrally in the page. English is, generally, its own metalanguage. An expression of English enclosed in matching quotation marks is another expression of English, as the quotation marks are parts of English too. This causes a potential problem of ambiguity if the expression quoted itself contains quotation marks. English allows us to talk about operations on English expressions, as in this example (64) An English word results from adding ‘ing’ or ‘ion’ to the expression ‘confus’. But this example is ambiguous. On one reading, it is discussing the expressions ‘confusing’ and ‘confusion’, and saying truly that they are both English words. But on another reading, the match‑ ing quotation marks are the one before ‘ing’ and the one after ‘ion’, and Example 64 is stating falsely that this unusual string of English letters and punctuation can be added to ‘confus’ to form an English word: ing’ or ‘ion To avoid this, we might introduce some mechanism for indicating which quotation marks are matched with each other. 7.3 Script Fonts, and Recursive Definitions Revisited However, we do not just want to talk about specific expressions of Sentential. We also want to be able to talk about any arbitrary sentence of Sentential. Indeed, I had to do this in §6, when I presented the recursive definition of a sentence of Sentential. I used upper case script font letters to do this, namely: 𝒜,ℬ,𝒞,𝒟,… These symbols do not belong to Sentential. Rather, they are part of our (augmented) metalan‑ guage that we use to talk about any expression of Sentential. To repeat the second clause of the recursive definition of a sentence of Sentential, we said: 2. If 𝒜is a sentence, then ¬𝒜is a sentence. This talks about arbitrary sentences. If we had instead offered: ›If ‘𝐴’ is a sentence, then ‘¬𝐴’ is a sentence. this would not have allowed us to determine whether ‘¬𝐵’ is a sentence. To emphasise, then: ‘𝒜’ is a symbol in augmented English, which we use to talk about any Sentential expression. ‘𝐴’ is a particular atomic sentence of Sentential. §7. USE AND MENTION 57 To come at this distinction a slightly different way, while ‘𝒜’ designates a sentence of Sentential, it can designate a different sentence on different occasions. It behaves a bit a like a pronoun. The pronoun ‘it’ always designates some object, but a different one in different circumstances of use. Likewise ‘𝒜’ can stand for different sentences of Sentential. By contrast, ‘𝐴’ always names just one atomic sentence of Sentential, the first letter of the English alphabet. This last example raises a further complication for our quotation conventions. I have not included any quotation marks in the clauses of our recursive definition of a sentence of Sentential in §6.2. Should I have done so? The problem is that the expression on the right‑hand‑side of most of our recursive clauses are not sentences of English, since they contain Sentential connectives, like ‘¬’. Consider clause 2. We might try to write: 2′. If 𝒜is a sentence, then ‘¬𝒜’ is a sentence. But this is no good: ‘¬𝒜’ is not a Sentential sentence, since ‘𝒜’ is a symbol of (augmented) English rather than a symbol of Sentential. What we really want to say is something like this: 2″. If 𝒜is any Sentential sentence, then the expression that consists of the symbol ‘¬’, fol‑ lowed immediately by the sentence 𝒜, is also a sentence. This is impeccable, but rather long‑winded. But we can avoid long‑windedness by creating our own conventions. We can perfectly well stipulate that an expression like ‘¬𝒜’ should simply be read as abbreviating the long‑winded account. So, officially, the metalanguage expression ‘¬𝒜’ simply abbreviates: the expression that consists of the symbol ‘¬’ followed by the sentence 𝒜 and similarly, for expressions like ‘(𝒜∧ℬ)’, ‘(𝒜∨ℬ)’, etc. The latter is the expression which con‑ sists of an opening parenthesis, followed by the sentence 𝒜, followed by the symbol ‘∨’, followed by the sentence ℬ, followed by a closing parenthesis. If you like, you can think of our recursive definition of a sentence as a schema standing for infin‑ itely many instances of each clause, one for each Sentential sentence. In the schematic clause for negation (‘If 𝒜is a sentence, ¬𝒜is also a sentence’), we can consider each instance involving ‘𝒜’ being replaced by some Sentential sentence surrounded by quotation marks in accordance with our conventions. So ‘¬𝒜’ is to be understood as abbreviating the expression consisting of a left quotation mark, a negation sign, the same Sentential sentence as 𝒜, and a right quotation mark. Hence if if 𝒜is ‘𝑃’, ¬𝒜just is ‘¬𝑃’. 7.4 Quotation Conventions for Arguments One of our main purposes for using Sentential is to study arguments, and that will be our con‑ cern in chapter 3. In English, the premises of an argument are often expressed by individual sentences, and the conclusion by a further sentence. Since we can symbolise English sentences, we can symbolise English arguments using Sentential. Thus we might ask whether the argu‑ ment whose premises are the Sentential sentences ‘𝐴’ and ‘𝐴→𝐶’, and whose conclusion is the 58 THE LANGUAGE OF SENTENTIAL LOGIC Sentential sentence ‘𝐶’, is valid. However, it is quite a mouthful to write that every time. So instead I shall introduce another bit of abbreviation. This: 𝒜1,𝒜2,…,𝒜𝑛∴𝒞 abbreviates: the argument with premises 𝒜1,𝒜2,…,𝒜𝑛and conclusion 𝒞 To avoid unnecessary clutter, we shall not regard this as requiring quotation marks around it. This is a name of an argument, not an argument itself. (Note, then, that ‘∴’ is a symbol of our augmented metalanguage, and not a new symbol of Sentential.) 7.5 Pedantry in Practice Having been precise about use and mention, you can now relax! If you’ve understood this sec‑ tion, you know how to do things properly. In exercises and practice problems, unless explicit instructions otherwise are given, you will be expected to do things properly and respect the dis‑ tinction between use and mention. But your understanding of the topic means that you can prob‑ ably do things a bit more sloppily elsewhere – safe in the knowledge you can fix them up if you need to. As the great twentieth century philosopher David Lewis said of the way he presented his account of the word ‘knows’ in his paper ‘Elusive Knowledge’,2 I could have said my say fair and square, bending no rules. It would have been tire‑ some, but it could have been done…. I could have taken great care to distinguish between (1) the language I use when I talk about knowledge, or whatever, and (2) the second language that I use to talk about the semantic and pragmatic workings of the first language. If you want to hear my story told that way, you probably know enough to do the job for yourself. Wise words. In the end, the distinction between use and mention is intended to remove poten‑ tial confusion. But sometimes over‑eager application of it can prove just as big an obstacle to communication. 2David Lewis (1996) ‘Elusive Knowledge’, Australasian Journal of Philosophy 74, pp. 549–67, at pp. 566–7. §7. USE AND MENTION 59 Key Ideas in §7 ›It is crucial to distinguish between use and mention – between talking about the world, and talking about expressions. ›We use Sentential to represent sentences and arguments. But we use English – augmented with some additional vocabulary – to talk about Sentential. ›We introduced a slightly unusual convention for understanding quoted expressions involving script font letters: ‘(𝒜→ℬ)’, to take a representat‑ ive example, is to be interpreted as the Sentential expression consisting of a left parenthesis, followed by whatever Sentential expression 𝒜rep‑ resents, followed by ‘→’, followed by whatever Sentential expression ℬ represents, followed by a closing right parenthesis. Practice exercises A. For each of the following: Are the quotation marks correctly used, strictly speaking? If not, propose a corrected version. 1. Snow is not a sentence of English. 2. ‘𝒜→𝒞’ is a sentence of Sentential. 3. ‘¬𝒜’ is the expression consisting of the symbol ‘¬’ followed by the upper case script letter ‘A’. 4. If ‘𝒜’ is a sentence, so is ‘(𝒜∨𝒜)’. 5. ‘𝒜’ has the same number of characters as ‘𝐴’. B. Example 64 was ambiguous because it was unclear which pairs of quotation marks were matched with each other. Can you come up with a proposal for how we might indicate matching quotation marks to avoid this potential ambiguity? Chapter 3 Truth Tables 8 Truth‑Functional Connectives 8.1 Functions So much for the grammar or syntax of Sentential. We turn now to the meaning of Sentential sen‑ tences. For technical reasons, it is best to start with the intended interpretation of the connectives. As a preliminary, we need to have the concept of a (mathematical) function. Frequently, we refer to things not by name, but by the relations they have to other things. You can refer to Barack Obama by name, but one could equally refer to him in relation to his role, as ‘the 44th President of the United States’, or in relation to his family, as ‘the husband of Michelle and father of Malia and Sasha’. These kinds of referring expressions are known as descriptions, and we will look at them in more detail in §19. Our interest in them now is in the relations they involve. Barack Obama is denoted by ‘the biological father of Malia Obama’ or ‘the biological father of Sasha Obama’, in relation to his children. But we can consider ‘the biological father of …’ in relation to other individuals: so ‘the biological father of Ivanka Trump’ denotes Donald, ‘the biological father of Daisy Turnbull’ denotes Malcolm, and so on. We can summarise this information in a table like this: Input ‘the biological father of …’ Malia Obama Barack Sasha Obama Barack Ivanka Trump Donald Daisy Turnbull Malcolm … … In this table we have an input, on the left, which is related to the output on the right. The relation which maps the things in the left column to their corresponding outputs on the right is known as a function – in this case, the ‘biological father of’ function. More precisely, a FUNCTION is a relation between the members of some collection 𝐴and some collection 𝐵(which may be the same as 𝐴), such that each input to the function is a member (or some members) of 𝐴, each output of the function is a member of 𝐵, and, crucially, each input members of 𝐴are associated with at most 61 62 TRUTH TABLES one member of 𝐵. So ‘the biological father of …’ is a function from the set of people (living or dead) to itself, and associates each person with their biological father. We assume that ‘biological father’ permits each person to be associated with a unique father. If we consider other notions of fatherhood, such as paternal figure, those would not yield a function, because many people have two or more paternal figures in their lives. Note that while everyone is associated with a unique biological father by this function (no input is associated with more than one output), the converse does not hold. Some outputs are linked to more than one input: for example, Barack Obama is the common output of this function when it is given Malia Obama and Sasha Obama as inputs. The ’biological father of’ function, like all functions, yields one output. It also takes only one input. But some functions have more than one input. Consider the ’taller of’ function that takes two people as input, and selects the taller of the two as output. So ’the taller of Ezi Magbegor and Alyssa Healy’ would describe Ezi Magbegor. And we could consider functions with more inputs – for example, a function that takes 7 numbers as input and spits out their average. (Though for functions like ’the average of’, it is more usual to let the input be a single set of numbers, and the output to be a single number, rather than letting there be multiple inputs – nothing really turns on this choice.) If a function takes one input, we’ll say it is a ONE‑PLACE FUNCTION; if two inputs, a TWO‑PLACE FUNCTION; and so on. A function is a map from some inputs to some outputs. To completely specify a function it is also important to identify what those inputs and outputs are. The English expression ‘the biological father of’ could be understood as a function from human beings to human beings; or from living creatures to living creatures. One could even consider it as a function that takes any input whatever. In the latter case, the function will be undefined for many possible inputs: the description, ‘the biological father of the Torrens footbridge’ fails to refer to anything. If a function does not associate an output with every possible input, it is a PARTIAL FUNCTION; if it is well‑defined for every input, it is a TOTAL FUNCTION. These notions are relative to the inputs in question: ‘the biological father of …’ is a total function when the possible inputs are human beings, but only a partial function when anything could be a possible input. Again, this doesn’t just apply to single‑input functions: ’the eldest child of … and …’ is a two‑place partial function that takes two people as input, and yields as output their eldest child – if they have one. Common examples of functions occur in mathematics: we can consider the two‑place function ‘the sum of 𝑥and 𝑦’, which takes two numbers as input, and spits out their unique sum, 𝑥+𝑦. This is again a function from a set to itself – this time all the inputs and outputs are drawn from the set of integers. There are functions which are from one set to another: consider, ‘the number of …’s children’, which is a one‑place function from people to numbers, mapping each person to the number of children they have, zero or more. (Even if they have more than one child, there is still a unique number characterising how many they have, and that is what this function spits out.) For many of the functions we’ve considered so far, the order of inputs doesn’t matter. But for some functions it does. Consider the two‑place subtraction function, ‘−’. Given inputs 7and 4, in that order, yields the output 3(i.e., 7−4=3); but those same inputs, in a different order, yield –3.1 1There is an order‑insensitive function which is closely related, the ’difference between’ function, which yields 3 when given those inputs in any order. §8. TRUTH‑FUNCTIONAL CONNECTIVES 63 There are many relations which are not functions. While Barack Obama can be characterised as ‘the father of Malia’, Malia Obama cannot be characterised as ‘the child of Barack’, since that attempted description would apply equally to her sister. A function is a special kind of rela‑ tion: one where each thing relates to some unique output. We treat the idea of a relation more thoroughly in §21.6. 8.2 The Idea of Truth‑Functionality The relevance of the notion of a function is that we are going to identify the meanings of the Sentential sentence connectives with a certain class of functions. A valid argument is one with a structure that guarantees the truth of the conclusion, given the truth of the premises (§2.5). Our interest in valid arguments leads us to be interested in the truth or falsity of sentences. Sentential gives rules governing the construction of complex sentences from smaller constituents for each sentence connective. The meanings of the sentence connect‑ ives in Sentential are likewise going to allow us to ‘construct’, or determine, the truth‑value of a complex sentence as the result of the truth values of its constituent sentences and the main connective of that sentence. This is an important idea about Sentential sentence connectives. They are each associated with a rule that fixes the truth value of a complex sentence of which they are the main connective, given the truth values of the constituent sentences. But such a rule is just a function: a func‑ tion that takes one or two (or more) truth values as input, and yields a truth value as output. Such a function is called a TRUTH‑FUNCTION. We can now summarise our important insight about Sentential: A connective is TRUTH‑FUNCTIONAL iff the truth value of a sentence with that connective as its main connective is uniquely determined by the truth value(s) of the constituent sentence(s), i.e., its meaning is a truth‑function. Every connective in Sentential is truth‑functional. It turns out that we don’t need to know anything more about the atomic sentences of Sentential than their truth values to assign a truth value to those nonatomic, or COMPOUND, sentences. More generally, the truth value of any compound sentence depends only on the truth value of the subsentences that comprise it. In order to know the truth value of ‘(𝐷∧𝐸)’, for instance, you only need to know the truth value of ‘𝐷’ and the truth value of ‘𝐸’. In order to know the truth value of ‘((𝐷∧𝐸)∨𝐹)’, you need only know the truth value of ‘(𝐷∧𝐸)’ and ‘𝐹’. And so on. This is in fact a good part of the reason why we chose these connectives, and chose their English near‑ equivalents as our structural words. To determine the truth value of some Sentential sentence, we only need to know the truth value of its components. This is why the study of Sentential is termed truth‑functional logic. While all of our connectives are truth‑functional, not every truth function has a corresponding connective. For example, the following is a one‑place truth function: Input Output T T F F 70 TRUTH TABLES (65) If Mitt Romney had won the 2012 election, then he would have been the 45th President of the USA. (66) If Mitt Romney had won the 2012 election, then he would have turned into a helium‑filled balloon and floated away into the night sky. Sentence 65 is true; sentence 66 is false. But both have false antecedents and false consequents. So the truth value of the whole sentence is not uniquely determined by the truth value of the parts. This use of ‘if’ fails our test for truth‑functionality. Do not just blithely assume that you can adequately symbolise an English ‘if …, then …’ with Sentential’s ‘→’. The crucial point is that sentences 65 and 66 employ SUBJUNCTIVE conditionals, rather than IN‑ DICATIVE conditionals. Subjunctive conditionals are also sometimes known as COUNTERFACTUALS. They ask us to imagine something contrary to what we are assuming as fact – that Mitt Romney lost the 2012 election – and then ask us to evaluate what would have happened in that case. The classic illustration of the difference between the indicative and subjunctive conditional comes from pairs like these: (67) If a dingo didn’t take Azaria Chamberlain, something else did. (68) If a dingo hadn’t taken Azaria Chamberlain, something else would have. The indicative conditional in 67 is true, given the actual historical fact that she was taken, and given that we are not assuming at this point anything about how she was taken. But is the sub‑ junctive in 68 also true? It seems not. She was not destined to be taken by something or other, and if the dingo hadn’t intervened, she wouldn’t have disappeared at all.4 The point to take away from this is that subjunctive conditionals cannot be tackled using ‘→’. This is not to say that they cannot be tackled by any formal logical language, only that Sentential is not up to the job.5 So the ‘→’ connective of Sentential is at best able to model the indicative conditional of English, as in 67. In fact there remain difficulties even with indicatives in Sentential. One family of difficulties arises from consideration of the or‑to‑if argument. The argument seems compelling in cases like this: (69) Either the butler or the gardener did it; So: If it wasn’t the butler, it was the gardener. But what if our confidence in the premise 69 derives from our confidence in just one disjunct? Suppose that we are certain it was the butler, and certain that the gardener has an airtight alibi and wasn’t anywhere near the manor at the time of the murder? Because we are certain it was the butler, we might be equally certain of 69, that it was either the butler or the gardener (this inference from a disjunct to a disjunction seems odd, but it is surely valid). But we might also be sure that if it wasn’t the butler, it was the valet – he was the only other person with motive. In this sort of case, we might have confidence in the premise 69 of this or‑to‑if argument and reject its conclusion. Yet the Sentential analogue of the or‑to‑if argument is valid: as we’ll see after we 4If we are assuming as fact that a dingo took her, then when we consider what would have happened had the dingo not been involved, we imagine a situation in which all the actual consequences of the dingo’s action are removed. 5There are in fact logical treatments of counterfactuals, the most influential of which is David Lewis (1973) Counter‑ factuals, Blackwell. §8. TRUTH‑FUNCTIONAL CONNECTIVES 71 introduce the concept of validity for Sentential in §11), ‘𝐴∨𝐵 ∴¬𝐴→ 𝐵’ turns out to be valid. This mismatch suggests that ‘if’ and ‘→’ aren’t a perfect match. I shall say a little more about other difficulties for the material conditional analysis of indicatives in §11.5 and in §30.1. For now, I shall content myself with the observation that ‘→’ is the only plausible candidate for a truth‑functional conditional. Our working hypothesis is that many uses of ‘if’ can be adequately approximated by ‘→’. Many English conditionals cannot be represented adequately using ‘→’. Sentential is an intrinsically limited language. But this is only a problem if you try to use it to do things it wasn’t designed to do. Key Ideas in §8 ›The connectives of Sentential are all truth‑functional, and have their meanings specified by the truth‑tables laid out in §8.3. ›When we treat a sentence of Sentential as symbolising an English sen‑ tence, we need only say that as far as truth value is concerned and truth‑ functional structure is concerned, they are alike. ›English has many nontruth‑functional connectives. Some uses of the con‑ ditional ‘if’ are nontruth‑functional. But as long as we remain aware of the limitations of Sentential, it can be a very powerful tool for modelling a significant class of arguments. Practice exercises A. Which of the following arguably may not characterise a function, where 𝑥is the input and 𝑦 is the output: 1. 𝑦is the product of 𝑥and itself; 2. 𝑦is the square root of 𝑥; 3. 𝑦is a child of 𝑥; 4. 𝑦is a child of 𝑥and younger than any other child of 𝑥; 5. 𝑦is taller than 𝑥; 6. 𝑦is as tall as 𝑥. B. True or false: if the main connective of some sentence is truth‑functional, then the truth value of the sentence uniquely determines the truth values of any constituents. C. Suppose †is some English one‑place connective, so that ‘†𝒜’ is a grammatical sentence. How can we test if it is not truth‑functional? D. How many possible two‑place truth‑functions are there, i.e., how many different functions take a pair of truth values as input, and yield a truth value as output? 9 Complete Truth Tables 9.1 Valuations So far, we have considered assigning truth values to Sentential sentences indirectly. We have said, for example, that a Sentential sentence such as ‘𝐵’ is to take the same truth value as the English sentence ‘Big Ben is in London’ (whatever that truth value may be). But we can also assign truth values directly. We can simply stipulate that ‘𝐵’ is to be true, or stipulate that it is to be false – at least for present purposes. A VALUATION is any assignment of truth values to some atomic sentences of Sentential. It assigns exactly one truth value, either True or False, to each of the sentences in question. A valuation is thus a function from atomic sentences to truth values. So this is a valuation: 𝐴,𝐺,𝑃,𝐺7↦T 𝐹,𝑅,𝑍↦F. There is no requirement that a valuation be a TOTAL FUNCTION, that is, that it assign a truth value to every atomic sentence. To fix the truth value of a sentence 𝒜of Sentential a valuation must assign a truth value to every atomic sentence 𝒜contains. A valuation is a temporary assignment of ‘meanings’ to Sentential sentences, in much the same way as a symbolisation key might be. (It only assigns truth values, the only dimension of meaning that Sentential is sensitive to.) What is distinctive about Sentential is that almost all of its basic vocabulary – the atomic sentences – only get their meanings in this temporary fashion. The only parts of Sentential that get their meanings permanently are the connectives, which always have a fixed interpretation. This is rather unlike English, where most words have their meanings on a permanent basis. But there are some words in English – like pronouns (‘he’, ‘she’, ‘it’) and demonstratives (‘this’, ‘that’) – that get their meaning assigned temporarily, and then can be reused with a different meaning 72 §9. COMPLETE TRUTH TABLES 73 in another context. Such expressions are called CONTEXT SENSITIVE. In this sense, all the atomic sentences of Sentential are context sensitive expressions. Of course we don’t have anything so explicit and deliberate as a valuation or a symbolisation key in English to assign a meaning to a particular use of ‘this’ or ‘that’ – the circumstances of a conversation automatically assign an appropriate object (usually). In Sentential, however, we need to explicitly set out the interpret‑ ations of the atomic sentences we are concerned with. 9.2 Truth Tables We introduced schematic truth tables in §8.3. These showed what truth value a compound sen‑ tence with a certain structure was determined to have by the truth values of its subsentences, whatever they might be. We now introduce a closely related idea, that of a truth table. This shows how a specific compound sentence has its truth value determined by the truth values of its specific atomic subsentences, across all the possible ways that those atomic subsentences might be assigned True and False. You will no doubt have realised that a way of assigning True and False to atomic sentences is a valuation. So we can say: a TRUTH TABLE summarises how the truth value of a compound sentence depends on the possible valuations of its atomic subsentences. Each row of a truth table represents a possible valuation. The entire complete truth table represents all possible valuations. And the truth table provides us with a means to calculate the truth value of complex sentences, on each possible valuation. This is pretty abstract. So it might be easiest to explain with an example. 9.3 A Worked Example Consider the sentence ‘(𝐻∧𝐼) → 𝐻’. There are four possible ways to assign True and False to the atomic sentences ‘𝐻’ and ‘𝐼’: both true, both false, ‘𝐻’ true and ‘𝐼’ false, and ‘𝐼’ true and ‘𝐻’ false. So there are four possible valuations of these two atomic sentences. We can lay out these valuations as follows: 𝐻 𝐼 (𝐻∧𝐼)→𝐻 T T T F F T F F To calculate the truth value of the entire sentence ‘(𝐻∧𝐼) → 𝐻’, we first copy the truth values for the atomic sentences and write them underneath the letters in the sentence: 𝐻 𝐼 (𝐻∧𝐼)→𝐻 T T T T T T F T F T F T F T F F F F F F 74 TRUTH TABLES Now consider the subsentence ‘(𝐻∧𝐼)’. This is a conjunction, (𝒜∧ℬ), with ‘𝐻’ as 𝒜and with ‘𝐼’ as ℬ. The schematic truth table for conjunction gives the truth conditions for any sentence of the form (𝒜∧ℬ), whatever 𝒜and ℬmight be. It summarises the point that a conjunction is true iff both conjuncts are true. In this case, our conjuncts are just ‘𝐻’ and ‘𝐼’. They are both true on (and only on) the first row of the truth table. Accordingly, we can calculate the truth value of the conjunction on all four rows. 𝒜∧ℬ 𝐻 𝐼 (𝐻∧𝐼)→𝐻 T T T T T T T F T F F T F T F F T F F F F F F F Now, the entire sentence that we are dealing with is a conditional, 𝒞 → 𝒟, with ‘(𝐻∧𝐼)’ as 𝒞 and with ‘𝐻’ as 𝒟. On the second row, for example, ‘(𝐻 ∧𝐼)’ is false and ‘𝐻’ is true. Since a conditional is true when the antecedent is false, we write a ‘T’ in the second row underneath the conditional symbol. We continue for the other three rows and get this: 𝒞 →𝒟 𝐻 𝐼 (𝐻∧𝐼)→𝐻 T T T T T T F F T T F T F T F F F F T F The conditional is the main logical connective of the sentence. And the column of ‘T’s underneath the conditional tells us that the sentence ‘(𝐻∧𝐼) → 𝐻’ is true regardless of the truth values of ‘𝐻’ and ‘𝐼’. They can be true or false in any combination, and the compound sentence still comes out true. Since we have considered all four possible assignments of truth and falsity to ‘𝐻’ and ‘𝐼’ – since, that is, we have considered all the different valuations – we can say that ‘(𝐻∧𝐼)→𝐻’ is true on every valuation. In this example, I have not repeated all of the entries in every column in every successive table. When actually writing truth tables on paper, however, it is impractical to erase whole columns or rewrite the whole table for every step. Although it is more crowded, the truth table can be written in this way: 𝐻 𝐼 (𝐻∧𝐼)→𝐻 T T T T T TT T F T F F TT F T F F T TF F F F F F TF §9. COMPLETE TRUTH TABLES 75 Most of the columns underneath the sentence are only there for bookkeeping purposes. The column that matters most is the column underneath the main connective for the sentence, since this tells you the truth value of the entire sentence. I have emphasised this, by putting this column in bold. When you work through truth tables yourself, you should similarly emphasise it (perhaps by drawing a box around the relevant column). 9.4 Building Complete Truth Tables A COMPLETE TRUTH TABLE has a row for every possible assignment of True and False to the relevant atomic sentences. Each row represents a valuation, and a complete truth table has a row for all the different valuations. The size of the complete truth table depends on the number of different atomic sentences in the table. A sentence that contains only one atomic sentence requires only two rows, as in the schematic truth table for negation. This is true even if the same letter is repeated many times, as in the sentence ‘((𝐶 ↔𝐶)→𝐶)∧¬(𝐶 →𝐶)’. The complete truth table requires only two rows because there are only two possibilities: ‘𝐶’ can be true or it can be false. The truth table for this sentence looks like this: 𝐶((𝐶↔𝐶)→𝐶)∧¬(𝐶→𝐶) T T T T T T FF T T T F F T F F F FF F T F Looking at the column underneath the main connective, we see that the sentence is false on both rows of the table; i.e., the sentence is false regardless of whether ‘𝐶’ is true or false. It is false on every valuation. A sentence that contains two atomic sentences requires four rows for a complete truth table, as in the schematic truth tables, and as in the complete truth table for ‘(𝐻∧𝐼)→𝐻’. A sentence that contains three atomic sentences requires eight rows: 𝑀 𝑁 𝑃 𝑀∧(𝑁∨𝑃) T T T T TT T T T T F T TT T F T F T T TF T T T F F T FF F F F T T F FT T T F T F F FT T F F F T F FF T T F F F F FF F F From this table, we know that the sentence ‘𝑀∧(𝑁∨𝑃)’ can be true or false, depending on the truth values of ‘𝑀’, ‘𝑁’, and ‘𝑃’. 76 TRUTH TABLES A complete truth table for a sentence that contains four different atomic sentences requires 16 rows. Five letters, 32 rows. Six letters, 64 rows. And so on. To be perfectly general: If a complete truth table has 𝑛different atomic sentences, then it must have 2𝑛rows.1 In order to fill in the columns of a complete truth table, begin with the right‑most atomic sentence and alternate between ‘T’ and ‘F’. In the next column to the left, write two ‘T’s, write two ‘F’s, and repeat. For the third atomic sentence, write four ‘T’s followed by four ‘F’s. This yields an eight row truth table like the one above. For a 16 row truth table, the next column of atomic sentences should have eight ‘T’s followed by eight ‘F’s. For a 32 row table, the next column would have 16 ‘T’s followed by 16 ‘F’s. And so on. Key Ideas in §9 ›A valuation of some atomic sentences associates each of them with ex‑ actly one of our truth values; it is like an extremely stripped down ver‑ sion of a symbolisation key. In there are 𝑛atomic sentences, there are 2𝑛 valuations of them. ›A truth table lays out the truth values of a particular Sentential sentence in each of the distinct possible valuations of its constituent atomic sen‑ tences. Practice exercises A. How does a schematic truth table differ from a regular truth table? What is a complete truth table? B. Offer complete truth tables for each of the following: 1. 𝐴→𝐴 2. 𝐶 →¬𝐶 3. (𝐴↔𝐵)↔¬(𝐴↔¬𝐵) 4. (𝐴→𝐵)∨(𝐵→𝐴) 5. (𝐴∧𝐵)→(𝐵∨𝐴) 6. ¬(𝐴∨𝐵)↔(¬𝐴∧¬𝐵) 7. ((𝐴∧𝐵)∧¬(𝐴∧𝐵))∧𝐶 8. ((𝐴∧𝐵)∧𝐶)→𝐵 9. ¬((𝐶∨𝐴)∨𝐵) If you want additional practice, you can construct truth tables for any of the sentences and argu‑ ments in the exercises for Chapter 2. 1Since the values of atomic sentences are independent of each other, each new atomic sentence 𝒜𝑛+1 we consider is capable of being true or false on every existing valuation on 𝒜1,…,𝒜𝑛, and so there must be twice as many valuations on 𝒜1,…,𝒜𝑛,𝒜𝑛+1 as on 𝒜1,…,𝒜𝑛. 10 Semantic Concepts In §9.1, we introduced the idea of a valuation and showed how to determine the truth value of any Sentential sentence on any valuation using a truth table in the remainder of the chapter. In this section, we shall introduce some related ideas, and show how to use truth tables to test whether or not they apply. 10.1 Logical Truths and Falsehoods In §3, I explained necessary truth and necessary falsity. Both notions have close but imperfect sur‑ rogates in Sentential. We shall start with a surrogate for necessary truth. 𝒜is a LOGICAL TRUTH iff it is true on every valuation (among those valuations on which it has a truth value). We need the parenthetical clause because of the way we have defined valuations. A given valu‑ ation might only assign truth values to some atomic sentences and not all. For any sentence 𝒜 which contains an atomic sentence to which a valuation doesn’t assign a truth value, 𝒜will not have any truth value according to that valuation. Logical truths in Sentential are sometimes called TAUTOLOGIES. We can determine whether a sentence is a logical truth just by using truth tables. If the sentence is true on every row of a complete truth table, then it is true on every valuation for its constituent atomic sentences, so it is a logical truth. In the example of §9, ‘(𝐻∧𝐼)→𝐻’ is a logical truth. This is only, though, a surrogate for necessary truth. There are some necessary truths that we cannot adequately symbolise in Sentential. An example is ‘2+2 = 4’. This must be true, but if we try to symbolise it in Sentential, the best we can offer is an atomic sentence, and no atomic sentence is a logical truth.1Still, if we can adequately symbolise some English sentence using 1At the risk of repeating myself: 2+2=4is necessarily true, but it is not necessarily true in virtue of its structure. A necessary truth is true, with its actual meaning, in every possible situation. A Sentential‑logical truth is true in the actual situation on every possible way of interpreting its atomic sentences. These are interestingly different notions. 77 78 TRUTH TABLES aSentential sentence which is a logical truth, then that English sentence expresses a necessary truth. We have a similar surrogate for necessary falsity: 𝒜is a LOGICAL FALSEHOOD iff it is false on every valuation (among those on which it has a truth value). We can determine whether a sentence is a logical falsehood just by using truth tables. If the sentence is false on every row of a complete truth table, then it is false on every valuation, so it is a logical falsehood. In the example of §9, ‘((𝐶 ↔𝐶)→𝐶)∧¬(𝐶 →𝐶)’ is a logical falsehood. A logical falsehood is sometimes called a CONTRADICTION, though it is perhaps even more common to reserve that term for those logical falsehoods which have the form (𝒜∧¬𝒜). 10.2 Logical Equivalence Here is a similar, useful notion: 𝒜and ℬare LOGICALLY EQUIVALENT iff they have the same truth value on every valuation among those which assign both of them a truth value. It is easy to test for logical equivalence using truth tables. Consider the sentences ‘¬(𝑃∨𝑄)’ and ‘¬𝑃∧¬𝑄’. Are they logically equivalent? To find out, we may construct a truth table. 𝑃 𝑄 ¬(𝑃∨𝑄) ¬𝑃∧¬𝑄 T T FT T T F T FF T T F FT T F F T FT F F T FF T T T F FF T F F TF F F T F TT F Look at the columns for the main connectives; negation for the first sentence, conjunction for the second. On the first three rows, both are false. On the final row, both are true. Since they match on every row, the two sentences are logically equivalent. 10.3 More Parenthetical Conventions Consider these two sentences: ((𝐴∧𝐵)∧𝐶) (𝐴∧(𝐵∧𝐶)) These have the same truth table, and are logically equivalent. Consequently, it will never make any difference from the perspective of truth value – which is all that Sentential cares about (see §8) – which of the two sentences we assert (or deny). And since the order of the parentheses does §10. SEMANTIC CONCEPTS 79 not matter, I shall allow us to drop them. In short, we can save some ink and some eyestrain by writing: 𝐴∧𝐵∧𝐶 The general point is that, if we just have a long list of conjunctions, we can drop the inner paren‑ theses. (I already allowed us to drop outermost parentheses in §6.) The same observation holds for disjunctions. Since the following sentences are logically equivalent: ((𝐴∨𝐵)∨𝐶) (𝐴∨(𝐵∨𝐶)) we can simply write: 𝐴∨𝐵∨𝐶 And generally, if we just have a long list of disjunctions, we can drop the inner parentheses. But be careful. These two sentences have different truth tables, so are not logically equivalent: ((𝐴→𝐵)→𝐶) (𝐴→(𝐵→𝐶)) So if we were to write: 𝐴→𝐵→𝐶 it would be dangerously ambiguous. So we must not do the same with conditionals. Equally, these sentences have different truth tables: ((𝐴∨𝐵)∧𝐶) (𝐴∨(𝐵∧𝐶)) So if we were to write: 𝐴∨𝐵∧𝐶, it would be dangerously ambiguous. Never write this. (This is the ambiguity discussed in §6.4.) The moral is: you can drop parentheses when dealing with a long list of conjunctions, or when dealing with a long list of disjunctions. But that’s it. 10.4 Consistency In §3, I said that sentences are jointly consistent iff it is possible for all of them to be true at once. We can offer a surrogate for this notion too: 𝒜1,𝒜2,…,𝒜𝑛are JOINTLY CONSISTENT iff there is some valuation which makes them all true. 86 TRUTH TABLES 11.5 The Limits of these Tests We have reached an important milestone: a test for the validity of arguments! But, we should not get carried away just yet. It is important to understand the limits of our achievement. There are three sorts of limitations I want to discuss: 1. Some valid arguments are overlooked by our test; 2. Some invalid arguments are misclassified by a naive application of our test to an inad‑ equate symbolisation; and 3. There are some putative examples of sentences that cannot be symbolised because of some assumptions that Sentential makes. I shall illustrate these limits with three examples. First, consider the argument: (75) Daisy is a small cow. So, Daisy is a cow. To symbolise this argument in Sentential, we would have to use two different atomic sentences – perhaps ‘𝑆’ and ‘𝐶’ – for the premise and the conclusion respectively. Now, it is obvious that ‘𝑆’ does not entail ‘𝐶’. But the English argument surely seems valid – the structure of ‘Daisy is a small cow’ guarantees that ‘Daisy is a cow’ is true. Note that a small cow might still be rather large, so we cannot fudge things by symbolising ‘Daisy is a small cow’ as a conjunction of ‘Daisy is small’ and ‘Daisy is a cow’. (We’ll return to this sort of case in §16, where we will see how to symbolise 75 as a valid argument in Quantifier.) But our Sentential‑based test for validity in English will have some false negatives: it will classify some valid English arguments as invalid. This is because some valid arguments are valid in virtue of structure which is not truth‑functional. We’ll see more examples of this in §15. Second, consider the following arguments: (76) It’s not the case that the Crows will win by a lot, if they win. So the Crows will win. (77) It’s not the case that, if God exists, She answers malevolent prayers. So God exists. Both of these arguments have the same structure. Let’s focus on the second, example 77. Sym‑ bolising it in Sentential, we would offer something like ‘¬(𝐺 → 𝑀) ∴ 𝐺’. Now, as can easily be checked with a truth table, this is a correct entailment in Sentential. So if we symbolise the argument 77 in Sentential in this way, the conditional premise entails that God exists. But that’s strange: surely even the atheist can accept sentence 77, without contradicting herself! Some say that 77 would be better symbolised by ‘(𝐺→¬𝑀)’, even though that doesn’t reflect the apparent form of the English sentence. ‘(𝐺→¬𝑀)’ does not entail 𝐺. This symbolisation does a better job of reflecting the intuitive consequences of the English sentence 77, but at the cost of abandoning a straightforward correspondence between the structure of English sentences and their Sentential symbolisations. A better alternative might be to think that the conditional ‘if God exists, she answers malevolent prayers’ is not to be symbolised by ‘→’. This conditional is false, many think: they may think, for §11. ENTAILMENT AND VALIDITY 87 example, that it is part of the concept of God that God is good, and hence would not grant prayers with evil intent. Even the atheist might accept this, maybe because they accept the subjunctive conditional ‘even if God were to exist, God would not answer malevolent prayers’ (§8.6). This sort of example has motivated many philosophers to offer nontruth‑functional accounts of the English ‘if’, including some that make it behave rather like a subjunctive conditional.3 The cases in 76 and 77 are examples of the (so‑called) paradoxes of material implication. They high‑ light a limitation of Sentential, in that it appears not to have a conditional connective that ad‑ equately models these uses of the English ‘if’. But it is also a limitation of our tests for validity in English, because the test is only as good as the symbolisations we come up with as part of it. If we are not careful, we might end up mistakenly using an inadequate symbolisation, and giving the wrong verdict about some argument, thinking it valid when it is not. (I will return one final time to the relation between ‘if’ and →in §30.1.) Finally, consider the sentence: (78) Jan is neither bald nor not‑bald. To symbolise this sentence in Sentential, we would offer something like ‘¬(𝐽 ∨ ¬𝐽)’. This a logical falsehood (check this with a truth‑table). But sentence 78 does not itself seem like a logical falsehood; for we might have happily go on to add ‘Jan is on the borderline of baldness’! Making this point another way: as may easily be seen by truth tables, ‘¬(𝐽∨¬𝐽)’ is logically equivalent to ‘(¬𝐽∧𝐽)’. This latter sentence symbolises an obvious logical falsehood in English: (79) Jan is both not‑bald and also bald. Is it equally obvious, though, that 78 is synonymous with 79? It seems like it may not be, even though our test will classify any English argument from one to the other as valid (since both are symbolised as logical falsehoods, which degenerately entail anything). Because of the way we have defined valuations, every sentence of Sentential is assigned either True or False in any valuation which makes it meaningful by assigning truth values to its atomic sentences. This property of Sentential is known as BIVALENCE: that every sentence has exactly one of the two possible truth values. The case of Jan’s baldness (or otherwise) raises the general question of what logic we should use when dealing with vague discourse, predicates like ‘bald’ or ‘tall’ which seem to have borderline cases. Many think it plausible that a borderline case of some predicate ℱis neither a case of ℱ, nor does it fail to be a case of ℱ. Hence they have been tempted to deny bivalence for English: ‘Jan is bald’, they say, is neither True nor False! If 𝑝is neither true nor false, then it is hardly surprising that ‘𝑝or not‑𝑝’ turns out to be untrue. If these thinkers are right that vagueness in English leads to the denial of bivalence, while Sentential is bivalent, this will give rise to mismatches between English and Sentential. These mismatches will not involve inadequate symbolisation, but a more fundamental disagreement about the background frame‑ work – here, a disagreement about the nature of truth. We will return to the topic of vagueness (though still briefly) at the end of this book, in §39.2. 3Edgington discusses Stalnaker’s version of such an account in ‘Nearest Possible Worlds’, §4.1 of her ‘Indicative Conditionals’, cited above (plato.stanford.edu/entries/conditionals/#Sta). 88 TRUTH TABLES In different ways, these three examples highlight some of the limits of working with a language like Sentential that can only handle truth‑functional connectives. Moreover, these limits give rise to some interesting questions in philosophical logic. Part of the purpose of this course is to equip you with the tools to explore these questions of philosophical logic. But we have to walk before we can run; we have to become proficient in using Sentential, before we can adequately discuss its limits, and consider alternatives. It is important to recognise that these are limits to Sentential only in its role as a framework to model validity in English and other natural languages. They are not problems for Sentential as a formal language. Moreover, as I have emphasised already, these limitations are merely manifestations of the fact that Sentential is being used as a model of natural language. Models are typically not designed or intended to capture every aspect of what they model. Their utility derives often from being simpler than the complex things they are representing. The limitations we have noted indicate that Sentential may not model English perfectly in these cases. But Sentential remains an adequate model of English in many other cases. Key Ideas in §11 ›If every valuation which makes some sentences all true is also one that makes some further sentence true, then those sentences entail the further sentence. We use the symbol ‘⊨’ for entailment. ›We can test for entailment using truth tables, in the same sort of way that we test for consistency. ›If an argument when symbolised turns out to be an entailment, then the original argument is valid in virtue of its truth‑functional structure. So we can test for validity using the truth table tests for entailment. ›These tests nevertheless have limitations: not every valid argument can be symbolised as a Sentential entailment. These limitations are typical of using simpler models to represent complex things. Practice exercises A. What does it mean to say that sentences 𝒜1,𝒜2,…,𝒜𝑛of Sentential entail a further sentence 𝒞? B. If 𝒜1,𝒜2,…,𝒜𝑛⊨𝒞, what can you say about the argument with premises 𝒜1,𝒜2,…,𝒜𝑛and conclusion 𝒞? C. Use truth tables to determine whether each argument is valid or invalid. 1. 𝐴→𝐴∴𝐴 2. 𝐴→(𝐴∧¬𝐴)∴¬𝐴 3. 𝐴∨(𝐵 →𝐴)∴¬𝐴→¬𝐵 4. 𝐴∨𝐵,𝐵∨𝐶,¬𝐴∴𝐵∧𝐶 5. (𝐵∧𝐴)→𝐶,(𝐶∧𝐴)→𝐵∴(𝐶∧𝐵)→𝐴 D. Answer each of the questions below and justify your answer. §11. ENTAILMENT AND VALIDITY 89 1. Suppose that 𝒜and ℬare logically equivalent. What can you say about 𝒜↔ℬ? 2. Suppose that (𝒜∧ℬ)→𝒞is neither a logical truth nor a logical falsehood. What can you say about whether 𝒜,ℬ∴𝒞is valid? 3. Suppose that 𝒜,ℬand 𝒞are jointly inconsistent. What can you say about (𝒜∧ℬ∧𝒞)? 4. Suppose that 𝒜is a logical falsehood. What can you say about whether 𝒜,ℬ⊨𝒞? 5. Suppose that 𝒞is a logical truth. What can you say about whether 𝒜,ℬ⊨𝒞? 6. Suppose that 𝒜and ℬare logically equivalent. What can you say about (𝒜∨ℬ)? 7. Suppose that 𝒜and ℬare not logically equivalent. What can you say about (𝒜∨ℬ)? E. If two sentences of Sentential,𝒜and 𝒟, are logically equivalent, what can you say about (𝒜→𝒟)? What about the argument 𝒜∴𝒟? F. Consider the following principle: ›Suppose 𝒜and ℬare logically equivalent. Suppose an argument contains 𝒜(either as a premise, or as the conclusion). The validity of the argument would be unaffected, if we replaced 𝒜with ℬ. Is this principle correct? Explain your answer. 12 Truth Table Shortcuts With practice, you will quickly become adept at filling out truth tables. In this section, I want to give you some permissible shortcuts to help you along the way. 12.1 Working through Truth Tables You will quickly find that you do not need to copy the truth value of each atomic sentence, but can simply refer back to them. So you can speed things up by writing: 𝑃 𝑄 (𝑃∨𝑄)↔¬𝑃 T T T FF T F T FF F T T TT F F F FT You also know for sure that a disjunction is true whenever one of the disjuncts is true. So if you find a true disjunct, there is no need to work out the truth values of the other disjuncts. Thus you might offer: 𝑃 𝑄 (¬𝑃∨¬𝑄)∨¬𝑃 T T F F F FF T F F T T TF F T TT F F TT Equally, you know for sure that a conjunction is false whenever one of the conjuncts is false. So if you find a false conjunct, there is no need to work out the truth value of the other conjunct. Thus you might offer: 90 §12. TRUTH TABLE SHORTCUTS 91 𝑃 𝑄 ¬(𝑃∧¬𝑄)∧¬𝑃 T T FF T F FF F T T F TT F F T F TT A similar short cut is available for conditionals. You immediately know that a conditional is true if either its consequent is true, or its antecedent is false. Thus you might present: 𝑃 𝑄 ((𝑃→𝑄)→𝑃)→𝑃 T T T T F T F T T F T F F T F T So ‘((𝑃 → 𝑄) → 𝑃) → 𝑃’ is a logical truth. In fact, it is an instance of Peirce’s Law, named after Charles Sanders Peirce. 12.2 Testing for Validity and Entailment When we use truth tables to test for validity or entailment, we are checking for bad rows: rows where the premises are all true and the conclusion is false. Note: ›Any row where the conclusion is true is not a bad row. ›Any row where some premise is false is not a bad row. Since all we are doing is looking for bad rows, we should bear this in mind. So: if we find a row where the conclusion is true, we do not need to evaluate anything else on that row: that row definitely isn’t bad. Likewise, if we find a row where some premise is false, we do not need to evaluate anything else on that row. With this in mind, consider how we might test the following claimed entailment: ¬𝐿→(𝐽∨𝐿),¬𝐿⊨𝐽. The first thing we should do is evaluate the conclusion on the right of the turnstile. If we find that the conclusion is true on some row, then that is not a bad row. So we can simply ignore the rest of the row. So at our first stage, we are left with something like: 𝐽 𝐿 ¬𝐿→(𝐽∨𝐿) ¬𝐿 𝐽 T T T T F T F T ? ? F F F ? ? F 92 TRUTH TABLES where the blanks indicate that we are not going to bother doing any more investigation (since the row is not bad) and the question‑marks indicate that we need to keep investigating. The easiest premise on the left of the turnstile to evaluate is the second, so we next do that: 𝐽 𝐿 ¬𝐿→(𝐽∨𝐿) ¬𝐿 𝐽 T T T T F T F T F F F F ? T F Note that we no longer need to consider the third row on the table: it will not be a bad row, because (at least) one of premises is false on that row. And finally, we complete the truth table: 𝐽 𝐿 ¬𝐿→(𝐽∨𝐿) ¬𝐿 𝐽 T T T T F T F T F F F F T FF T F The truth table has no bad rows, so this claimed entailment is genuine. (Any valuation on which all the premises are true is a valuation on which the conclusion is true.) It might be worth illustrating the tactic again, this time for validity. Let us check whether the following argument is valid 𝐴∨𝐵,¬(𝐴∧𝐶),¬(𝐵∧¬𝐷)∴(¬𝐶∨𝐷). So we need to check whether the premises entail the conclusion. At the first stage, we determine the truth value of the conclusion. Since this is a disjunction, it is true whenever either disjunct is true, so we can speed things along a bit. We can then ignore every row apart from the few rows where the conclusion is false. §12. TRUTH TABLE SHORTCUTS 93 𝐴 𝐵 𝐶 𝐷 𝐴∨𝐵 ¬(𝐴∧𝐶) ¬(𝐵∧¬𝐷) (¬𝐶∨𝐷) T T T T T T T T F ? ? ? F F T T F T T T T F F T T T F T T T T F T F ? ? ? F F T F F T T T F F F T T F T T T T F T T F ? ? ? F F F T F T T F T F F T T F F T T T F F T F ? ? ? F F F F F T T F F F F T T We must now evaluate the premises. We use shortcuts where we can: 𝐴 𝐵 𝐶 𝐷 𝐴∨𝐵 ¬(𝐴∧𝐶) ¬(𝐵∧¬𝐷) (¬𝐶∨𝐷) T T T T T T T T F T F T F F T T F T T T T F F T T T F T T T T F T F T F T F F T F F T T T F F F T T F T T T T F T T F T T FFT T F F F T F T T F T F F T T F F T T T F F T F FFF F F F T T F F F F T T If we had used no shortcuts, we would have had to write 256 ‘T’s or ‘F’s on this table. Using shortcuts, we only had to write 37. We have saved ourselves a lot of work. By the notion of a bad rows – a potential counterexample to a purported entailment – you can save yourself a huge amount of work in testing for validity. There is still lots of work involved in symbolising any natural language argument into Sentential, but once that task is undertaken it is a relatively automatic process to determine whether the symbolisation is an entailment. 94 TRUTH TABLES Key Ideas in §12 ›Some shortcuts are available in constructing truth tables. For example, if a conjunction has one false conjunct, we needn’t check the truth value of the other in order to determine that the whole conjunction is false. ›When applying our test for entailment, we need only check those rows on which all the premises are true to see if the conclusion is false on those rows. So we needn’t check any row where the conclusion is true, or where a premise is false. Practice exercises A. Using shortcuts, determine whether each sentence is a logical truth, a logical falsehood, or neither. 1. ¬𝐵∧𝐵 2. ¬𝐷∨𝐷 3. (𝐴∧𝐵)∨(𝐵∧𝐴) 4. ¬(𝐴→(𝐵→𝐴)) 5. 𝐴↔(𝐴→(𝐵∧¬𝐵)) 6. ¬(𝐴∧𝐵)↔𝐴 7. 𝐴→(𝐵∨𝐶) 8. (𝐴∧¬𝐴)→(𝐵∨𝐶) 9. (𝐵∧𝐷)↔(𝐴↔(𝐴∨𝐶)) 13 Partial Truth Tables Sometimes, we do not need to know what happens on every row of a truth table. Sometimes, just a single row or two will do. 13.1 Direct Uses of Partial Truth Tables Logical Truth In order to show that a sentence is a logical truth (tautology), we need to show that it is true on every valuation. That is to say, we need to know that it comes out true on every row of the truth table. So, it seems, we need a complete truth table. To show that a sentence is not a logical truth, however, we only need one valuation, correspond‑ ing to a truth table row on which the sentence is false. Therefore, in order to show that some sentence is not a logical truth, it is enough to provide a single valuation – a single row of the truth table – which makes the sentence false. Suppose that we want to show that the sentence ‘(𝑈 ∧𝑇) → (𝑆∧𝑊)’ is not a logical truth. We set up a PARTIAL TRUTH TABLE: 𝑆 𝑇 𝑈 𝑊 (𝑈∧𝑇)→(𝑆∧𝑊) F We have only left space for one row, rather than 16, since we are only looking for one valuation on which the sentence is false. For just that reason, we have filled in ‘F’ for the entire sentence. A partial truth table is a device for ‘reverse engineering’ a valuation, given a truth value assigned to a complex sentence. We work backward from that truth value to what the valuation must or could be. The main connective of the sentence is a conditional. In order for the conditional to be false, the antecedent must be true and the consequent must be false. So we fill these in on the table: 𝑆 𝑇 𝑈 𝑊 (𝑈∧𝑇)→(𝑆∧𝑊) TFF 95 14 Expressiveness of Sentential When we introduced the idea of truth‑functionality in §8.2, we observed that every sentence connective in Sentential was truth‑functional. As we noted, that property allows us to represent complex sentences involving only these connectives using truth tables. 14.1 Other Truth‑Functional Connectives Are there other truth functional connectives than those in Sentential? If there were, they would have schematic truth tables that differ from those for any of our connectives. And it is easy to see that there are. Consider this proposed connective: The Sheffer stroke For any sentences 𝒜and ℬ,𝒜 ↓ ℬis true if and only if both 𝒜and ℬare false. We can summarize this in the schematic truth table for the Sheffer Stroke: 𝒜 ℬ 𝒜↓ℬ T T F T F F F T F F F T Inspection of the schematic truth tables for ∧,∨, etc., shows that their truth tables are different from this one, and hence the Sheffer Stroke is not one of the connectives of Sentential. It is a connective of English however: it is the ‘neither … nor …’ connective that features in ‘Siya is neither an archer nor a jockey’, which is false iff she is either. ‘Whether or not’ The connective ‘… whether or not …’, as in the sentence ‘Sam is happy whether or not she’s rich’ seems to have this schematic truth table: 102 §14. EXPRESSIVENESS OF Sentential 103 𝒜 ℬ 𝒜whether or not ℬ T T T T F T F T F F F F This too corresponds to no existing connective of Sentential. In fact, it can be shown that there are many other potential truth‑functional connectives that are not included in the language Sentential.1 14.2 The Expressive Power of Sentential Should we be worried about this, and attempt to add new connectives to Sentential? It turns out we already have enough connectives to say anything we wish to say that makes use of only truth‑ functional connectives.2The connectives that are already in Sentential are TRUTH‑FUNCTIONALLY COMPLETE: For any truth‑functional connective in any language – that is, one which has a truth‑table like ours – there is a schematic sentence of Sentential which has the same truth table. Remember that a schematic sentence is something like 𝒜∨¬ℬ, where arbitrary Sentential sen‑ tences can fill the places indicated by 𝒜and ℬ. This is actually not very difficult to show. We’ll start with an example. Suppose we have the Eng‑ lish sentence ‘Siya is exactly one of an archer and a jockey’. This sentence features the connective ‘Exactly one of … and …’. In this case, the simpler sentences which are connected to form the complex sentence are ‘Siya is an archer’ and ‘Siya is a jockey’. The schematic truth table for this connective is as follows: 𝒜 ℬ ‘Exactly one of 𝒜and ℬ’ T T F T F T F T T F F F 1There are in fact sixteen truth‑functional connectives that join two simpler sentences into a complex sentence, but Sentential only includes four. (Why sixteen? Because there are four rows of the schematic truth table for such a connective, and each row can have either a T or an F recorded against it, independent of the other rows, so there are 2×2×2×2=16ways of constructing such a truth‑table.) 2I should flag a potential limitation here: this result needs our assumption that True and False are the only truth values. Some MANY‑VALUED LOGICS include further ‘truth values’; for example a third truth value Indeterminate. (It is dubious whether that is a truth value, or whether a sentence having it just reflects our ignorance of its truth or falsity.) Such logics can have truth‑functional connectives that don’t behave like any connective of Sentential. One motivating example for a third truth value, neither true nor false, are cases of presupposition failure we will look at in §19.4. 104 TRUTH TABLES We want now to find a schematic Sentential sentence that has this same truth table. So we shall want the sentence to be true on the second row, and true on the third row, and false on the other rows. In other words, we want a sentence which is true on either the second row or the third row. Let’s begin by focusing on that second row, or rather the family of valuations corresponding to it. Those valuations include only those that make 𝒜true and and ℬfalse. These are the only valuations among those we are considering which make 𝒜true and ℬfalse. So they are the only valuations which make both 𝒜and ¬ℬ true. So we can construct a sentence which is true on valuations in that family, and those valuations alone: the conjunction of 𝒜and ¬ℬ,(𝒜∧¬ℬ). Now look at the third row and its associated family of valuations. Those valuations make 𝒜false and ℬtrue. They are the only valuations among those we are considering which make 𝒜false and ℬtrue. So they are the only valuations among those we are considering which make both of ¬𝒜and ℬtrue. So we can construct a schematic sentence which is true on valuations in that family, and only those valuations: the conjunction of ¬𝒜and ℬ,(¬𝒜∧ℬ). Our target sentence, the one with the same truth table as ‘Exactly one of 𝒜and ℬ’, is true on either the second or third valuations. So it is true if either (𝒜∧¬ℬ)is true or if (¬𝒜∧ℬ)is true. And there is of course a schematic Sentential sentence with just this profile: (𝒜∧¬ℬ)∨(¬𝒜∧ℬ). Let us summarise this construction by adding to our truth table: 𝒜 ℬ ‘Exactly one of 𝒜and ℬ’(𝒜∧¬ℬ)∨(¬𝒜∧ℬ) T T FFFF T F TTTF F T TFTT F F FFFF As we can see, we have come up with a schematic Sentential sentence with the intended truth table. 14.3 The Disjunctive Normal Form Procedure The procedure sketched above can be generalised: 1. First, identify the truth table of the target connective; 2. Then, identify which families of valuations (schematic truth table rows) the target sentence is true on, and for each such row, construct a conjunctive schematic Sentential sentence true on that row alone. (It will be a conjunction of schematic letters sentences of those schematic letters which are true on the valuation, and negated schematic letters, for those schematic letters false on the valuation). ›What if the target connective is true on no valuations? Then let the schematic Sentential sentence (𝒜∧¬𝒜)represent it – it too is true on no valuation. 3. Finally, the schematic Sentential sentence will be a disjunction of those conjunctions, be‑ cause the target sentence is true according to any of those valuations. §14. EXPRESSIVENESS OF Sentential 105 ›What if there is only one such conjunction, because the target sentence is true in only one valuation? Then just take that conjunction to be the Sentential rendering of the target sentence. Logicians say that the schematic sentences that this procedure spits out are in DISJUNCTIVE NOR‑ MAL FORM. This procedure doesn’t always give the simplest schematic Sentential sentence with a given truth table, but for any truth table you like this procedure gives us a schematic Sentential sentence with that truth table. Indeed, we can see that the Sentential sentence ¬(𝒜 ↔ ℬ)has the same truth table as our target sentence too. The procedure can be used to show that there is some redundancy in Sentential itself. Take the connective ↔. Our procedure, applied to the schematic truth table for 𝒜 ↔ ℬ, yields the following schematic sentence: (𝒜∧ℬ)∨(¬𝒜∧¬ℬ). This schematic sentence says the same thing as the original schematic sentence with the bicon‑ ditional as its main connective, without using the biconditional. This could be used as the basis of a program to remove the biconditional from the language. But that would make Sentential more difficult to use, and we will not pursue this idea further. Key Ideas in §14 ›There are truth‑functional connectives, such as ‘neither … nor …’, which don’t correspond to any of the official connectives of Sentential. ›Nevertheless for any sentence structure 𝒜 ⊕ ℬ, where ‘⊕’ is a truth‑ functional connective, it is possible to construct a Sentential schematic sentence which has the same truth table. (Indeed, one can do this making use only of negation, conjunction, and disjunction.) ›So Sentential is in fact able to express any truth‑functional connective. Practice exercises A. For each of columns (i), (ii) and (iii) below, use the procedure just outlined to find a Sentential sentence that has the truth table depicted: 𝒜 ℬ (i) (ii) (iii) T T T T F T F T T T F T T F F F F T T T B. Can you find Sentential schematic sentences which have the same truth table as these English connectives? 106 TRUTH TABLES 1. ‘𝒜whether or not ℬ’; 2. ‘Not both 𝒜and ℬ’; 3. ‘Neither 𝒜nor ℬ, but at least one of them’; 4. ‘If 𝒜then ℬ, else 𝒞’. Chapter 4 The Language of Quantified Logic 15 Building Blocks of Quantifier 15.1 The Need to Decompose Sentences Consider the following argument, which is obviously valid in English: Willard is a logician. All logicians wear funny hats. So Willard wears a funny hat. To symbolise it in Sentential, we might offer a symbolisation key: 𝐿: Willard is a logician. 𝐴: All logicians wear funny hats. 𝐹: Willard wears a funny hat. And the argument itself becomes: 𝐿,𝐴∴𝐹 This is invalid in Sentential – there is a valuation on which the premises are true and the conclu‑ sion false. But the original English argument is clearly valid. The problem is not that we have made a mistake while symbolising the argument. This is the best symbolisation we can give in Sentential. The problem lies with Sentential itself. ‘All lo‑ gicians wear funny hats’ is about both logicians and hat‑wearing. By not retaining this in our symbolisation, we lose the connection between Willard’s being a logician and Willard’s wearing a hat. Another example. This argument is also intuitively valid: John loves James; So: John loves someone. Again, the best Sentential symbolisation is the invalid 𝑃 ∴ 𝑄. The validity of this argument depends on the internal structure of the sentences, and specifically the connection between the name ‘James’ and the phrase ‘someone’. 108 §15. BUILDING BLOCKS OF Quantifier 109 The basic units of Sentential are atomic sentences, and Sentential cannot decompose these. None of the sentences in the arguments above have any truth‑functional connectives, so must be symbolised as atomic sentences of Sentential. To symbolise arguments like the preceding, we will have to develop a new logical language which will allow us to split the atom. We will call this language Quantifier, and the study of this language and its features is quantified logic. The details of Quantifier will be explained throughout this chapter, but here is the basic idea about how to split the atom(ic sentence). The key insight is that many natural language sen‑ tences have subject‑predicate structure, and some arguments are valid in virtue of this structure. Quantifier adds to Sentential some resources for modelling this structure – or perhaps more ac‑ curately, it allows us to model the predicate‑name structure of many sentences, along with any truth‑functional structure. Names First, we have names. In Quantifier, we indicate these with lower case italic letters. For instance, we might let ‘𝑏’ stand for Bertie, or let ‘𝑖’ stand for Willard. The names of Quantifier correspond to proper names in English, like ‘Willard’ or ‘Elyse’, which also stand for the things they name. Predicates Second, we have predicates. English predicates are expressions like ‘⎵is a dog’ or ‘⎵is a logician’. These are not complete sentences by themselves. In order to make a complete sentence, we need to fill in the gap. We need to say something like ‘Bertie is a dog’ or ‘Willard is a logician’. In Quantifier, we indicate predicates with upper case italic letters. For instance, we might let the Quantifier predicate ‘𝐷’ symbolise the English predicate ‘⎵is a dog’. Then the expression ‘𝐷𝑏’ will be a sentence in Quantifier, which symbolises the English sentence ‘Bertie is a dog’. Equally, we might let the Quantifier predicate ‘𝐿’ symbolise the Eng‑ lish predicate ‘⎵is a logician’. Then the expression ‘𝐿𝑖’ will symbolise the English sentence ‘Willard is a logician’. Quantifiers Third, we have quantifier phrases. These tell us how much. In English there are lots of quantifier phrases. But in Quantifier we will focus on just two: ‘all’/‘every’ and ‘there is at least one’/‘some’. So we might symbolise the English sentence ‘there is a dog’ with the Quantifier sentence ‘∃𝑥𝐷𝑥’, which we would naturally read aloud as ‘there is at least one thing, 𝑥, such that 𝑥is a dog’. That is the general idea. But Quantifier is significantly more subtle than Sentential. So we will come at it slowly. 15.2 Singular Terms In English, a SINGULAR TERM is a noun phrase that refers to a specific person, place, or thing. The word ‘dog’ is not a singular term, because there are a great many dogs. The phrase ‘Bertie’ is a singular term, because it refers to a specific terrier. Here are some further examples: ›Proper Names, e.g., ‘Bertie enjoys playing fetch’; ‘Scott Morrison breached the human rights of children seeking asylum’; 110 THE LANGUAGE OF QUANTIFIED LOGIC ›Definite Descriptions, e.g., ‘The oldest person is a woman’; ‘Ortcutt is the shortest spy’; ›Possessives, e.g., ‘Antony’s eldest child loves coding’; ›Pronouns, e.g., ‘She (points) plays violin’; ‘Icecream is delicious. Everyone loves it’; ›Demonstratives, e.g., ‘That loud dog is so annoying’. The clearest cases of singular terms are proper names, and these occupy a distinct syntactic cat‑ egory in Quantifier. Most of the other types of English singular terms are modelled in more or less indirect ways in Quantifier. Definites and possessives are handled principally by paraphrase. We treat possessives as dis‑ guised definite descriptions, paraphrasing ‘Antony’s eldest child’ as something like ‘the eldest child of Antony’. In turn, definites are handled as complex constructions in Quantifier, as we’ll see when we take them up in §19. Even trickier in some ways are singular term uses of pronouns and demonstratives. Both of these constructions rely on the CONVERSATIONAL CONTEXT to fix a determinate reference. I might need to gesture, or rely on our previous utterances, to understand who ‘she’ refers to in ‘she plays violin’. Likewise, ‘that loud dog’ might vary in which dog it refers to from conversation to conversation. There are approaches that attempt to model the role of context in determining the meaning of an expression, but we will not attempt to do this here. Quantifier is not designed to model every aspect of English. If we are forced to try to represent some English sentences involving context‑ sensitive singular terms in Quantifier, we will need to resort to paraphrases that are not fully adequate (e.g., treating demonstratives as definite descriptions, despite their differences). Moreover, pronouns are not singular terms in every use. Compare the uses of the pronoun ‘she’ in ‘She plays violin’ and ‘Every girl thinks she deserves icecream’. The first refers to a specific individual, perhaps with some contextual cues like gestures to help identify which. The second, however, doesn’t refer to a specific girl – rather, it ranges over all girls. Perhaps surprisingly, Quantifier takes this second kind of use of pronouns to be primary. We will discuss how Quantifier represents pronouns when we introduce variables in §15.5. 15.3 Names PROPER NAMES are a particularly important kind of singular term. These are expressions that label individuals without describing them. The name ‘Emerson’ is a proper name, and the name alone does not tell you anything about Emerson. Of course, some names are traditionally given to boys and other are traditionally given to girls. If ‘Hilary’ is used as a singular term, you might guess that it refers to a woman. You might, though, be guessing wrongly. Indeed, the name does not necessarily mean that what is referred to is even a person: Hilary might be a giraffe, for all you could tell just from the name ‘Hilary’. In English, the use of certain names triggers our knowledge of these conventions, so that (for example) the use of name ‘Fido’ might well trigger an expectation that the thing named is a dog. However, while it would violate convention to name your child ‘Fido’, once the you manage to assign the name it can perfectly well refer to a human child. §15. BUILDING BLOCKS OF Quantifier 111 In Quantifier, there are no conventions around its category of NAMES. These are pure names, whose only role is to designate some specific individual. Names in Quantifier are represented by lower‑case letters ‘𝑎’ through to ‘𝑟’. We can add subscripts if we want to use some letter more than once, if we have a complicated discourse with many different names. So here are some names in Quantifier:𝑎,𝑏,𝑐,…,𝑟,𝑎1,𝑓32,𝑗390,𝑚12. These should be thought of along the lines of proper names in English. But with some differ‑ ences. First, ‘Antony Eagle’ is a proper name, but there are a number of people with this name. (Equally, there are at least two people with the name ‘P.D. Magnus’ and several people named ‘Tim Button’.) We live with this kind of ambiguity in English, allowing context to determine that some particular utterance of the name ‘Antony Eagle’ refers to one of the contributors to this book, and not some other guy. In Quantifier, we do not tolerate any such ambiguity. Each name must pick out exactly one thing. (However, two different names may pick out the same thing.) Second, names (and predicates) in Quantifier are assigned their meaning, or interpreted, only temporarily. (This is just like the way that atomic sentences are only assigned a truth value in Sentential temporarily, relative to a valuation.) As with Sentential, we provide symbolisation keys. These indicate, temporarily, what a name shall pick out. So we might offer: 𝑒: Elsa 𝑔: Gregor 𝑚: Marybeth Again, what we are saying in using natural language names is that the thing referred to by the Quantifier name ‘𝑒’ is stipulated – for now – to be the same thing referred to by the English proper noun ‘Elsa’. We are not saying that ‘𝑒’ and ‘Elsa’ are synonyms. For all we know, perhaps there is additional nuance to the meaning of the name ‘Elsa’ other than what it refers to. If there is, it is not preserved in the Quantifier name ‘𝑒’, which has no nuance in its meaning other than the thing it denotes. The Quantifier name ‘𝑒’ might be stipulated to denote Elsa on one occasion, and Eddie on another. You may have been taught in school that a noun is a ‘naming word’. It is safe to use names in Quantifier to symbolise proper nouns. But dealing with common nouns is a more subtle matter. Even though they are often described as ‘general names’, common nouns actually function as names relatively rarely. Some common nouns which do often function as names are NATURAL KIND TERMS like ‘gold’ and ‘tiger’. These are common nouns which really do name a kind of thing. Consider this example: Gold is scarce; Nothing scarce is cheap; So: Gold isn’t cheap. This argument is valid in virtue of its form. The form of the first premise has the phrase ‘⎵ is scarce’ being predicated of ‘gold’, in which case, ‘gold’ must be functioning as a name in this argument. But notice that ‘⎵is gold’ is a perfectly good predicate. So we cannot simply treat all natural kind terms as proper names of general kinds. 118 THE LANGUAGE OF QUANTIFIED LOGIC 2. Silver is useful in film photography; 3. Mary’s ring is silver; 4. Kurt and Alonzo lift a couch; 5. The biggest threat to life on earth is carbon dioxide. B. Identify the possible predicates that can be found by replacing singular terms with gaps in these sentences: 1. He dislikes Joel; 2. Andrew Leigh was a professor of economics at the ANU; 3. The professor of genomics was elected president of the Academy. C. Make use of this symbolisation key to symbolise the following sentences into Quantifer, com‑ menting on any difficulties: domain: cities and towns in South Australia 𝑎: Adelaide 𝑚: Mount Gambier 𝑇:⎵is a town 𝑈:⎵is ugly 𝐿:⎵is large 1. Adelaide is big and ugly. 2. Mount Gambier is not large. 3. If Mount Gambier is large, then Adelaide definitely is! 4. Mount Gambier is a large village. 5. Every city and town is ugly. D. Can this argument be adequately symbolised in Quantifier? Comment on any difficulties. 1. She is tall; 2. Bob likes her; So: Bob likes someone tall. 16 Sentences with One Quantifier We now have the basic pieces of Quantifier. Symbolising many sentences of English will only be a matter of knowing the right way to combine predicates, names, quantifiers, and the truth‑ functional connectives. There is a knack to this, and there is no substitute for practice. 16.1 Common Quantifier Phrases As in Sentential (recall §5), we will give canonical symbolisations for certain common English quantificational structures. Consider these sentences: (101) Every coin in my pocket is a 20¢ piece. (102) Some coin on the table is a dollar. (103) Not all the coins on the table are dollars. (104) None of the coins in my pocket are dollars. In providing a symbolisation key, we need to specify a domain. Since we are talking about coins in my pocket and on the table, the domain must at least contain all of those coins. Since we are not talking about anything besides coins, we let the domain be all coins. Since we are not talking about any specific coins, we do not need to need to deal with any names. So here is our key: domain: all coins 𝑃:⎵is in my pocket 𝑇:⎵is on the table 𝑄:⎵is a 20¢ piece 𝐷:⎵is a dollar Sentence 101 is most naturally symbolised using a universal quantifier. The universal quantifier says something about everything in the domain, not just about the coins in my pocket. So if we want to talk just about coins in my pocket, we will need to restrict the quantifier, by imposing a condition on the things we are saying are 20¢ pieces. That is: something in the domain is 119 120 THE LANGUAGE OF QUANTIFIED LOGIC claimed to be a 20¢ piece only if it meets the restricting condition. That leads us to this conditional paraphrase: (105) For any (coin): if that coin is in my pocket ⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵ restriction ,then it is a 20¢ piece. So we can symbolise it as ‘∀𝑥(𝑃𝑥→𝑄𝑥)’. Since sentence 101 is about coins that are both in my pocket and that are 20¢ pieces, it might be tempting to translate it using a conjunction. However, the sentence ‘∀𝑥(𝑃𝑥 ∧ 𝑄𝑥)’ would symbolise the sentence ‘every coin is both a 20¢ piece and in my pocket’. This obviously means something very different than sentence 101. And so we see: A sentence can be symbolised as ∀𝑥(ℱ𝑥 → 𝒢𝑥) if it can be paraphrased in English as ‘every F is G’ or ‘all Fs are Gs’. Example 102 uses the quantifier phrase ‘some’. The same thought could be expressed using different quantifier phrases: (106) At least one coin on the table is a dollar. (107) There is a coin on the table that is a dollar. These phrases all indicate an existential quantifier. In these examples, the class of coins on the table is being related to the class of dollar coins, and it is claimed that at least one member of the former class is also in the latter class – that there is overlap. This is represented in Quantifier following the example of this paraphrase: (108) There is something (a coin): it is in both the class of things on the table, and in the class of dollar coins. That is symbolised using a conjunction, ‘∃𝑥(𝑇𝑥∧𝐷𝑥)’. We know from Sentential that the order of conjuncts doesn’t matter: ‘(𝑃∧𝑄)’ is logically equi‑ valent to ‘(𝑄∧𝑃)’. Likewise in Quantifier, ‘∃𝑥(𝑇𝑥∧𝐷𝑥)’ is logically equivalent to ‘∃𝑥(𝐷𝑥∧𝑇𝑥)’. This fits well with the English sentences we are symbolising, because overlap is itself a symmet‑ rical relation between classes. We see this also in the fact that we can paraphrase example 102 as ‘Some dollar coin is on the table’. Notice that we needed to use a conditional with the universal quantifier, but we used a conjunc‑ tion with the existential quantifier. Suppose we had instead written ‘∃𝑥(𝑇𝑥→𝐷𝑥)’. That would mean that there is some object in the domain such that if it is 𝑇, then it is also 𝐷. For this to be true, we just need something to not be 𝑇. So it is very easy for ‘∃𝑥(𝑇𝑥 → 𝐷𝑥)’ to be true. Given our symbolisation, it will be true if some coin is not on the table. Of course there is a coin that is not on the table: there are coins lots of other places. That is rather less demanding than the claim that something is both 𝑇and 𝐷. A conditional will usually be the natural connective to use with a universal quantifier, but a conditional within the scope of an existential quantifier tends to say something very weak indeed. As a general rule of thumb, do not put conditionals in the scope of existential quantifiers unless you are sure that you need one. §16. SENTENCES WITH ONE QUANTIFIER 121 A sentence can be symbolised as ∃𝑥(ℱ𝑥∧𝒢𝑥)if it can be paraphrased in English as ‘some F is G’, ‘at least one F is G’, or ‘There is an F which is G’. Sentence 103 can be paraphrased as, ‘It is not the case that every coin on the table is a dollar’. So we can symbolise it by ‘¬∀𝑥(𝑇𝑥 → 𝐷𝑥)’. You might look at sentence 103 and paraphrase it instead as, ‘Some coin on the table is not a dollar’. You would then symbolise it by ‘∃𝑥(𝑇𝑥∧¬𝐷𝑥)’. Although it is probably not immediately obvious yet, these two sentences are logically equivalent. (This is due to the logical equivalence between ¬∀𝑥𝒜and ∃𝑥¬𝒜, mentioned in §15, along with the logical equivalence between ¬(𝒜→ℬ)and 𝒜∧¬ℬ.) Sentence 104 can be paraphrased as, ‘It is not the case that there is some dollar in my pocket’. This can be symbolised by ‘¬∃𝑥(𝑃𝑥∧𝐷𝑥)’. It might also be paraphrased as, ‘Everything in my pocket is a nondollar’, and then could be symbolised by ‘∀𝑥(𝑃𝑥→¬𝐷𝑥)’. Again the two symbolisations are logically equivalent. Both are correct symbolisations of sentence 104. 16.2 Empty Predicates In §15, I emphasised that a name must pick out exactly one object in the domain. However, a predicate need not apply to anything in the domain. A predicate that applies to nothing in a domain is called an EMPTY predicate (relative to that domain). This is worth exploring. Suppose we want to symbolise these two sentences: (109) Every monkey knows sign language (110) Some monkey knows sign language It is possible to write the symbolisation key for these sentences in this way: domain: animals 𝑀:⎵is a monkey. 𝑆:⎵knows sign language. Sentence 109 can now be symbolised by ‘∀𝑥(𝑀𝑥 → 𝑆𝑥)’. Sentence 110 can be symbolised as ‘∃𝑥(𝑀𝑥∧𝑆𝑥)’. It is tempting to say that sentence 109 entails sentence 110. That is, we might think that it is impossible for it to be the case that every monkey knows sign language, without it’s also being the case that some monkey knows sign language. But this would be a mistake. It is possible for the sentence ‘∀𝑥(𝑀𝑥→𝑆𝑥)’ to be true even though the sentence ‘∃𝑥(𝑀𝑥∧𝑆𝑥)’ is false. How can this be? The answer comes from considering whether these sentences would be true or false if there are no monkeys. If there are no monkeys at all (in some domain), then ‘∀𝑥(𝑀𝑥 → 𝑆𝑥)’ would be vacuously true. Take the domain of reptiles. Look at the domain, and pick any monkey you like – it knows sign language!1There is certainly no counterexample to the claim 1Remember this is not a counterfactual claim (8.6); even if ‘All monkeys know sign language’ is vacuously true in the domain of reptiles, that wouldn’t mean that ‘Any monkey would know sign language’ is true. 122 THE LANGUAGE OF QUANTIFIED LOGIC available in this domain. And because of the role of the conditional in our symbolisation, it turns out that a universally quantified claim with an unsatisfied restricting condition will also be true. In Quantifier, a universally quantified sentence of the form ∀𝓍(𝒜𝓍 → ℬ𝓍)is false only if we can find something which is 𝒜without being ℬ. If we can’t find such a thing, perhaps because we can’t find anything which is 𝒜in the first place, then the sentence will be true (since truth is just lack of falsity, and this sentence isn’t false because we can’t find a case that falsifies it). This derives ultimately from the feature of Sentential we have already acknowledged to be questionably analogous to English, namely, the fact that a conditional is false only if there is a counterexample, a case where the antecedent is true and the consequent false. Another example will help to bring this home. Suppose we extend the above symbolisation key, by adding: 𝑅:⎵is a refrigerator Now consider the sentence ‘∀𝑥(𝑅𝑥 → 𝑀𝑥)’. This symbolises ‘every refrigerator is a monkey’. And this sentence is true, given our symbolisation key. This is counterintuitive, since we do not want to say that there are a whole bunch of refrigerator monkeys. It is important to remember, though, that ‘∀𝑥(𝑅𝑥 → 𝑀𝑥)’ is true iff any member of the domain that is a refrigerator is a monkey. Since the domain is animals, there are no refrigerators in the domain. Again, then, the sentence is vacuously true. If you were actually dealing with the sentence ‘All refrigerators are monkeys’, then you would most likely want to include kitchen appliances in the domain. Then the predicate ‘𝑅’ would not be empty and the sentence ‘∀𝑥(𝑅𝑥→𝑀𝑥)’ would be false. Remember, though, that a predicate is empty only relative to a particular domain. When ℱis an empty predicate relative to a given domain, a sentence ∀𝑥(ℱ𝑥→ …)will be vacuously true of that domain. 16.3 Picking a Domain The appropriate symbolisation of an English language sentence in Quantifier will depend on the symbolisation key. Choosing a key can be difficult. Suppose we want to symbolise the English sentence: (111) Every rose has a thorn. We might offer this symbolisation key: 𝑅:⎵is a rose 𝑇:⎵has a thorn It is tempting to say that sentence 111 should be symbolised as ‘∀𝑥(𝑅𝑥→𝑇𝑥)’. But we have not yet chosen a domain. If the domain contains all roses, this would be a good symbolisation. Yet §16. SENTENCES WITH ONE QUANTIFIER 123 if the domain is merely things on my kitchen table, then ‘∀𝑥(𝑅𝑥→𝑇𝑥)’ would only come close to covering the fact that every rose on my kitchen table has a thorn. If there are no roses on my kitchen table, the sentence would be trivially true. This is not what we want. To symbolise sentence 111 adequately, we need to include all the roses in the domain. But now we have two options. First, we can restrict the domain to include all roses but only roses. Then sentence 111 can, if we like, be symbolised with ‘∀𝑥𝑇𝑥’. This is true iff everything in the domain has a thorn; since the domain is just the roses, this is true iff every rose has a thorn. By restricting the domain, we have been able to symbolise our English sentence with a very short sentence of Quantifier. So this approach can save us trouble, if every sentence that we want to deal with is about roses. Second, we can let the domain contain things besides roses: rhododendrons; rats; rifles; whatevers. And we will certainly need to include a more expansive domain if we simultaneously want to symbolise sentences like: (112) Every cowboy sings a sad, sad song. Our domain must now include both all the roses (so that we can symbolise sentence 111) and all the cowboys (so that we can symbolise sentence 112). So we might offer the following symbol‑ isation key: domain: people and plants 𝐶:⎵is a cowboy 𝑆:⎵sings a sad, sad song 𝑅:⎵is a rose 𝑇:⎵has a thorn Now we will have to symbolise sentence 111 with ‘∀𝑥(𝑅𝑥→𝑇𝑥)’, since ‘∀𝑥𝑇𝑥’ would symbolise the sentence ‘every person or plant has a thorn’. Similarly, we will have to symbolise sentence 112 with ‘∀𝑥(𝐶𝑥→𝑆𝑥)’. In general, the universal quantifier can be used to symbolise the English expression ‘everyone’ if the domain only contains people. If there are people and other things in the domain, then ‘everyone’ must be treated as ‘every person’. If you choose a narrow domain, you can make the task of symbolisation easier. If we are at‑ tempting to symbolise example 96, ‘every girl thinks that she deserves icecream’, we can pick the domain to be girls and then we only need to introduce a predicate ‘⎵thinks that they themselves deserve icecream’, symbolised ‘𝐷’. The symbolisation is then the simple ‘∀𝑥𝐷𝑥’. But our options are limited if the conversation goes on to talk about things other than girls. On the other hand, if you pick an expansive domain (such as everything whatsoever), you can always just impose an appropriate restriction. In this case, we could introduce the predicate ‘𝐺’ to stand for ‘⎵is a girl’, and symbolise the sentence as ‘∀𝑥(𝐺𝑥→𝐷𝑥)’. When choosing an expansive domain, you must take some care with implicitly restricted predic‑ ates. It would not be appropriate to paraphrase ‘all the beer has been drunk’ relative to a very expansive domain as ‘For everything: if it is beer, it has been drunk’. To adequately capture the intent, we shall need to make the implicit contextual restriction of the predicate explicit, in something like this paraphrase ‘For everything: if it is beer at the party, it has been drunk’. 124 THE LANGUAGE OF QUANTIFIED LOGIC 16.4 The Utility of Paraphrase When symbolising English sentences in Quantifier, it is important to understand the structure of the sentences you want to symbolise. What matters is the final symbolisation in Quantifier, and sometimes you will be able to move from an English language sentence directly to a sentence of Quantifier. Other times, it helps to paraphrase the sentence one or more times. Each success‑ ive paraphrase should move from the original sentence closer to something that you can finally symbolise directly in Quantifier. For the next several examples, we will use this symbolisation key: domain: women 𝐵:⎵is a bassist. 𝑅:⎵is a rock star. 𝑘: Kim Deal Now consider these sentences: (113) If Kim Deal is a bassist, then she is a rock star. (114) If any woman is a bassist, then she is a rock star. The same words appear as the consequent in sentences 113 and 114 (‘…she is a rock star’), but they mean very different things (recall §15.5). To make this clear, it often helps to paraphrase the original sentences into a more unusual but clearer form. Sentence 113 can be paraphrased as, ‘Consider Kim Deal: if she is a bassist, then she is a rockstar’. The bare pronoun ‘she’ gets to denote Kim Deal because of our initial ‘Consider Kim Deal’ remark. This then says something about one particular person, and can obviously be symbolised as ‘𝐵𝑘→ 𝑅𝑘’. Sentence 114 gets a very similar paraphrase, with the same embedded conditional: ‘Consider any woman: if she is a bassist, then she is a rockstar’. The difference in the ‘Consider …’ phrase how‑ ever forces a very different intepretation for the sentence as a whole. Replacing the English pro‑ nouns by variables, the Quantifier equivalent of a pronoun, we get this awkward quasi‑English paraphrase: ‘For any woman x, if x is a bassist, then x is a rockstar’. Now this can be symbolised as ‘∀𝑥(𝐵𝑥 → 𝑅𝑥)’. This is the same sentence we would have used to symbolise ‘Every woman who is a bassist is a rock star’. And on reflection, that is surely true iff sentence 114 is true, as we would hope. Consider these further sentences, and let us consider the same interpretation as above, though in a domain of all people. (115) If anyone is a bassist, then Kim Deal is a rock star. (116) If anyone is a bassist, then they are a rock star. The same words appear as the antecedent in sentences 115 and 116 (‘If anyone is a bassist…’). But it can be tricky to work out how to symbolise these two uses. Again, paraphrase will come to our aid. §16. SENTENCES WITH ONE QUANTIFIER 125 Sentence 115 can be paraphrased, ‘If there is at least one bassist, then Kim Deal is a rock star’. It is now clear that this is a conditional whose antecedent is a quantified expression; so we can symbolise the entire sentence with a conditional as the main connective: ‘∃𝑥𝐵𝑥→𝑅𝑘’. Sentence 116 can be paraphrased, ‘For all people x, if x is a bassist, then x is a rock star’. Or, in more natural English, it can be paraphrased by ‘All bassists are rock stars’. It is best symbolised as ‘∀𝑥(𝐵𝑥→𝑅𝑥)’, just like sentence 114. The word ‘any’ is particularly tricky, because it can sometimes mean ‘every’ and sometimes ‘at least one’! Think about the two occurrences of ‘any’ in this sentence: (117) Any student will be happy if they have any money. For every student: if there exists some money they possess, then they will be happy. This can be symbolised ‘∀𝑥(𝑆𝑥→(∃𝑦(𝑀𝑦∧𝑃𝑥𝑦)→𝐻𝑥))’, where the first occurrence of ‘any’ is represented by a universal quantifier, and the second by an existential quantifier. The moral is that the English words ‘any’ and ‘anyone’ should typically be symbolised using quantifiers. And if you are having a hard time determining whether to use an existential or a universal quantifier, try paraphrasing the sentence with an English sentence that uses words besides ‘any’ or ‘anyone’.2 16.5 Quantifiers and Scope Continuing the example, suppose I want to symbolise these sentences: (118) If everyone is a bassist, then Tim is a bassist (119) Everyone is such that, if they are a bassist, then Tim is a bassist. To symbolise these sentences, I shall have to add a new name to the symbolisation key, namely: 𝑏: Tim Sentence 118 is a conditional, whose antecedent is ‘everyone is a bassist’. So we will symbolise it with ‘(∀𝑥𝐵𝑥 → 𝐵𝑏)’. This sentence is necessarily true: if everyone is indeed a bassist, then take any one you like – for example Tim – and he will be a bassist. 2The story about ‘any’ and ‘anyone’ is actually rather interesting. It is well‑known to linguists that ‘any’ has at least two readings: so‑called FREE CHOICE ‘ANY’, which is more or less like a universal quantifier (‘Any friend of Jessica’s is a friend of mine!’), and NEGATIVE POLARITY ₍NPI₎ ‘ANY’, which only occurs in ‘negative’ contexts, like negation (‘I don’t want any peas!’), where it functions more or less like an existential quantifier (‘It is not the case that: there exist peas that I want’). Interestingly, the antecedent of a conditional is a negative environment (being equivalent to ¬𝒜∨𝒞), and so we expect that ‘any’ in the antecedent of a conditional will have an existential interpretation. And it does: ‘If anyone is home, they will answer the door’ means something like: ‘If someone is home, then that person will answer the door’. It does not mean ‘If everyone is home, then they will answer the door’. This is what we see in 115. But 116 is not a conditional – its main connective is a quantifier. So here free choice ‘any’ is the natural interpretation, so we use the universal quantifier. We see the same thing with the quantifier ‘someone’. In ‘if someone is a bassist, Kim Deal is’, someone gets symbol‑ ised by an existential quantifier in the scope of a conditional. But in ‘If someone is a bassist, they are a musician’ it should be symbolised by a universal taking scope over a conditional. 126 THE LANGUAGE OF QUANTIFIED LOGIC Sentence 119, by contrast, might best be paraphrased by ‘every person x is such that, if x is a bassist, then Tim is a bassist’. This is symbolised by ‘∀𝑥(𝐵𝑥 → 𝐵𝑏)’. And this sentence is false. Kim Deal is a bassist. So ‘𝐵𝑘’ is true. But Tim is not a bassist, so ‘𝐵𝑏’ is false. Accordingly, ‘𝐵𝑘→𝐵𝑏’ will be false. So ‘∀𝑥(𝐵𝑥→𝐵𝑏)’ will be false as well. In short, ‘(∀𝑥𝐵𝑥 → 𝐵𝑏)’ and ‘∀𝑥(𝐵𝑥 → 𝐵𝑏)’ are very different sentences. We can explain the difference in terms of the scope of the quantifier. The scope of quantification is very much like the scope of negation, which we considered when discussing Sentential (§6.3), and it will help to explain it in this way. We define quantifier scope officially in §20, but we also return to it in a preliminary way in §17.2. In the sentence ‘(¬𝐵𝑘 → 𝐵𝑏)’, the scope of ‘¬’ is just the antecedent of the conditional. We are saying something like: if ‘𝐵𝑘’ is false, then ‘𝐵𝑏’ is true. Similarly, in the sentence ‘(∀𝑥𝐵𝑥→𝐵𝑏)’, the scope of ‘∀𝑥’ is just the antecedent of the conditional. We are saying something like: if ‘𝐵’ is true of everything, then ‘𝐵𝑏’ is also true. In the sentence ‘¬(𝐵𝑘 → 𝐵𝑏)’, the scope of ‘¬’ is the entire sentence. We are saying something like: ‘(𝐵𝑘→𝐵𝑏)’ is false. Similarly, in the sentence ‘∀𝑥(𝐵𝑥→𝐵𝑏)’, the scope of ‘∀𝑥’ is the entire sentence. We are saying something like: ‘(𝐵𝑥→𝐵𝑏)’ is true of everything. Scope can make a drastic difference in meaning. Reconsider these examples: (120) (Scope of ‘∀𝑥’ ⎴ ∀𝑥𝐵𝑥 →𝐵𝑏) If everything is 𝐵, then 𝑏is too. Trivially true (121) Scope of ‘∀𝑥’ ⎴⎴⎴⎴⎴⎴⎴⎴⎴ ∀𝑥(𝐵𝑥→𝐵𝑏) All 𝐵s are such that 𝑏is 𝐵.False if 𝑏isn’t 𝐵but there are some 𝐵s The moral of the story is simple. When you are using quantifiers and conditionals, be very careful to make sure that you have sorted out the scope correctly. 16.6 Dealing with Complex Adjectives When we encounter a sentence like (122) Herbie is a white car, we can paraphrase this as ‘Herbie is white and Herbie is a car’. We can then use a symbolisation key like: 𝑊:⎵is white 𝐶:⎵is a car ℎ: Herbie This allows us to symbolise sentence 122 as ‘𝑊ℎ∧𝐶ℎ’. But now consider: (123) Julia Gillard is a former prime minister. (124) Julia Gillard is prime minister. §16. SENTENCES WITH ONE QUANTIFIER 127 Following the case of Herbie, we might try to use a symbolisation key like: 𝐹:⎵is former 𝑃:⎵is Prime Minister 𝑗: Julia Gillard. Then we would symbolise 123 by ‘𝐹𝑗∧𝑃𝑗’, and symbolise 124 by ‘𝑃𝑗’. That would however be a mistake, since that symbolisation suggests that the argument from 123 to 124 is valid, because the symbolisation of the premise does logically entail the symbolisation of the conclusion. ‘White’ is a INTERSECTIVE adjective, which is a fancy way of saying that the white 𝐹s are among the 𝐹sand among the white things: any white car is a car and white, just like any successful lawyer is a lawyer and successful, and a one tonne rhinoceros is both a rhino and a one tonne thing. But ‘former’ is a PRIVATIVE adjective, which means that any former 𝐹is not now among the 𝐹s. Other privative adjectives occur in phrases such as ‘fake diamond’, ‘Deputy Lord Mayor’, and ‘mock trial’. When symbolising these sentences, you cannot treat them as a conjunction. So you will need to symbolise ‘⎵is a fake diamond’ and ‘⎵is a diamond’ using completely different predicates, to avoid a spurious entailment between them. The moral is: when you see an adjectivally modified predicate like ‘white car’, you need to ask yourself carefully whether the modifier is intersective, and can be symbolised as a conjunctive predicate, or not.3 Things are a bit more complicated, however. Recall this example from page 86: Daisy is a small cow. We note that a small cow is definitely a cow, and so it seems we might treat ‘small’ as an intersect‑ ive adjective. We might formalise this sentence like this: ‘𝑆𝑑∧𝐶𝑑’, assuming this symbolisation key: 𝑆:⎵is small 𝐶:⎵is a cow 𝑑: Daisy But note that our symbolisation would suggest that this argument is valid: (125) Daisy is a small cow; so Daisy is small. The symbolised argument, 𝑆𝑑∧𝐶𝑑∴𝑆𝑑, is clearly valid. But the original argument 125 is not valid. Even a small cow is still rather large. (Likewise, even a short basketball player is still generally well above average height.) The point is that ‘⎵is a small cow’ denotes the property something has when it is small for a cow, while ‘⎵is small’ denotes the property of being a small thing. (In ordinary speech we tend to keep the ‘for an 𝐹’ 3This caution also applies to adjectives which are neither intersective nor privative, like ‘alleged’ in ‘alleged murderer’. These ought not be symbolised by conjunction either. 134 THE LANGUAGE OF QUANTIFIED LOGIC These are TWO‑PLACE PREDICATES. They need to be filled in with two terms (names or pronouns, most commonly) in order to make a sentence. Conversely, if we start with an English sentence containing many singular terms, we can remove two singular terms, to obtain different two‑place predicates. Consider the sentence ‘Vinnie borrowed the family car from Nunzio’. By deleting two singular terms, we can obtain any of three different two‑place predicates: Vinnie borrowed ⎵from ⎵; ⎵borrowed the family car from ⎵; ⎵borrowed ⎵from Nunzio. And by removing all three singular terms, we obtain a THREE‑PLACE PREDICATE: ⎵borrowed ⎵from ⎵. Indeed, there is no in principle upper limit on the number of gaps or places that our predicates may contain. Now there is a little problem with the above. I have used the same symbol, ‘⎵’, to indicate a gap formed by deleting a term from a sentence. However (as Frege emphasised), these are different gaps. To obtain a sentence, we can fill them in with the same term, but we can equally fill them in with different terms, and in various different orders. The following are all perfectly good sentences, obtained by filling in the gaps in ‘⎵loves ⎵’, but they mean quite different things: Karl loves Karl; Karl loves Imre; Imre loves Karl; Imre loves Imre. The point is that we need some way of keeping track of the gaps in predicates, so that we can keep track of how we are filling them in. Another way to put the point: when it comes to two‑(or more)‑place predicates, sometimes the order matters. ‘Shaq is taller than Jordan’ doesn’t mean the same thing as ‘Jordan is taller than Shaq’. It matters whose name fills the first gap in the predicate, and whose name fills the second. To keep track of the gaps, we shall label them. The labelling conventions I adopt are best ex‑ plained by example. Suppose I want to symbolise the following sentences: (128) Karl loves Imre. (129) Imre loves himself. (130) Karl loves Imre, but not vice versa. (131) Karl is loved by Imre. I will start with the following symbolisation key: §17. MULTIPLE GENERALITY 135 domain: people 𝑖: Imre 𝑘: Karl 𝐿:⎵ 1loves ⎵ 2 ›Sentence 128 will now be symbolised by ‘𝐿𝑘𝑖’. ›Sentence 129 can be paraphrased as ‘Imre loves Imre’. It can now be symbolised by ‘𝐿𝑖𝑖’. ›Sentence 130 is a conjunction. We might paraphrase it as ‘Karl loves Imre, and Imre does not love Karl’. It can now be symbolised by ‘𝐿𝑘𝑖∧¬𝐿𝑖𝑘’. ›Sentence 131 might be paraphrased by ‘Imre loves Karl’. It can then be symbolised by ‘𝐿𝑖𝑘’. Of course, this erases the difference in tone between the active and passive voice; such nuances are lost in Quantifier. This last example highlights something important. Suppose we add to our symbolisation key the following: 𝑀:⎵ 2loves ⎵ 1 Here, we have used the same English word (‘loves’) as we used in our symbolisation key for ‘𝐿’. However, we have swapped the order of the gaps around (just look closely at those little subscripts!) So ‘𝑀𝑘𝑖’ and ‘𝐿𝑖𝑘’ now both symbolise ‘Imre loves Karl’. ‘𝑀𝑖𝑘’ and ‘𝐿𝑘𝑖’ now both symbolise ‘Karl loves Imre’. Since love can be unrequited, these are very different claims. The moral is simple. When we are dealing with predicates with more than one place, we need to pay careful attention to the order of the places. With these examples in hand, I can now give the official account of how we understand Quantifier symbolisations. Suppose we have an Quantifier expression 𝒜𝑡1…𝑡𝑘, where each 𝑡𝑖is a name or a variable, symbolising a singular term, and where 𝒜symbolises a 𝑘‑place predicate. The 𝑖‑th term is to be interpreted as filling the gap labelled ‘𝑖’. So consider the following symbolisation key: domain: places 𝑎: Adelaide 𝑏: Alice Springs 𝑐: Coober Pedy 𝐵:⎵ 2is between ⎵ 1and ⎵ 3 𝐾:⎵ 1is between ⎵ 2and ⎵ 3 . Then if we want to symbolise ‘Coober Pedy is between Adelaide and Alice Springs’, we can do so using either ‘𝐵𝑎𝑐𝑏’ or ‘𝐾𝑐𝑎𝑏’. The difference is in how the symbolisation key instructs us to fill the gaps we have established in the predicate as we take steps to represent it symbolically. There is no ‘right’ answer here: either can be good. The representation using ‘𝐵’ graphically represents which item is between the other two in the syntax itself, while the representation using ‘𝐾’ is more faithful to the original English. 136 THE LANGUAGE OF QUANTIFIED LOGIC Suppose we add to our symbolisation key the following: 𝑆:⎵ 1thinks only of ⎵ 1 𝑇:⎵ 1thinks only of ⎵ 2 𝑎: Alice As in the case of ‘𝐿’ and ‘𝑀’ above, the difference between these examples is only in how the gaps in the construction ‘… thinks only of …’ are labelled. In ‘𝑇’, we have labelled the two gaps differently. They do not need to be filled with different names or variables, but there is always the potential to put different names in those different gaps. In the case of ‘𝑆’, the gaps have the same label. In some sense, there is only one gap in this sentence, which is why the symbolisation key associates it with a one‑place predicate – it means something like ‘𝑥thinks only of themself’. The second predicate is more flexible. Take something we can say with the predicate ‘𝑆’, such as ‘𝑆𝑎’, ‘Alice thinks only of herself’. We can express pretty much the same thought using the two‑place predicate ‘𝑇’: ‘𝑇𝑎𝑎’. We have introduced a potential ambiguity in our treatment of predicates. (See also §20.2.) There is nothing overt in our language that distinguishes the one‑place predicate ‘𝐴’ (such that ‘𝐴𝑏’ is grammatical) from the two‑place predicate ‘𝐴’ (such that ‘𝐴𝑏’ is ungrammatical, but ‘𝐴𝑏𝑘’ is grammatical). We are, in effect, just letting context disambiguate how many argument places there are in a given predicate, by assuming that in any expression of Quantifier we write down, the number of names or variables following a predicate indicates how many places it has. We could introduce a system to disambiguate: perhaps adding a superscripted ‘1’ to all one‑place predicates, a superscripted ‘2’ to all two‑place predicates, etc. Then ‘𝐴1𝑏’ is grammatical while ‘𝐴1𝑏𝑘’ is not; conversely, ‘𝐴2𝑏’ is ungrammatical and ‘𝐴2𝑏𝑘’ is grammatical. This system of su‑ perscripts would be effective but cumbersome. We will thus keep to our existing practice, letting context disambiguate. What you should not do, however, is make use of the same capital letter to symbolise two different predicates in the same symbolisation key. If you do that, context will not disambiguate, and you will have failed to give an interpretation of the language at all. 17.2 Scope and Nested Quantifiers Once we have two (or more) gaps in a predicate, we can fill them with different things. We’ve so far seen cases where multiple names are slotted into a many‑place predicate. But we can also insert other terms, like variables. So to continue our example using the predicate ‘𝑇’, ‘⎵ 1 thinks only of ⎵ 2’, we can put a variable in the first gap, and a name in the second, if we wish: ‘𝑇𝑥𝑎’. This isn’t a sentence, because no quantifer tells us how to understand that variable. (The sentence might be representing ‘they think only of Alice’, but without context there is no determinate referent for the pronoun ‘they’.) Introduce a quantifier, and we have an interpretable sentence: (132) ∀𝑥𝑇𝑥𝑎; Everyone thinks only of Alice. The fact that we can fill the two gaps of two‑place predicates with different things, or even with the same thing, gives us a reason to favour the two‑place predicate symbolisation of ‘Alice thinks §17. MULTIPLE GENERALITY 137 only of themself’ as ‘𝑇𝑎𝑎’. That allows us to symbolise certain arguments that cannot be ad‑ equately symbolised using a one‑place predicate. For example: ‘Alice thinks only of herself; so there is someone who is the only person Alice thinks of’. The symbolisation of this argument might be: ‘𝑇𝑎𝑎∴∃𝑥𝑇𝑎𝑥’. This might have some prospect of being valid, whereas ‘𝑆𝑎∴∃𝑥𝑇𝑎𝑥’ will not be valid. The real power of many‑place predicates comes when we consider examples in which both gaps in the predicate are filled by variables governed by different quantifiers. In cases where the quan‑ tifier expressions interact, we can express things we cannot say even when we allow logically complex combinations of one‑quantifier sentences. With this power comes potential confusion too. So let’s proceed carefully. Consider the sentence ‘everyone loves someone’. This illustrates our goal, as two quantifier ex‑ pressions occur in this sentence: ‘everyone’ and ‘someone’. But it also illustrates the potential pitfalls, as there is a possible ambiguity in this sentence. It might mean either of the following: (133) For every person, there is some person that they love (134) There is some particular person whom every person loves It is fairly straightforward to see that these don’t mean the same thing. The first would be true as long as everybody has somebody they love. One sort of case in which 133 is true is the cyclic central love triangle in Twelfth Night, where Viola loves Duke Orsino, the Duke loves Olivia, and Olivia loves Viola (who, disguised as a young man, is the Duke’s go‑between with Olivia). In the Twelfth Night situation, 134 is not true. It could only be true if everybody loves the same person, e.g., if the Duke, Viola, and Olivia herself all love Olivia. How can we symbolise these two different disambiguations of our original sentence? (Remem‑ ber: one of the strengths of symbolic logic is that it is supposed to be able to clearly represent that which would be ambiguous in natural language.) Let’s paraphrase a little more formally as we step towards a fully symbolic representation. As our sentence has two quantifiers, I will use numbers to link pronouns in our paraphrase with the quantifier expressions which govern them. Using this device, our sentences can be paraphrased as follows: (135) Everyone1is such that there is someone2such that: they1love them2. (136) There is someone2such that everyone1is such that: they1love them2. (Take a moment to convince yourself that these paraphrases succeed.) You can see immediately that the difference in these paraphrases lies in the order of the quan‑ tifier expressions, and the remainder of the paraphrase, ‘they1love them2’, is the same in each sentence. Using variables to symbolise pronouns, and choosing the variable ‘𝑥’ for ‘they1’ and ‘𝑦’ for ‘them2’, we can symbolise this ‘𝐿𝑥𝑦’, where ‘𝐿’ stands for the two‑place predicate ‘⎵ 1 loves ⎵ 2’. The quantifier order in the paraphrases governs how they interact. As we saw in §16.5, the scope of a quantifier is roughly the Quantifier expression in which that quantifier is the main connective. (Later on we will be a little more precise about the way that quantifier scope functions 138 THE LANGUAGE OF QUANTIFIED LOGIC in Quantifier: see §20.) So in ‘∀𝑥∃𝑦𝐿𝑥𝑦’, the scope of ‘∀𝑥’ is the whole sentence, while the scope of ‘∃𝑦’ is just ‘∃𝑦𝐿𝑥𝑦’. The following guides us in interpreting these ‘nested’ quantifiers, in which one falls in the scope of another: When one quantifier occurs in the scope of another, the narrower scope quan‑ tifier should be understood with respect to the value assigned to a variable by the wider scope quantifier. Let’s apply this to our example. In 135 ‘everyone’ comes first, and ‘someone’ comes next. The intended interpretation is that this is true iff for any person 𝑥that you pick, with respect to that choice you can then find someone 𝑦who 𝑥loves. If you had chosen someone else as the value of 𝑥, then parasitic on that different choice you may end up needing to find a different value for 𝑦. Compare the reversed quantifier scope in 136. That is true iff there is someone 𝑦such that, with respect to that particular choice for 𝑦, any person 𝑥you pick, 𝑥loves 𝑦. With respect to a different initial choice for 𝑦, there may be values for 𝑥that do not satisfy 𝑥loves 𝑦, but as the initial choice is governed by an existential quantifier, that won’t undermine the truth of the sentence. This gives us our two different symbolisations: ›Sentence 133 can be symbolised by ‘∀𝑥∃𝑦𝐿𝑥𝑦’. Return to our example love triangle between Duke Orsino, Viola, and Olivia. For any of the three people you might choose, you can find another person in the domain who they love. So sentence 133 is true. ›Sentence 134 is symbolised by ‘∃𝑦∀𝑥𝐿𝑥𝑦’. Sentence 134 is not true in the Twelfth Night situation. For each of the people in the domain, you can find someone who doesn’t love them, and hence no one is universally beloved. If, instead, each person loved Olivia, then we could find someone (Olivia), such that everyone else we examined turns out to love them. In that case, 134 would be true. This example, besides giving some indication of how to read sentences with multiple quantifiers, illustrates that quantifier scope matters a great deal. Indeed, the mistake that arises when one illegitimately switches them around even has a special name: a quantifier shift fallacy. Here is a real life example from Aristotle:2 Suppose, then, that [A] the things achievable by action have some end that we wish for because of itself, and because of which we wish for the other things, and that we do not choose everything because of something else – for if we do, it will go on without limit, so that desire will prove to be empty and futile[; c]learly, [B] this end will be the good, that is to say, the best good. (Aristotle, Nichomachean Ethics 1094a18‑22) Setting aside Aristotle’s subsidiary argument about desire, this argument seems to involve the following pattern of inference: 2Note that it is hotly contested whether Aristotle actually commits a fallacy here, given the compressed nature of his prose. See, inter alia, J L Ackrill (1999), ‘Aristotle on eudaimonia’, pp. 57–77 in N Sherman, ed., Aristotle’s Ethics: Critical Essays, Rowman & Littlefield. §17. MULTIPLE GENERALITY 139 Every action aims at some end which is desired because of itself. (∀∃) So: There is end desired because of itself which is the aim of every action, the best good. (∃∀) This argument form is obviously invalid. It’s just as bad as:3 Every dog has its day. (∀∃) So: There is a day for all the dogs. (∃∀) The moral is: take great care with the scope of quantification. 17.3 Stepping Stones to Symbolisation Once we have the possibility of multiple quantifiers and many‑place predicates, representation in Quantifier can quickly start to become a bit tricky. When you are trying to symbolise a complex sentence, I recommend laying down several stepping stones. As usual, this idea is best illustrated by example. Consider this representation key: domain: people and dogs 𝐷:⎵ 1is a dog 𝐹:⎵ 1is a friend of ⎵ 2 𝑂:⎵ 1owns ⎵ 2 𝑔: Geraldo And now let’s try to symbolise these sentences: (137) Geraldo is a dog owner. (138) Someone is a dog owner. (139) All of Geraldo’s friends are dog owners. (140) Every dog owner is the friend of a dog owner. (141) Every dog owner’s friend owns a dog of a friend. Sentence 137 can be paraphrased as, ‘There is a dog that Geraldo owns’. This can be symbolised by ‘∃𝑥(𝐷𝑥∧𝑂𝑔𝑥)’. Sentence 138 can be paraphrased as, ‘There is some y such that y is a dog owner’. Dealing with part of this, we might write ‘∃𝑦(𝑦is a dog owner)’. Now the fragment we have left as ‘𝑦is a dog owner’ is much like sentence 137, except that it is not specifically about Geraldo. (We chose the variable ‘𝑦’ with this in mind, to avoid a clash with the variable ‘𝑥’ in our symbolisation of 137 – see below.) So we can symbolise sentence 138 by: ∃𝑦∃𝑥(𝐷𝑥∧𝑂𝑦𝑥). I need to pause to clarify something here. In working out how to symbolise the last sentence, we wrote down ‘∃𝑦(𝑦is a dog owner)’. To be very clear: this is neither aQuantifier sentence nor an English sentence: it uses bits of Quantifier (‘∃’, ‘𝑦’) and bits of English (‘dog owner’). It is really 3Thanks to Rob Trueman for the example. 140 THE LANGUAGE OF QUANTIFIED LOGIC is just a stepping‑stone on the way to symbolising the entire English sentence with a Quantifier sentence, a bit of rough‑working‑out. Sentence 139 can be paraphrased as, ‘Everyone who is a friend of Geraldo is a dog owner’. Using our stepping‑stone tactic, we might write ∀𝑥(𝐹𝑥𝑔→𝑥is a dog owner) Now the fragment that we have left to deal with, ‘𝑥is a dog owner’, is structurally just like sen‑ tence 137. But it would be a mistake for us simply to put ‘𝑥’ in place of ‘𝑔’ from our symbolisation of 137, yielding ∀𝑥(𝐹𝑥𝑔→∃𝑥(𝐷𝑥∧𝑂𝑥𝑥)). Here we have a CLASH OF VARIABLES. The scope of the universal quantifier, ‘∀𝑥’, is the entire conditional. But ‘𝐷𝑥’ also falls within the scope of the existential quantifier ‘∃𝑥’. Which quantifier has priority and governs the interpretation of the variable? In Quantifier, if a variable 𝓍occurs in an Quantifier sentence, it is always governed by the quantifier which has the narrowest scope which includes that occurrence of 𝓍. So in the sentence above, the quantifier ‘∃𝑥’ governs every occurrence of ‘𝑥’ in ‘(𝐷𝑥∧𝑂𝑥𝑥)’. Given this, the symbolisation does not mean what we intended. It says, roughly, ‘everyone who is a friend of Geraldo is such that there is a self‑owning dog’. This is not at all the meaning of the English sentence we are aiming to symbolise. To provide an adequate symbolisation, then, we must avoid clashing variables. We can do this easily enough. There was no requirement to use ‘𝑥’ as the variable in our symbolisation of 137, so we can easily choose some different variable for our existential quantifier. That will give us something like this, which adequately symbolises sentence 139: ∀𝑥(𝐹𝑥𝑔→∃𝑧(𝐷𝑧∧𝑂𝑥𝑧)). Sentence 140 can be paraphrased as ‘For any x that is a dog owner, there is a dog owner who is a friend of x’. Using our stepping‑stone tactic, this becomes ∀𝑥(𝑥is a dog owner →∃𝑦(𝑦is a dog owner ∧𝐹𝑦𝑥)) Completing the symbolisation, we end up with ∀𝑥(∃𝑧(𝐷𝑧∧𝑂𝑥𝑧)→∃𝑦(∃𝑧(𝐷𝑧∧𝑂𝑦𝑧)∧𝐹𝑦𝑥)) Note that we have used the same variable, ‘𝑧’, in both the antecedent and the consequent of the conditional, but that these are governed by two different quantifiers. This is ok: there is no potential confusion here, because it is obvious which quantifier governs each variable. We might graphically represent the scope of the quantifiers thus: scope of ‘∀𝑥’ ⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴ ∀𝑥(scope of 1st ‘∃𝑧’ ⎴⎴⎴⎴⎴⎴⎴⎴⎴ ∃𝑧(𝐷𝑧∧𝑂𝑥𝑧)→ scope of ‘∃𝑦’ ⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴⎴ ∃𝑦( scope of 2nd ‘∃𝑧’ ⎴⎴⎴⎴⎴⎴⎴⎴⎴ ∃𝑧(𝐷𝑧∧𝑂𝑦𝑧)∧𝐹𝑦𝑥)) Even in this case, however, you might want to choose different variables for every quantifier just as a practical matter, preventing any possibility of confusion for your readers. §17. MULTIPLE GENERALITY 141 Sentence 141 is the trickiest yet. First we paraphrase it as ‘For any x that is a friend of a dog owner, x owns a dog which is also owned by a friend of x’. Using our stepping‑stone tactic, this becomes: ∀𝑥(𝑥is a friend of a dog owner →𝑥owns a dog which is owned by a friend of 𝑥). Breaking this down a bit more: ∀𝑥(∃𝑦(𝐹𝑥𝑦∧𝑦is a dog owner)→∃𝑦(𝐷𝑦∧𝑂𝑥𝑦∧𝑦is owned by a friend of 𝑥)). And a bit more: ∀𝑥(∃𝑦(𝐹𝑥𝑦∧∃𝑧(𝐷𝑧∧𝑂𝑦𝑧))→∃𝑦(𝐷𝑦∧𝑂𝑥𝑦∧∃𝑧(𝐹𝑧𝑥∧𝑂𝑧𝑦))). And we are done! 17.4 Sentence Structure and Levels of Analysis As I emphasised in §4.3, a single English sentence has many structures, depending on how fine‑ grained one is in the analysis of that sentence. We could symbolise ‘Antony owns a car’ in a number of different ways given the resources we have so far, in increasingly finer detail: 1. We could symbolise it as an atomic sentence of Sentential like ‘𝐴’, ignoring all internal structure of the sentence (because it has no internal truth‑functional structure). This is also a zero‑place predicate, so is also a sentence of Quantifier. 2. We could symbolise it as another atomic sentence of Quantifier, but one which does re‑ cognise the internal subject‑predicate struture of the English sentence. For example, we could symbolise it as ‘𝑊𝑎’, where ‘𝑊’ symbolises ‘⎵ 1owns a car’ and ‘𝑎’ symbolises ‘Antony’. 3. Or we could symbolise it as a complex quantified sentence ‘∃𝑦(𝐶𝑦∧𝑂𝑎𝑦)’, where ‘𝑎’ is as before, but ‘𝐶’ means ‘⎵ 1is a car’ and ‘𝑂’ is ‘⎵ 1owns ⎵ 2’. 4. We could imaginably symbolise it at even finer levels of structure, breaking down the predicate ‘is a car’ into the verb phrase ‘is’ and the indefinite noun phrase ‘a car’ (which is itself complex). This would go beyond the representational resources even of Quantifier. Note that any structure identified is preserved: the name ‘𝑎’ continues to appear in more fine‑ grained symbolisations once it has appeared. A one‑place predicate, having appeared in the second symbolisation, still appears indirectly in the third symbolisation. For the open sentence ‘∃𝑦(𝐶𝑦∧𝑂𝓍𝑦)’ can be understood a representing a complex one‑place predicate: ‘𝓍’ is not asso‑ ciated with any quantifier, and can be replaced by a name to form a grammatical sentence. How to symbolise the structure of a sentence very much depends on what purpose you have in symbolising. What matters is that you manage to represent enough structure to determine whether the target argument you are symbolising is valid. Valid arguments can be symbolised as invalid arguments, if you don’t attend to the relevant structure, or don’t have resources in your 142 THE LANGUAGE OF QUANTIFIED LOGIC language to represent that structure. But we just observed that any more fine‑grained analysis of the structure of a sentence retains the coarser structure (as it just adds more detailed substructure). So if you can show an argument is valid (conclusive in virtue of its structure) at some level of analysis, it will remain valid according to any more fine‑grained understanding of the structure of that sentence. So you should aim to symbolise just enough structure in an argument to be able to demonstrate its validity – if indeed it is valid. Key Ideas in §17 ›The true power of Quantifier comes from its ability to handle multiple generality. ›But with great power comes additional complexity: we need to keep track of different places in our predicates, and different quantifiers that may govern those places. Even small alterations in scope or order can drastically change the meaning of the symbolisation we produce. ›Use of paraphrases and hybrid English‑Quantifier sentences can be use‑ ful in figuring out how to symbolise a complex claim featuring multiple quantifiers and complex many‑place predicates. Practice exercises A. Using this symbolisation key: domain: all animals 𝐴:⎵ 1is an alligator 𝑀:⎵ 1is a monkey 𝑅:⎵ 1is a reptile 𝑍:⎵ 1lives at the zoo 𝐿:⎵ 1loves ⎵ 2 𝑎: Amos 𝑏: Bouncer 𝑐: Cleo symbolise each of the following sentences in Quantifier: 1. If Cleo loves Bouncer, then Bouncer is a monkey. 2. If both Bouncer and Cleo are alligators, then Amos loves them both. 3. Cleo loves a reptile. 4. Bouncer loves all the monkeys that live at the zoo. 5. All the monkeys that Amos loves love him back. 6. Every monkey that Cleo loves is also loved by Amos. 7. There is a monkey that loves Bouncer, but sadly Bouncer does not reciprocate this love. B. Using the following symbolisation key: §17. MULTIPLE GENERALITY 143 domain: all animals 𝐷:⎵ 1is a dog 𝑆:⎵ 1likes samurai movies 𝐿:⎵ 1is larger than ⎵ 2 𝑏: Bertie 𝑒: Emerson 𝑓: Fergus symbolise the following sentences in Quantifier: 1. Bertie is a dog who likes samurai movies. 2. Bertie, Emerson, and Fergus are all dogs. 3. Emerson is larger than Bertie, and Fergus is larger than Emerson. 4. All dogs like samurai movies. 5. Only dogs like samurai movies. 6. There is a dog that is larger than Emerson. 7. If there is a dog larger than Fergus, then there is a dog larger than Emerson. 8. No animal that likes samurai movies is larger than Emerson. 9. No dog is larger than Fergus. 10. Any animal that dislikes samurai movies is larger than Bertie. 11. There is an animal that is between Bertie and Emerson in size. 12. There is no dog that is between Bertie and Emerson in size. 13. No dog is larger than itself. 14. Every dog is larger than some dog. 15. There is an animal that is smaller than every dog. 16. If there is an animal that is larger than any dog, then that animal does not like samurai movies. C. Using this symbolisation key, domain: all animals 𝐿:⎵ 1is larger than ⎵ 2. 𝐹:⎵ 1is friendlier than ⎵ 2. 𝐷:⎵ 1is a dog. 𝑏: Bertie. 𝑒: Emerson. 𝑓: Fergus. render the following into natural English, commenting on any difficulties: 1. (𝐿𝑒𝑏∧𝐿𝑓𝑒); 2. ∃𝑥(𝐷𝑥∧𝐿𝑥𝑒); 3. (∃𝑥(𝐷𝑥∧𝐿𝑥𝑓)→∃𝑦(𝐷𝑦∧𝐹𝑦𝑒)); 4. ∀𝑥(𝐷𝑥→¬𝐿𝑥𝑓); 5. ∃𝑦((𝐿𝑦𝑏∧𝐿𝑒𝑦)∨(𝐿𝑦𝑒∧𝐿𝑏𝑦)); 6. ∀𝑥(𝐷𝑥→∃𝑦(𝐷𝑦∧𝐹𝑥𝑦)); 7. ∀𝑥∀𝑦(𝐿𝑥𝑦→∃𝑧(𝐷𝑧∧(𝐹𝑦𝑧∧𝐹𝑧𝑥))). 246 NATURAL DEDUCTION FOR SENTENTIAL Notice that ℬcan be any sentence of Sentential whatsoever. So the following is a perfectly good proof: 1 𝑀 2(𝑀∨(((𝐴↔𝐵)→(𝐶∧𝐷))↔(𝐸∧𝐹))) ∨I, 1 Using a truth table to show this would have taken 128 lines. Here is an example, to show our rules in action: 1 (𝐴∧𝐵) 2 𝐴 ∧E, 1 3 𝐵 ∧E, 1 4 (𝐴∨𝐶) ∨I, 2 5 (𝐵∨𝐶) ∨I, 3 6 ((𝐴∨𝐶)∧(𝐵∨𝐶)) ∧I, 4,5 This disjunction rule is supported by the following valid Sentential argument forms: ›𝒜⊨(𝒜∨ℬ); and ›ℬ⊨(𝒜∨ℬ). The rule of disjunction introduction is one place where ‘natural’ deduction doesn’t seem to live up to its name. Is this implication really one we would naturally make? Despite appearances, maybe we do sometimes reason like this. Consider this argument: ‘I won’t ever eat meat again. Well, either I won’t, or it will be an accident!’ (But perhaps this is better thought of as a retraction of my initial over‑bold claim, rather than an inference from it.) Nevertheless, even if the rule is artificial, that doesn’t make it incorrect. We can see, clearly, that it corresponds to a valid argument. So perhaps the problem with it as a piece of reasoning is due to something other than invalidity. ›Sometimes disjunction introduction looks like you are ‘throwing away’ information that you already have – you know enough to treat ‘𝑃’ as a premise, but you end up assenting only to the weaker claim ‘𝑃∨𝑄’. But can this be the full story? Conjunction elimination seems to involve the same sort of inference from a logically stronger to a logically weaker claim, and that doesn’t arouse nearly as much animosity as disjunction introduction. ›An alternative explanation: maybe the introduced disjunct seems irrelevant, because the content of the sentence to which the rule is applied has in general nothing to do with the disjunct introduced. This is not the case with conjunction elimination, where the result of applying the rule is clearly related to the sentence it is applied to. §28. BASIC RULES FOR Sentential: RULES WITHOUT SUBPROOFS 247 These considerations of relevance or information value lie beyond logic proper. They concern what we, as thinkers and hearers, can conclude about the speaker’s state of mind, given that they have said something with a particular content. This is the domain of that part of linguist‑ ics known as PRAGMATICS, the study of meaning in context. Most theories of pragmatics predict that disjunction introduction is valid but often conversationally inappropriate. So Paul Grice, for example, says that if you are in a position to contribute the information of a claim 𝒫to a conver‑ sation, and you are a cooperative speaker, then you will not contribute the weaker information ‘𝒫 or 𝒬’, even though you should still regard it as true.2 28.6 Reiteration The last natural deduction rule in this category is REITERATION (R). This just allows us to repeat an assumption or claim 𝒜we have already established, so long as the repeated sentence remains in the range of any assumption which the original was in the range of. 𝑚 𝒜 ⋮ 𝒜R, 𝑚 Such a rule is obviously legitimate; but one might well wonder how such a rule could ever be useful. Here is an example of it in action: 1 𝑃 2 ((𝑃∧𝑃)→𝑄) 3 𝑃 R, 1 4 (𝑃∧𝑃) ∧I, 1,3 5 𝑄 →E, 2,4 This rule is unnecessary at this point in the proof (we could have applied conjunction introduc‑ tion and cited line 1 twice in our commentary), but it can be easier in practice to have two distinct lines to which to apply conjunction introduction. The real benefits of reiteration come when we have multiple subproofs, as we will see in the following section (§29.1) – particularly when it comes to the negation rules. But we will also see later in §33.1 that, strictly speaking, we don’t need the reiteration rule – though for convenience we will keep it. And once we are able to discharge assumptions, reiteration carries some risks (§29.3). 2Grice’s discussion of disjunction is at pp. 44–6 in H P Grice (1989) Studies in the Way of Words, Harvard University Press; see also §4 of Maria Aloni (2016) ‘Disjunction’ in Edward N Zalta, ed., The Stanford Encyclopedia of Philosophy plato.stanford.edu/entries/disjunction/#DisjConv. 248 NATURAL DEDUCTION FOR SENTENTIAL Key Ideas in §28 ›Natural deduction gives us rules that tell us how to infer from a sentence with a certain main connective (elimination rules), and rules that tell us how to infer to a sentence with a certain main connective (introduction rules). ›Proofs using the rules so far retain all the assumptions made during the course of the proof, and so correspond to an argument with those as‑ sumptions as premises and the final line of the proof as a conclusion. ›Reiteration is an optional rule, but useful for housekeeping in proofs. Practice exercises A. The following ‘proof’ is incorrect. Explain the mistakes it makes. 1 𝐴∧(𝐵∧𝐶) 2 ((𝐵∨𝐶)→𝐷) 3 𝐵 ∧E, 1 4 (𝐵∨𝐶) ∨I, 3 5 𝐷 →E, 4,2 B. Is the following purported proof correct? 1 𝐴 2 𝐵 3 𝐴 R, 1 C. The following proof is missing its commentary. Please supply the correct annotations on each line that needs one: 1 (𝑃∧𝑆) 2 𝑃 3 𝑆 4 (𝑆→𝑅) 5 𝑅 6 (𝑅∨𝐸) §28. BASIC RULES FOR Sentential: RULES WITHOUT SUBPROOFS 249 D. Give natural deduction proofs for the following arguments: 1. 𝑃∴((𝑃∨𝑄)∧𝑃); 2. ((𝑃∧𝑄)∧(𝑅∧𝑃))∴((𝑅∧𝑄)∧𝑃); 3. (𝐴→(𝐴→𝐵)),𝐴∴𝐵; 4. (𝐵↔(𝐴↔𝐵)),𝐵∴𝐴. 29 Basic Rules for Sentential: Rules with Subproofs We’ve already seen in §27 how to start a proof by making assumptions. But the true power of natural deduction relies on its rules governing when you can make additional assumptions during the course of the proof, and how you can discharge those assumptions when you no longer need them. 29.1 Additional Assumptions and Subproofs In natural deduction, both making and discharging additional assumptions are handled using SUBPROOFS. These are subsidiary proofs within the main proof, which encapsulate that part of a larger proof that depends on an assumption that is not among the premises. (Conversely, we can think of the premises of an argument as those assumptions left undischarged at the conclusion of a proof.) When we start a subproof, we draw another vertical line (to the right of any existing assumption lines) to indicate that we are no longer in the main proof. Then we write in the assumption upon which the subproof will be based. A subproof can be thought of as essentially posing this question: what could we show, if we also make this additional assumption? We’ve already seen this in action earlier (§27), when we said that we could indicate the range of several premises in an argument by either attaching them all to one vertical assumption line, or introducing a new vertical line for each new assumption. In that case, we never got rid of the new assumptions: they remained as premises. What will be new in this section is that some rules take us back out of a subproof. So the rules we will now consider are quite different from the rules covered in §28, none of which have this feature of being able to escape from a previously introduced assumption. When we are working within a subproof, we can refer to the additional assumption that we made in introducing the subproof, and to anything that we obtained from our original assumptions. (After all, those original assumptions are still in effect.) But at some point, we shall want to stop working with the additional assumption: we shall want to return from the subproof to the main 250 §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 251 proof. To indicate that we have returned to the main proof, the vertical line for the subproof comes to an end. At this point, we say that the subproof is CLOSED. Having closed a subproof, we have set aside the additional assumption, so it will be illegitimate to draw upon anything that depends upon that additional assumption. This has been implicit in our discussion all along, but it is good to make it very clear: Any point in a natural deduction proof is in the range of some (zero or more) currently active assumptions, and the natural deduction rules can be applied to extend the proof from that point only by appealing to prior sentences which rely at most on those same assumptions (or perhaps fewer). Equivalently, any rule can be applied to any earlier lines in a proof, except for those lines which occur within a closed subproof. Closing a subproof is called DISCHARGING the assumptions of that subproof. So we can put the point this way: at no stage of a proof can you apply a rule to a sentence that occurs only in the range of an already discharged assumption. Subproofs, then, allow us to think about what we could show, if we made additional assumptions. The point to take away from this is not surprising – in the course of a proof, we have to keep very careful track of what assumptions we are making, at any given moment. Our proof system does this very graphically, with those vertical assumption lines that indicate the range of an assumption. (Indeed, that’s precisely why we have chosen to use this proof system.) When you discharge an assumption, closing a subproof, you generally introduce some further new sentence. That sentence can be thought of as a summary of the subproof, in the context of the other undischarged assumptions. This is particularly evident if you think about the conditional introduction rule we are about to introduced. When can we begin a new subproof? Whenever we want. That is the upshot of the New Assump‑ tion rule from §27.3. At any stage in a proof it is legitimate to assume something new, as long as we begin keeping track of what in our proof rests on this new assumption. We don’t need any reason to justify making an assumption, but that doesn’t mean it’s a good idea to introduce them haphazardly. Some guidelines to help decide when it might be particularly appropriate to make a new assumption are discussed in §32. The idea of opening subproofs by making new assumptions can be used to illustrate a remark I made above about when a claim depends on assumptions (p. 234). Consider this proof: 1 (𝑃∧𝑄) 2 𝑄 ∧E, 1 3 𝑅 4 𝑄 R, 2 In this proof, even though the occurrence of ‘𝑄’ on line 4 occurs within the range of the assump‑ tion ‘𝑅’, it does not intuitively depend on it. We used reiteration to show that ‘𝑄’ is still derivable from active assumptions at line 4, but it does not follow that ‘𝑄’ depends in any robust way on every assumption that is active at line 4. 252 NATURAL DEDUCTION FOR SENTENTIAL 29.2 Conditional Introduction To illustrate the use of subproofs, we will begin with the rule of conditional introduction. It is fairly easy to motivate informally. The following argument in English should be valid: Ludwig is reactionary. Therefore if Ludwig is libertarian, then Ludwig is both reactionary and libertarian. If someone doubted that this was valid, we might try to convince them otherwise by explaining ourselves as follows: Assume that Ludwig is reactionary. Now, additionally assume that Ludwig is liber‑ tarian. Then by conjunction introduction, it follows that Ludwig is both reactionary and libertarian. Of course, that only follows conditional on the assumption that Lud‑ wig is libertarian. But this just means that, if Ludwig is libertarian, then he is both reactionary and libertarian – at least, that follows given our initial assumption that he is reactionary. This kind of reasoning is vital for understanding conditional claims. As the Cambridge philo‑ sopher Frank Ramsey pointed out: If two people are arguing ‘If 𝒫, will 𝒬?’ and are both in doubt as to 𝒫, they are adding 𝒫hypothetically to their stock of knowledge and arguing on that basis about 𝒬….1 Ramsey’s idea is that if we can reach the conclusion that 𝒞on the basis of the hypothetical sup‑ position that 𝒜(generally together with some other assumptions) then we would be entitled to judge, given the other assumptions alone, that if 𝒜turns out to be true, then 𝒞will also turn out to be true – for short, that if 𝒜then 𝒞. This observation of Ramsey’s – that conditionals embody the categorical content of hypothetical reasoning – has been important for many accounts of the English conditional, not all of them wholly congenial to the idea that ‘if’ is to be understood as ‘→’. Yet the essence of his idea motivates the conditional introduction rule of natural deduction. Transferred into natural deduction format, here is the pattern of reasoning that we just used. We started with one premise, ‘Ludwig is reactionary’, symbolised ‘𝑅’. Thus: 1 𝑅 The next thing we did is to make an additional assumption (‘Ludwig is libertarian’), for the sake of argument. To indicate that we are no longer dealing merely with our original assumption (‘𝑅’), but with some additional assumption, we continue our proof as follows: 1F P Ramsey (1929), ‘General Propositions and Causality’, at p. 155 in F P Ramsey (1990) Philosophical Papers, D H Mellor, ed., Cambridge University Press. §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 253 1 𝑅 2 𝐿 We are not claiming, on line 2, to have proved ‘𝐿’ from line 1. We are just making another as‑ sumption. So we do not need to write in any justification for the additional assumption on line 2. We do, however, need to mark that it is an additional assumption. We do this in the usual way, by drawing a line under it (to indicate that it is an assumption) and by indenting it with a further assumption line (to indicate that it is additional). With this extra assumption in place, we are in a position to use ∧I. So we could continue our proof: 1 𝑅 2 𝐿 3 𝑅∧𝐿 ∧I, 1,2 The two vertical lines to the left of line 3 show that ‘𝑅∧𝐿’ is in the range of both assumptions, and indeed depends on them collectively. So we have now shown that, on the additional assumption, ‘𝐿’, we can obtain ‘𝑅∧𝐿’. We can therefore conclude that, if ‘𝐿’ obtains, then so does ‘𝑅∧𝐿’. Or, to put it more briefly, we can conclude ‘𝐿→(𝑅∧𝐿)’: 1 𝑅 2 𝐿 3 𝑅∧𝐿 ∧I, 1,2 4 𝐿→(𝑅∧𝐿) →I, 2–3 Observe that we have dropped back to using one vertical line. We are no longer relying on the additional assumption, ‘𝐿’, since the conditional itself follows just from our original assumption, ‘𝑅’. The use of conditional introduction has discharged the temporary assumption, so that the final line of this proof relies only on the initial assumption ‘𝑅’ – we made use of the assumption ‘𝐿’ only in the nested subproof, and the range of that assumption is restricted to sentences in that subproof. Note that the conditional sentence ‘𝐿→(𝑅∧𝐿)’ is a summary of what went on in the subproof, given the undischarged assumption ‘𝑅’: if you made the additional assumption 𝐿, then you could derive ‘(𝑅∧𝐿)’. The general pattern at work here is the following. We first make an additional assumption, A; and from that additional assumption, we prove B. In that case, we have established the following: If is does in fact turn out that A, then it also turns out that B. This is wrapped up in the rule for CONDITIONAL INTRODUCTION: 254 NATURAL DEDUCTION FOR SENTENTIAL 𝑖 𝒜 ⋮ 𝑗 ℬ ⋮ (𝒜→ℬ) →I, 𝑖–𝑗 There can be as many or as few lines as you like between lines 𝑖and 𝑗. Notice that in our presenta‑ tion of the rule, discharging the assumption 𝒜takes us out of the subproof in which ℬis derived from 𝒜. If 𝒜is the initial assumption of a proof, then discharging it may well leave us with a conditional claim that depends on no undischarged assumptions at all. We see an example in this proof, where the main proof, marked by the leftmost vertical line, features no horizontal line marking an assumption: 1 𝑃∧𝑃 2 𝑃 ∧E, 1 3 (𝑃∧𝑃)→𝑃 →I, 1–2 It might come as no surprise that the conclusion of this proof – being provable from no undis‑ charged assumptions at all – turns out to be a logical truth. It will help to offer a further illustration of →I in action. Suppose we want to consider the follow‑ ing: 𝑃→𝑄,𝑄→𝑅∴𝑃 →𝑅. We start by listing both of our premises. Then, since we want to arrive at a conditional (namely, ‘𝑃→𝑅’), we additionally assume the antecedent to that conditional. Thus our main proof starts: 1 𝑃→𝑄 2 𝑄→𝑅 3 𝑃 Note that we have made ‘𝑃’ available, by treating it as an additional assumption. But now, we can use →E on the first premise. This will yield ‘𝑄’. And we can then use →E on the second premise. So, by assuming ‘𝑃’ we were able to prove ‘𝑅’, so we apply the →I rule – discharging ‘𝑃’ – and finish the proof. Putting all this together, we have: §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 255 1 𝑃→𝑄 2 𝑄→𝑅 3 𝑃 4 𝑄 →E, 1,3 5 𝑅 →E, 2,4 6 𝑃→𝑅 →I, 3–5 Let’s consider another example, this one demonstrating why reiteration can be so useful in sub‑ proofs. We know that 𝑃 ∴𝑄→𝑃is a valid argument, from truth‑tables. This is a proof: 1 𝑃 2 𝑄 3 𝑃 R, 1 4 (𝑄→𝑃) →I, 2–3 Note that strictly speaking we needn’t have used reiteration here: the assumption of ‘𝑃’ remains active at line 2, so technically we could apply →I to close the subproof and introduce the condi‑ tional immediately after line 2. But the use of reiteration makes it much clearer what is going on in the proof – even though all it does it repeat the earlier assumption and remind us that it is still an active assumption. We now have all the rules we need to show that the argument on page 229 is valid. Here is the six line proof, some 175,000 times shorter than the corresponding truth table: 1 𝐴1→𝐶1 2 (𝐴1∧𝐴2∧𝐴3∧𝐴4∧𝐴5∧𝐴6∧𝐴7∧𝐴8∧𝐴9∧𝐴10) 3 𝐴1∧E, 2 4 𝐶1→E, 1,3 5 (𝐶1∨𝐶2∨𝐶3∨𝐶4∨𝐶5∨𝐶6∨𝐶7∨𝐶8∨𝐶9∨𝐶10) ∨I, 4 6 (𝐴1∧𝐴2∧𝐴3∧𝐴4∧𝐴5∧𝐴6∧𝐴7∧𝐴8∧𝐴9∧𝐴10)→(𝐶1∨𝐶2∨𝐶3∨𝐶4∨𝐶5∨𝐶6∨𝐶7∨𝐶8∨𝐶9∨𝐶10) →I, 2–5 Import‑Export Our rules so far can be used to demonstrate two important principles governing the conditional. The principle of IMPORTATION is the claim that from ‘if 𝑃then, if also 𝑄, then 𝑅’ it follows that ‘if 𝑃and also 𝑄, then 𝑅’. The principle of EXPORTATION is the converse, that from ‘if 𝑃and also 𝑄, then 𝑅’ it follows that ‘if 𝑃, then if also 𝑄, then 𝑅’. First, we prove importation holds for our conditional: 262 NATURAL DEDUCTION FOR SENTENTIAL Another informative example demonstrates the logical equivalence of ‘((𝑃∧𝑄)→𝑅)’ and ‘(𝑃 → (𝑄→𝑅))’ given importation and exportation (page 255). We will re‑use both of our earlier proofs, stitching them together using biconditional introduction in the final line: 1 (𝑃→(𝑄→𝑅)) 2 (𝑃∧𝑄) 3 𝑃 ∧E, 2 4 (𝑄→𝑅) →E, 1,3 5 𝑄 ∧E, 2 6 𝑅 →E, 4,5 7 ((𝑃∧𝑄)→𝑅) →I, 2–6 8 ((𝑃∧𝑄)→𝑅) 9 𝑃 10 𝑄 11 (𝑃∧𝑄) ∧I, 9,10 12 𝑅 →E, 8,11 13 (𝑄→𝑅) →I, 10–12 14 (𝑃→(𝑄→𝑅)) →I, 9–13 15 ((𝑃∧𝑄)→𝑅)↔(𝑃→(𝑄→𝑅)))↔I, 8–14,1–7 Note the small gap between the nested vertical lines between lines 7 and 8 – that shows we have two subproofs here, not one. (That would also be indicated by the fact that the sentence on line 8 has a horizontal line under it – no vertical assumption line has two markers of where the assumptions cease.) The acceptability of our proof rules is grounded in the fact that they will never lead us from truth to falsehood. The acceptability of the biconditional introduction rule is demonstrated by the following correct entailment: ›If 𝒞1,…,𝒞𝑛,𝒜⊨ℬand 𝒞1,…,𝒞𝑛,ℬ⊨𝒜, then 𝒞1,…,𝒞𝑛⊨𝒜↔ℬ. 29.7 Disjunction Elimination The disjunction elimination rule is slightly trickier than those we’ve seen so far. Suppose that either Ludwig is reactionary or he is libertarian. What can you conclude? Not that Ludwig is reactionary; it might be that he is libertarian instead. And equally, not that Ludwig is libertarian; for he might merely be reactionary. It can be hard to draw a definite conclusion from a disjunction just by itself. §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 263 But suppose that we could somehow show both of the following: first, that Ludwig’s being re‑ actionary entails that he is an Austrian economist: second, that Ludwig’s being libertarian also entails that he is an Austrian economist. Then if we know that Ludwig is either reactionary or libertarian, then we know that, whichever he is, Ludwig is an Austrian economist. This we might call ‘no matter whether’ reasoning: if each of 𝒜and ℬimply 𝒞, then no matter whether 𝒜or ℬ, still 𝒞. Sometimes this kind of reasoning is called proof by cases, since you start with the assumption that either of two cases holds, and then show something follows no matter which case is actual. This insight can be expressed in the following rule, which is our DISJUNCTION ELIMINATION (∨E) rule: 𝑚 (𝒜∨ℬ) ⋮ 𝑖 𝒜 ⋮ 𝑗 𝒞 ⋮ 𝑘 ℬ ⋮ 𝑙 𝒞 ⋮ 𝒞 ∨E, 𝑚,𝑖–𝑗,𝑘–𝑙 This is obviously a bit clunkier to write down than our previous rules, but the point is fairly simple. Suppose we have some disjunction, 𝒜∨ℬ. Suppose we have two subproofs, showing us that 𝒞follows from the assumption that 𝒜, and that 𝒞follows from the assumption that ℬ. Then we can derive 𝒞itself. As usual, there can be as many lines as you like between 𝑖and 𝑗, and as many lines as you like between 𝑘and 𝑙. Moreover, the subproofs and the disjunction can come in any order, and do not have to be adjacent. Some examples might help illustrate the rule in action. Consider this argument: (𝑃∧𝑄)∨(𝑃∧𝑅)∴𝑃. An example proof might run thus: 264 NATURAL DEDUCTION FOR SENTENTIAL 1 (𝑃∧𝑄)∨(𝑃∧𝑅) 2 𝑃∧𝑄 3 𝑃 ∧E, 2 4 𝑃∧𝑅 5 𝑃 ∧E, 4 6 𝑃 ∨E, 1,2–3,4–5 An adaptation of the previous proof can be used to establish a proof for this argument: (𝑃∧𝑄)∨(𝑃∧𝑅)∴(𝑃∧(𝑄∨𝑅)). We begin the cases in the same way as above, but as we continue please note the use of the disjunc‑ tion introduction rule to get the last line of each subproof in the right format to use disjunction elimination. 1 (𝑃∧𝑄)∨(𝑃∧𝑅) 2 𝑃∧𝑄 3 𝑃 ∧E, 2 4 𝑄 ∧E, 2 5 (𝑄∨𝑅) ∨I, 4 6 𝑃∧(𝑄∨𝑅) ∧I, 3,5 7 𝑃∧𝑅 8 𝑃 ∧E, 7 9 𝑅 ∧E, 7 10 (𝑄∨𝑅) ∨I, 9 11 𝑃∧(𝑄∨𝑅) ∧I, 8,10 12 𝑃∧(𝑄∨𝑅) ∨E, 1,2–6,7–11 Don’t be alarmed if you think that you wouldn’t have been able to come up with this proof yourself. The ability to come up with novel proofs will come with practice. The key question at this stage is whether, looking at the proof, you can see that it conforms with the rules that we have laid down. And that just involves checking every line, and making sure that it is justified in accordance with the rules we have laid down. Another slightly tricky example. Consider: 𝐴∧(𝐵∨𝐶)∴(𝐴∧𝐵)∨(𝐴∧𝐶). Here is a proof corresponding to this argument: §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 265 1 𝐴∧(𝐵∨𝐶) 2 𝐴 ∧E, 1 3 𝐵∨𝐶 ∧E, 1 4 𝐵 5 𝐴∧𝐵 ∧I, 2,4 6 (𝐴∧𝐵)∨(𝐴∧𝐶) ∨I, 5 7 𝐶 8 𝐴∧𝐶 ∧I, 2,7 9 (𝐴∧𝐵)∨(𝐴∧𝐶) ∨I, 8 10 (𝐴∧𝐵)∨(𝐴∧𝐶) ∨E, 3,4–6,7–9 This disjunction rule is supported by the following valid Sentential argument form: ›If 𝒟1,…,𝒟𝑛,𝒜∨ℬ,𝒜⊨𝒞and 𝒟1,…,𝒟𝑛,𝒜∨ℬ,ℬ⊨𝒞, then 𝒟1,…,𝒟𝑛,𝒜∨ℬ⊨𝒞. 29.8 Negation Introduction Our negation rules are inspired by the form of reasoning known as reductio (recall page 222). In reductio reasoning, we make an assumption that 𝒜for the sake of argument, and show that some‑ thing contradictory follows from it. Then we can conclude that our assumption was false, and that its negation must be true. There are lots of approaches to negation in natural deduction, but all of them stem from this same basic insight: when an assumption that 𝒜goes awry, conclude ¬𝒜. We see reductio reasoning used a lot in mathematics. For example: Suppose there is a largest number, call it 𝑛. Since 𝑛is the largest number, 𝑛+1 ⩽ 𝑛. But then, subtracting 𝑛from both sides, 1⩽0. And that is absurd. So there is no largest number. Here, we make an assumption, for the sake of argument. We derive from it a claim, in this case that 1 ⩽ 0. And we note that claim is absurd, given what we already know. So we conclude that the negation of our assumption holds, and cease to rely on the problematic assumption. The only claims in logic that it is safe to say are absurd are logical falsehoods. So in a logical version of reductio reasoning, we will want to show that claims that contradict one another will be derivable in the range of an assumption, in order to prove the negation of that assumption. Our NEGATION INTRODUCTION rule fits this pattern very clearly: 266 NATURAL DEDUCTION FOR SENTENTIAL 𝑖 𝒜 ⋮ 𝑗 ℬ ⋮ 𝑘 ¬ℬ ¬𝒜 ¬I, 𝑖–𝑗,𝑖–𝑘 Here, we can prove a sentence and its negation both within the range of the assumption that 𝒜. So if 𝒜were assumed, something contradictory would be derivable under that assumption. (We could apply conjunction introduction to lines 𝑗and 𝑘to make the logical falsehood explicit, but that wouldn’t be strictly necessary.) Since logical falsehoods are fundamentally unacceptable as the termination of a chain of argument, we must have begun with an inappropriate starting point when we assumed 𝒜. So, in fact, we conclude, ¬𝒜, discharging our erroneous assumption that 𝒜. There is no need for the line with ℬon it to occur before the line with ¬ℬon it. Almost always the logical falsehood arises because of a clash between the claim we assume and some bit of prior knowledge – typically, some claim we have established earlier in the proof. We will thus make frequent use of the rule of reiteration in applications of negation introduction, to get the contradictory claims in the right place to make the rule easy to apply. Here is an example of the rule in action, showing that this argument is provable: 𝐴,¬𝐵 ∴¬(𝐴→𝐵). 1 𝐴 2 ¬𝐵 3 𝐴→𝐵 4 𝐵 →E, 3,1 5 ¬𝐵 R, 2 6 ¬(𝐴→𝐵) ¬I, 3–4,3–5 Another example, for practice. Let’s prove this argument: (𝐶 →¬𝐴)∴(𝐴→¬𝐶). §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 267 1 (𝐶 →¬𝐴) 2 𝐴 3 𝐶 4 ¬𝐴 →E, 1,3 5 𝐴 R, 2 6 ¬𝐶 ¬I, 3–5,3–4 7 (𝐴→¬𝐶) →I, 2–6 The correctness of the negation introduction rule is demonstrated by this valid Sentential argu‑ ment form: ›If 𝒞1,…,𝒞𝑛,𝒜⊨ℬand 𝒞1,…,𝒞𝑛,𝒜⊨¬ℬ, then 𝒞1,…,𝒞𝑛⊨¬𝒜. 29.9 Negation Elimination The rule of negation introduction is interesting, because it is almost its own elimination rule too! Consider this schematic proof: 𝑖 ¬𝒜 ⋮ 𝑗 ℬ ⋮ 𝑘 ¬ℬ 𝑘+1 ¬¬𝒜 ¬I, 𝑖–𝑗,𝑖–𝑘 This proof terminates with a sentence that is logically equivalent to 𝒜, discharging the assump‑ tion that ¬𝒜because it leads to contradictory conclusions. This looks awfully close to a rule of negation elimination – if only we could find a way to replace a doubly‑negated sentence ¬¬𝒜by the logically equivalent sentence 𝒜, which would have eliminated the negation from the prob‑ lematic assumption ¬𝒜. In our system, we approach this problem by the brute force method – we allow ourselves to use the derivation of contradictory sentences from a negated sentence to motivate the elimination of that negation. This leads to our rule of NEGATION ELIMINATION: 268 NATURAL DEDUCTION FOR SENTENTIAL 𝑖 ¬𝒜 ⋮ 𝑗 ℬ ⋮ 𝑘 ¬ℬ 𝒜 ¬E, 𝑖–𝑗,𝑖–𝑘 This is also reductio reasoning, though in this case from a negated assumption. But again, if the assumption of ¬𝒜goes awry and allows us to derive contradictory claims (perhaps given what we’ve already shown), that licenses us to conclude 𝒜. With the rule of negation elimination, we can prove some claims that are hard to prove directly. For example, suppose we wanted to prove an instance of the LAW OF EXCLUDED MIDDLE, that (𝒜∨ ¬𝒜)is true for any sentence 𝒜. Suppose we aim at proving the specific instance ‘(𝑃∨¬𝑃)’. (It’s easy to see that the proof we give can be adapted to any other instance of the law.) You might initially have thought: it is a disjunction, so should be proved by disjunction introduction. But we cannot prove either the sentence letter ‘𝑃’ or its negation from no assumptions – so we could not prove excluded middle from no assumptions if it was by disjunction introduction from one of its disjuncts. So we proceed indirectly: we show that supposing the negation of the law of excluded middle leads to logical falsehood, and conclude it by negation elimination: 1 ¬(𝑃∨¬𝑃) 2 𝑃 3 (𝑃∨¬𝑃) ∨I, 2 4 ¬(𝑃∨¬𝑃) R, 1 5 ¬𝑃 ¬I, 2–3,2–4 6 (𝑃∨¬𝑃) ∨I, 5 7 ¬(𝑃∨¬𝑃) R, 1 8 (𝑃∨¬𝑃) ¬E, 1–6,1–7 One interesting feature of this proof is that one of the contradictory sentences is the assumption itself. When the assumption that ¬𝒜goes wrong, it might be because we have the resources to prove 𝒜! Some interesting philosophical controversy surrounds proofs like this: see §30.3. To see our negation rules in action, consider: 𝑃∴(𝑃∧𝐷)∨(𝑃∧¬𝐷). Here is a proof corresponding with the argument: §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 269 1 𝑃 2 ¬((𝑃∧𝐷)∨(𝑃∧¬𝐷)) 3 𝐷 4 (𝑃∧𝐷) ∧I, 1,3 5 (𝑃∧𝐷)∨(𝑃∧¬𝐷) ∨I, 4 6 ¬𝐷 ¬I, 3–5,3–2 7 (𝑃∧¬𝐷) ∧I, 1,6 8((𝑃∧𝐷)∨(𝑃∧¬𝐷))∨I, 7 9 (𝑃∧𝐷)∨(𝑃∧¬𝐷) ¬E, 2–8 I make two comments. In line 6, the justification cites line 2 which lies outside the subproof. That is okay, since the application of the rule lies within the range of the assumption of line 2. In line 9, the justification only cites the subproof from 2 to 8, rather than two ranges of line numbers. This is because in this application of our rule, we have the special case where the sentence such that both it and its negation can be derived from the assumption is that assumption. It would be trivial to derive it from itself. The negation elimination rule is supported by this valid Sentential argument form: ›If 𝒞1,…,𝒞𝑛,¬𝒜⊨ℬand 𝒞1,…,𝒞𝑛,¬𝒜⊨¬ℬ, then 𝒞1,…,𝒞𝑛⊨𝒜; 29.10 Putting it All Together We have now explained all of the basic rules for the proof system for Sentential.Let’s return to some of the arguments from §27 with which we began our exploration of this system, to see how they can be proved. And I will give a third example of a complex proof that uses many of our rules. 1. One argument we considered earlier was this: ¬(𝐴∨𝐵)∴(¬𝐴∧¬𝐵). We can now see that the proof we began to construct can be completed can be proved as in Figure 29.1. 2. The second proof we began constructing earlier corresponded to this argument: (𝐴∨𝐵),¬(𝐴∧𝐶),¬(𝐵∧¬𝐷)∴(¬𝐶∨𝐷). The proof can be completed as in Figure 29.2 on page 271. 3. Finally, Figure 29.3 shows a long proof involving most of our rules in action (page 272). 270 NATURAL DEDUCTION FOR SENTENTIAL 1 ¬(𝐴∨𝐵) 2 𝐴 3 (𝐴∨𝐵) ∨I, 2 4 ¬(𝐴∨𝐵) R, 1 5 ¬𝐴 ¬I, 2–3,2–4 6 𝐵 7 (𝐴∨𝐵) ∨I, 6 8 ¬(𝐴∨𝐵) R, 1 9 ¬𝐵 ¬I, 6–7,6–8 10 (¬𝐴∧¬𝐵) ∧I, 5,9 Figure 29.1: Proof of ¬(𝐴∨𝐵)∴(¬𝐴∧¬𝐵) These three proofs are more complex than the others we’ve considered, because they involve multiple rules in tandem. You should make sure you understand why each rule applies where it does, and that the proofs are correct, before you move on. You probably won’t feel that you are able to construct a proof yourself as yet, and that is okay. It is important now to see that these are in fact proofs. Some ideas about how to go about constructing them yourself will be presented in §32. But you will also get a sense about how to construct complex proofs as you practice constructing simpler proofs and start to see how they can be slotted together to form larger proofs. There is no substitute for practice. Key Ideas in §29 ›The rules for our system are summarised on page 362. ›It is important that we keep track of restrictions on when we can make use of claims derived in a subproof, since those subproofs may be making use of assumptions we are no longer accepting. ›Our proof rules match the interpretation of Sentential we have given – they will not permit us to say that some claim is provable from some assumptions when that claim isn’t entailed by those assumptions. §29. BASIC RULES FOR Sentential: RULES WITH SUBPROOFS 271 1 𝐴∨𝐵 2 ¬(𝐴∧𝐶) 3 ¬(𝐵∧¬𝐷) 4 𝐴 5 𝐶 6 𝐴∧𝐶 ∧I, 4,5 7 ¬(𝐴∧𝐶) R, 2 8 ¬𝐶 ¬I, 5–6,5–7 9 (¬𝐶∨𝐷) ∨I, 8 10 𝐵 11 ¬𝐷 12 (𝐵∧¬𝐷) ∧I, 10,11 13 ¬(𝐵∧¬𝐷) R, 3 14 𝐷 ¬E, 11–12,11–13 15 (¬𝐶∨𝐷) ∨I, 14 16 (¬𝐶∨𝐷) ∨E, 1,4–9,10–15 Figure 29.2: Proof that (𝐴∨𝐵),¬(𝐴∧𝐶),¬(𝐵∧¬𝐷)∴(¬𝐶∨𝐷). 278 NATURAL DEDUCTION FOR SENTENTIAL 𝑚 𝒜 ⋮ 𝑛 𝒜tonk ℬtonk‑I, 𝑚 𝑚 𝒜tonk ℬ ⋮ 𝑛 ℬ tonk‑E, 𝑚 You will notice that ‘tonk’ has an introduction rule like ‘∨’, and an elimination rule like ‘∧’. Of course ‘tonk’ is a connective we would not like in a language, since pairing the introduction and elimination rules would allow us to prove any arbitrary sentence from any assumption what‑ sover: 1 𝑃 2 𝑃tonk 𝑄tonk‑I, 1 3 𝑄 tonk‑E, 2 If we are to rule out such deviant connectives as ‘tonk’, Prior argues, we have to accept that ‘an expression must have some independently determined meaning before we can discover whether inferences involving it are valid or invalid’ (Prior, op.cit., p. 38). We cannot, that is, accept the inferentialist position that the rules of implication come first and the meaning comes second. In‑ ferentialists have replied, but we must unfortunately leave this interesting debate here for now.3 30.3 Constructivism The proof of excluded middle we saw on p. 268 is an example of an INDIRECT PROOF: even though the main connective of our conclusion is a disjunction, we don’t establish it by disjunction intro‑ duction. This is typical in fact of reductio reasoning in general: we show something is true, by showing that an absurdity would be true if it were false. An influential group of mathematicians are worried by indirect proofs. There is a view known as CONSTRUCTIVISM which regards mathematical objects as constructed not discovered. The closely re‑ lated view known as INTUITIONISM agrees, while offering a particular account of the construction as fundamentally deriving from human perception of the passage of time. The Dutch mathem‑ atician L E J Brouwer is most famously associated with this view. He says this about the origins of our understanding of the natural numbers: …intuitionistic mathematics is an essentially languageless activity of the mind hav‑ ing its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics. (Brouwer 1981, 4–5) 4 3The interested reader might wish to start with this reply to Prior: Nuel D Belnap, Jr (1962) ‘Tonk, Plonk and Plink’, Analysis 22, pp. 130–34. 4L E J Brouwer (1981) Brouwer’s Cambridge lectures on intuitionism, D van Dalen, ed., Cambridge University Press, at pp. 4–5. §30. SOME PHILOSOPHICAL ISSUES ABOUT CONDITIONALS, MEANING, AND NEGATION 279 Regardless of your view of intuitionism, the idea that mathematical objects are not pre‑existing inhabitants of some Platonic realm has a lot of appeal. Constructivists of all stripes think we shouldn’t accept the law of excluded middle, because to think that any claim of the form 𝒜∨¬𝒜must be true is to think that there is a pre‑existing fact of the matter as to whether or not 𝒜– and there may not be until we have constructed the mathem‑ atical objects in question. A mathematical existence proof, for example showing that a number with a certain property exists, must – according to the constructivist – consist in a construction of the specific number in question. We cannot show that some number has a property just by showing that the supposition that no number has that property leads to contradiction. When you show that ¬𝒜leads to absurdity, you have constructed a proof of ¬¬𝒜, but that is not the same as a proof of 𝒜itself. Constructivists thus typically accept the ¬I rule. If you prove ¬𝒜 by showing that the assumption 𝒜leads to a contradiction, you have positively constructed an absurdity on the basis of that assumption, and that proof is acceptable. However, constructivists reject the ¬E rule. A proof showing that the assumption ¬𝒜leads to a contradiction amounts only to a positive construction of ¬¬𝒜– not a construction of 𝒜.5 Constructive logic has some interesting features that our logical system does not. For example, constructive logic has the DISJUNCTION PROPERTY: If 𝒜∨ℬcan be proved from no assumptions, then either 𝒜can be proved from no assumptions, or ℬcan be proved from no assumptions. Our logic Sentential clearly lacks the disjunction property, as the proof of excluded middle demonstrates: ‘𝑃∨¬𝑃’ can be proved but neither of its disjuncts is provable. Constructive and intuitionistic logic rest on an alternative conception of the nature of logic to that we have advocated. It is typically understood to involve recentering logic around provability rather than truth. So rather than saying ‘𝑃∨¬𝑃’ is false, constructivists deny that it is provable, and then add that mathematical language should be restricted to what is provable, rather than relying on a Platonistic notion of abstract unworldly truth. The philosophical potential of this alternative way of thinking about logic unfortunately takes us beyond the scope of the present course.6 5Nevertheless, constructivists accept the rule known as Explosion (§33.6). This is a derived rule in Sentential, but the derivation makes essential use of ¬E, so Explosion is added as a separate rule by constructivists. The system that arises when you simply omit the ¬E rule and add no alternative rules is known as MINIMAL LOGIC. 6A brief account of intuitionistic logic and the central role it gives to provability can be found in Joan Moschovakis (2024) ‘Intuitionistic Logic’, in Edward N. Zalta, ed., The Stanford Encyclopedia of Philosophy plato.stanford.edu/ archives/sum2024/entries/logic-intuitionistic/. 280 NATURAL DEDUCTION FOR SENTENTIAL Key Ideas in §30 ›The question of how to understand conditionals in natural language is a tricky one. The natural deduction rules we adopt are suitable to under‑ stand the logical conditional ‘→’, but this may only be an approximation of English ‘if’. ›The philosophical question of whether connectives are given meaning by their truth tables or by their natural deduction rules is an interesting one. ›Constructive (or intuitionistic) mathematics is often understood to call for a revision of our logical proof rules, in particular ¬E. 31 Proof‑Theoretic Concepts 31.1 Provability and the Deduction Theorem We shall introduce some new vocabulary and notation. If there is a proof conforming to our natural deduction rules which ends on a line containing 𝒞, such that the undischarged assumptions still in effect on that last line are all among 𝒜1,𝒜2,…,𝒜𝑛then we say that 𝒞is PROVABLE FROM 𝒜1,𝒜2,…,𝒜𝑛. This is abbreviated, in our metalanguage, like this: 𝒜1,𝒜2,…,𝒜𝑛⊢𝒞. Consider this proof: 1 𝐴 2 ¬𝐵→¬𝐴 3 ¬𝐵 4 ¬𝐴 →E, 2,3 5 𝐴 R, 1 6 𝐵 ¬E, 3–4,3–5 The undischarged assumptions are ‘𝐴’ and ‘¬𝐵 → ¬𝐴’ – the assumption ‘¬𝐵’ on line 3 is dis‑ charged by the application of negation elimination that leads to the last line, ‘𝐵’. So this proof shows that 𝐴,¬𝐵 →¬𝐴⊢𝐵. The symbol ‘⊢’ is known as the single turnstile. I want to emphasise that this is different from the double turnstile symbol (‘⊨’) that represents entailment (§23). 281 282 NATURAL DEDUCTION FOR SENTENTIAL ›The single turnstile, ‘⊢’, concerns the existence of a certain kind of formal proof – namely, 𝒜1,𝒜2,…,𝒜𝑛⊢ 𝒞 claims that there is a formal proof which terminates with 𝒞and has among its undischarged assumptions only sentences among 𝒜1,𝒜2,…,𝒜𝑛. ›The double turnstile, ‘⊨’, concerns the non‑existence of a certain kind of interpretation (or valuation, in the special case of Sentential) – namely, that there is no interpretation making each of 𝒜1,𝒜2,…,𝒜𝑛true while making 𝒞false. These are very different notions. However, if we’ve designed our proof system well, we shouldn’t be able to prove a conclusion from some assumptions unless that conclusion validly follows from those assumptions. And if we are really fortunate, we should be able to provide a proof corresponding to any valid argument. (More on this in §38.) But even if our two turnstiles agree on which sentences they relate to other sentences, they still mean different things. Recall the discussion of coextensive predicates in §21.6 – even if the extensions of ‘⊢’ and ‘⊨’ are the same, we apply them on quite different grounds. If they coincide despite being defined so differently, that is some evidence that we are uncovering a genuine and important relation between sentences, describable in a number of different ways. A key result, known as the DEDUCTION THEOREM, links the notion of provability with the condi‑ tional: 𝒜1,…,𝒜𝑛,ℬ⊢𝒞iff 𝒜1,…,𝒜𝑛⊢ℬ→𝒞. We can show this result by showing how to convert a proof showing 𝒜1,…,𝒜𝑛,ℬ ⊢ 𝒞 into a proof showing 𝒜1,…,𝒜𝑛⊢ℬ→𝒞, and vice versa. So first suppose we have the proof on the left, we can apply conditional introduction to discharge the occurence of ℬand prove a conditional with ℬas antecedent: 1 𝒜1 ⋮ 𝑛 𝒜𝑛 𝑛+1 ℬ ⋮ 𝑘 𝒞 1 𝒜1 ⋮ 𝑛 𝒜𝑛 𝑛+1 ℬ ⋮ 𝑘 𝒞 𝑘+1 ℬ→𝒞 →I, 𝑛+1–𝑘 The other direction is just as easy. Suppose we have the proof on the left, terminating in ℬ → 𝒞, we can make a new assumption of ℬand use conditional elimination to generate a proof terminating in 𝒞, with that new assumption remaining undischarged. §31. PROOF‑THEORETIC CONCEPTS 283 1 𝒜1 ⋮ 𝑛 𝒜𝑛 ⋮ 𝑘 ℬ→𝒞 1 𝒜1 ⋮ 𝑛 𝒜𝑛 ⋮ 𝑘 ℬ→𝒞 𝑘+1 ℬ 𝑘+2 𝒞 →E, 𝑘,𝑘+1 31.2 Other Proof‑Theoretic Notions We now introduce a few notions that can be defined in terms of provability. We write ⊢𝒜 to mean that there is a proof of 𝒜which ends up having no undischarged assumptions. (You can think of it having no claims on the left hand side of the turnstile – a proof which has all of its undischarged assumptions among no claims must have no undischarged assumptions!) We now define: 𝒜is a THEOREM iff ⊢𝒜. Just as provability is analogous to entailment (and is, we hope, coextensive with it), so theorem‑ hood corresponds to logical truth. To illustrate the idea, suppose I want to prove that ‘¬(𝐺 ∧¬𝐺)’ is a theorem. So I must start my proof without any assumptions. However, since I want to prove a sentence whose main con‑ nective is a negation, I shall want to immediately begin a subproof by making the additional assumption ‘𝐺∧¬𝐺’ for the sake of argument, and show that this leads to contradictory con‑ sequences. All told, then, the proof looks like this: 1 𝐺∧¬𝐺 2 𝐺 ∧E, 1 3 ¬𝐺 ∧E, 1 4 ¬(𝐺∧¬𝐺) ¬I, 1–2,1–3 We have therefore constructed a proof of ‘¬(𝐺∧¬𝐺)’ with no (undischarged) assumptions, show‑ ing that ‘¬(𝐺∧¬𝐺)’ is a theorem. This particular theorem is an instance of what is sometimes called the LAW OF NON‑CONTRADICTION, that for any 𝒜,¬(𝒜∧¬𝒜). You can see how the proof above could be adapted to demonstrate the theoremhood of any instance of the law of non‑ contradiction. Simply substitute any sentence 𝒜for every occurence of ‘𝐺’ in the above proof, 284 NATURAL DEDUCTION FOR SENTENTIAL and the transformed proof will remain correct (any internal sentence connectives featuring in 𝒜 aren’t addressed by the proof rules in that proof).1 Because every proof begins with an assumption, we can only obtain a proof of a theorem if we discharge that opening assumption with a rule which allows one to close a subproof: conditional or biconditional introduction, or either of the negation rules (introduction or elimination): 1 𝑄 2 𝑃 3 (𝑃∧𝑄) ∧I, 2,1 4 (𝑃∧𝑄) 5 𝑃 ∧E, 4 6 𝑃↔(𝑃∧𝑄) ↔I, 2–6,4–5 7 𝑄→(𝑃 ↔(𝑃∧𝑄)) →I, 1–6 There is a connection to the deduction theorem here too. Any correct proof of 𝒞with one un‑ discharged assumption 𝒜will demonstrate 𝒜⊢𝒞. The deduction theorem then assures us that ⊢𝒜→𝒞. We see just this in the last line of the above proof, where a proof that 𝑄⊢(𝑃 ↔(𝑃∧𝑄)) is converted to a proof showing that ⊢𝑄→(𝑃↔(𝑃∧𝑄)). But we cannot say that every theorem has a negation, a conditional or a biconditional as its main connective. For one thing, we could have started with a negated disjunction or conjunction. For another, once we have a proof of a theorem, we can apply disjunction or conjunction introduction to its last line: e.g., we could extend the above proof by conjunction introduction to show that ⊢((𝑄→(𝑃→𝑄))∧(𝑄→(𝑃 →𝑄))). To show that something is a theorem, you just have to find a suitable proof. It is typically much harder to show that something is not a theorem. To do this, you would have to demonstrate, not just that certain proof strategies fail, but that no proof is possible. Even if you fail in trying to prove a sentence in a thousand different ways, perhaps the proof is just too long and complex for you to make out. Perhaps you just didn’t try hard enough. Even if you come up with a systematic search strategy to show that some sentence ℬisn’t a theorem, there is no guarantee your strategy will yield a result. Suppose you tried to construct all well‑formed proofs terminating with ℬ from shortest to longest, aiming to show there is no proof in which all assumptions have been discharged. As there is no longest proof, there is no guarantee at any stage in this process that your failure to find such a proof shows there is no such proof. It might just be that the shortest such proof is longer than any you’ve yet considered. On the other hand, if one of the proofs you construct is a proof of ℬwith no undischarged assumptions, they you have shown conclusively that it is a theorem, and you can stop your search. Showing that something isn’t theorem can be harder than showing that it is, in terms of how many proofs you have to consider. (On the other hand, showing that something is a logical truth can be harder than showing that it is not, in terms of how many interpretations you need to consider.) Here is another new bit of terminology: 1We have already seen a proof showing an instance of the law of excluded middle is a theorem in §29.9, page 268. §31. PROOF‑THEORETIC CONCEPTS 285 Two sentences 𝒜and ℬare PROVABLY EQUIVALENT iff each can be proved from the other; i.e., both 𝒜⊢ℬand ℬ⊢𝒜. Here is a third new bit of terminology: The sentences 𝒜1,𝒜2,…,𝒜𝑛are JOINTLY CONTRARY iff a sentence and its neg‑ ation can be proved from them, i.e., for some ℬ,𝒜1,𝒜2,…,𝒜𝑛⊢ ℬ and 𝒜1,𝒜2,…,𝒜𝑛⊢ ¬ℬ. (Sometimes in this case the 𝒜𝑖s are said to be PROVABLY INCONSISTENT.) Equivalently, some sentences are jointly contrary if you can prove a contradiction from them: 𝒜1,𝒜2,…,𝒜𝑛⊢ℬ∧¬ℬ. It is straightforward to show that some sentences 𝒜1,𝒜2,…,𝒜𝑛are jointly contrary (if they are): you just need to provide two proofs, one terminating in ℬand the other in ¬ℬ, such that all of the undischarged assumptions in those proofs are among the 𝒜𝑖s. Showing that some sentences are not jointly contrary is much harder. It would require more than just providing a proof or two; it would require showing that no proof of a certain kind is possible. Some sentences are jointly contrary iff the negation of their conjunction is a theorem. Suppose we have these proofs showing the 𝒜𝑖s to be jointly contrary: 1 𝒜1 ⋮ 𝑛 𝒜𝑛 ⋮ 𝑘 ℬ 1 𝒜1 ⋮ 𝑛 𝒜𝑛 ⋮ 𝑘′¬ℬ These can be adapted to form part of a larger proof: 1 𝒜1∧…∧𝒜𝑛 2 𝒜1∧E, 1 ⋮ 𝑛+1 𝒜𝑛∧E, 1 ⋮ 𝑘+1 ℬ from, 2–𝑛+1 ⋮ 𝑘′+𝑖 ¬ℬ from, 2–𝑛+1 𝑘′+𝑖+1 ¬(𝒜1∧…∧𝒜𝑛) ¬I, 1–𝑘+1,1–𝑘′+𝑖 286 NATURAL DEDUCTION FOR SENTENTIAL Conversely, you can extract proofs of 𝒜1∧…∧𝒜𝑛⊢ ℬ and 𝒜1∧ …∧ 𝒜𝑛⊢ ¬ℬ from the above proof. The pattern is quite general, since any theorem which is a negated conjunction will be proved by some application of negation introduction on the original conjunction, which involves showing that original conjunction to include jointly contrary sentences. To establish whether these proof‑theoretic properties hold for some sentences requires us to con‑ struct one or two proofs, and to establish that they do not hold requires us to consider all possible proofs. Table 31.1 summarises the requirements for provability, contrariety, etc. 31.3 Structural Rules and the Theory of Proofs Our proofs get their main shape from the proof rules governing the connectives. But choices we have made about how to build proofs also contribute. These principles about how to construct proofs give rise to quite abstract and general features of the notion of provability represented by ‘⊢’, sometimes known as the structural rules governing the notion of provability.2 For example: in our proof system, it does not matter in what order we make assumptions. These two proofs, distinct in their structure, nevertheless both show that 𝐴,𝐵⊢(𝐴∧𝐵). 1 𝐴 2 𝐵 3 (𝐴∧𝐵) ∧I, 1,2 1 𝐵 2 𝐴 3 (𝐴∧𝐵) ∧I, 2,1 Recall that our definition of provability says that 𝒜1,…,𝒜𝑛⊢ ℬ just in case there is a proof whose undischarged assumptions are all among the 𝒜𝑖s. No mention is made of the order of those undischarged assumptions. So while both the proofs above show that 𝐴,𝐵 ⊢(𝐴∧𝐵), they also both show that 𝐵,𝐴⊢(𝐴∧𝐵). This feature, that you can permute the order of assumptions arbitrarily, is wholly general, and is known as the COMMUTATIVITY of assumptions. That is to say: 2For more on structural rules, and the various logics that don’t have all the structural features of our natural de‑ duction system, see Greg Restall (2018) ‘Substructural Logics’ in Edward N Zalta, ed., The Stanford Encyclopedia of Philosophy plato.stanford.edu/entries/logic-substructural/. Yes No theorem? one proof all possible proofs equivalent? two proofs all possible proofs jointly contrary? two proofs all possible proofs provable one proof all possible proofs Table 31.1: What we need to establish proof‑theoretic features. §31. PROOF‑THEORETIC CONCEPTS 287 A notion of provability satisfies commutativity just in case 𝒜1,…,ℬ,…,𝒞,…,𝒜𝑛,⊢𝒟iff 𝒜1,…,𝒞,…,ℬ,…,𝒜𝑛,⊢𝒟. Commutativity and the deduction theorem seem trivial. But they can be surprisingly powerful. Consider, for example, this trivial proof by reiteration that 𝑃→𝑄⊢𝑃→𝑄: 1 𝑃→𝑄 2 𝑃→𝑄 R, 1 We can then reason as follows: 1. 𝑃→𝑄⊢𝑃→𝑄; 2. 𝑃→𝑄,𝑃⊢𝑄(by the deduction theorem); 3. 𝑃,𝑃→𝑄⊢𝑄(by commutativity); 4. 𝑃⊢(𝑃→𝑄)→𝑄(by the deduction theorem). This argument doesn’t construct a formal proof; it just assures you that there will be one. (One of the exercises asks you to construct the formal proof.) Here’s another example of a structural feature of our proof system. We allow a given line of a proof to be reused multiple times, as long as the assumptions on which that line relies remain undischarged. See this proof that (𝑃→(𝑃→𝑄))⊢(𝑃→𝑄): 1 𝑃→(𝑃→𝑄) 2 𝑃 3 𝑃→𝑄 →E, 1,2 4 𝑄 →E, 3,2 5 𝑃→𝑄 →I, 2–4 Here we appeal to line 2 multiple times: in eliminating the conditional on line 1 and the con‑ ditional on line 3. No rule governing any of our connectives is associated with this behaviour: rather, it is built in to the way we allow all of our rules to appeal to any previous line (as long as the line doesn’t appear in a closed subproof), even if that line has been appealed to already by some other rule. It is fairly easy to see that the above proof cannot succed without multiple appeal to line 2. This feature of our proof system is known as CONTRACTION: if there is a proof in which any 𝒜 occurs as an undischarged assumption on two or more distinct lines, there is also a proof in which one of those assumptions of 𝒜is removed. More concisely: 294 NATURAL DEDUCTION FOR SENTENTIAL that, anything we can prove using the rule R, we can prove (with one more line) using just the basic rules of §§29–28. So we can describe the rule R as a derived rule, since its justification is derived from our basic rules. You might note that in lines 5–7 in the complicated proof in Figure 29.3, we in effect made use of this proof scheme, introducing a conjunction from prior lines only to immediately eliminate again, just to ensure that the relevant sentences appeared directly in the range of the assumption ‘𝑄’. We even have an explanation here about why you can’t reiterate a line from a closed subproof. If all applications of reiteration are in fact abbreviations of the above schema, then that restriction on reiteration derives from the more general restriction that we cannot appeal to a proof line that relies on an assumption that has been discharged. 33.2 Disjunctive Syllogism Here is a very natural argument form. Mitt is either in Massachusetts or in DC. He is not in DC. So, he is in Massachusetts. This inference pattern is called DISJUNCTIVE SYLLOGISM. We could add it to our proof system: 𝑚 (𝒜∨ℬ) 𝑛 ¬𝒜 ℬDS, 𝑚,𝑛 𝑚 (𝒜∨ℬ) 𝑛 ¬ℬ 𝒜DS, 𝑚,𝑛 This is, if you like, a new rule of disjunction elimination. But there is nothing fundamentally new here. We can emulate the rule of disjunctive syllogism using our basic proof rules, as the schematic proof in Figure 33.1 indicates. We have used the rule of reiteration in this schematic proof, but we already know that any uses of that rule can themselves be replaced by more roundabout proofs using conjunction introduction and elimination, if required. So adding disjunctive syllogism would not make any new proofs possible that were not already obtainable in our original system. 33.3 Modus tollens Another useful pattern of inference is embodied in the following argument: If Hillary won the election, then she is in the White House. She is not in the White House. So she did not win the election. This inference pattern is called MODUS TOLLENS. The corresponding rule is: §33. DERIVED RULES FOR Sentential 295 𝑚 𝒜∨ℬ 𝑛 ¬𝒜 ⋮ 𝑘 𝒜 𝑘+1 ¬ℬ 𝑘+2 𝒜 R, 𝑘 𝑘+3 ¬𝒜 R, 𝑛 𝑘+4 ℬ ¬E, 𝑘+1–𝑘+2,𝑘+1–𝑘+3 𝑘+5 ℬ 𝑘+6 ℬ R, 𝑘+5 𝑘+7 ℬ ∨E, 𝑚,𝑘–𝑘+4,𝑘+5–𝑘+6 Figure 33.1: Disjunctive syllogism is derivable in the standard proof system. 𝑚 (𝒜→ℬ) 𝑛 ¬ℬ ¬𝒜 MT, 𝑚,𝑛 This is, if you like, a new rule of conditional elimination. This rule is, again, a conservative addition to our stock of proof rules. Any application of it could be emulated by the form of proof using our original rules shown in Figure 33.2. Again, the schmatic proof makes a dispensible use of reiteration. 33.4 Double Negation Elimination In Sentential, the double negation ¬¬𝒜 is equivalent to 𝒜. In natural languages, too, double negations tend to cancel out – Malcolm is not unaware that his leadership is under threat iff he is aware that it is. That said, you should be aware that context and emphasis can prevent them from doing so. Consider: ‘Jane is not not happy’. Arguably, one cannot derive ‘Jane is happy’, since the first sentence should be understood as meaning the same as ‘Jane is not unhappy’. This is compatible with ‘Jane is in a state of profound indifference’. As usual, moving to Sentential forces us to sacrifice certain nuances of English expressions – we have, in Sentential, just one resource for translating negative expressions like ‘not’ and the suffix ‘un‑’, even if they are not synonyms in English. 296 NATURAL DEDUCTION FOR SENTENTIAL 𝑚 𝒜→ℬ 𝑛 ¬ℬ ⋮ 𝑘 𝒜 𝑘+1 ℬ →E, 𝑚,𝑘 𝑘+2 ¬ℬ R, 𝑛 𝑘+3 ¬𝒜 ¬I, 𝑘–𝑘+1,𝑘–𝑘+2 Figure 33.2: Modus tollens is derivable in the standard proof system. Obviously we can show that 𝒜⊢¬¬𝒜by means of the following proof: 1 𝒜 2 ¬𝒜 3 ¬¬𝒜 ¬I, 2–1,2–2 There is a proof rule that corresponds to the other direction of this equivalence, the rule of DOUBLE NEGATION ELIMINATION: 𝑖 ¬¬𝒜 ⋮ 𝒜 ¬¬E, 𝑖 This rule is redundant, given the proof rules of Sentential: 1 ¬¬𝒜 2 ¬𝒜 3 ¬¬𝒜 R, 1 4 ¬𝒜 R, 2 5 𝒜 ¬E, 2–4,2–3 Anything we can prove using the ¬¬E rule can be proved almost as briefly using just ¬E. §33. DERIVED RULES FOR Sentential 297 33.5 Tertium non datur Suppose that we can show that if it’s sunny outside, then Bill will have brought an umbrella (for fear of sunburn). Suppose we can also show that, if it’s not sunny outside, then Bill will have brought an umbrella (for fear of rain). Well, there is no third way for the weather to be. So, whatever the weather, Bill will have brought an umbrella. This line of thinking motivates the following rule: ⋮ 𝑖 𝒜 𝑗 ℬ 𝑘 ¬𝒜 𝑙 ℬ ℬTND, 𝑖–𝑗,𝑘–𝑙 The rule is sometimes called TERTIUM NON DATUR, which means roughly ‘no third way’. There can be as many lines as you like between 𝑖and 𝑗, and as many lines as you like between 𝑘and 𝑙. Moreover, the subproofs can come in any order, and the second subproof does not need to come immediately after the first. Tertium non datur is able to be emulated using just our original proof rules. Figure 33.3 contains a schematic proof which demonstrates this. Once again, a dispensible use of reiteration occurs in this proof just to make it more readable. 33.6 Explosion The colourfully named principle of EXPLOSION is the rule that from a contradiction, anything can be derived. The name reflects the disaster that it would be if a contradiction were true. This principle is sometimes called by its medieval Latin name EX FALSO QUODLIBET – ‘from a (logical) falsehood, anything’. The rule takes the following form: 298 NATURAL DEDUCTION FOR SENTENTIAL 𝑖 𝒜 𝑗 ℬ 𝑘 ¬𝒜 𝑙 ℬ ⋮ 𝑚 𝒜→ℬ →I, 𝑖–𝑗 𝑚+1 ¬𝒜→ℬ →I, 𝑘–𝑙 𝑚+2 ¬ℬ 𝑚+3 𝒜 𝑚+4 ℬ →E, 𝑚,𝑚+3 𝑚+5 ¬ℬ R, 𝑚+2 𝑚+6 ¬𝒜 ¬I, 𝑚+3–𝑚+5 𝑚+7 ℬ →E, 𝑚+1,𝑚+6 𝑚+8 ℬ ¬E, 𝑚+2–𝑚+7 Figure 33.3: Tertium non datur is derivable in the standard proof system. ⋮ 𝑗 𝒜 ⋮ 𝑘 ¬𝒜 ⋮ 𝑙 ℬ EX, 𝑗,𝑘 Again, explosion can be emulated using our existing rules (note, in light of §30.3, the use of ¬E): §33. DERIVED RULES FOR Sentential 299 1 𝒜 2 ¬𝒜 3 ¬ℬ 4 𝒜 R, 1 5 ¬𝒜 R, 2 6 ℬ ¬E, 3–4,3–5 The principle of explosion has been controversial, along with its semantic cousin that inconsistent premises entail anything: 𝒜,¬𝒜⊨ℬ. It is easy enough to see that these principles are acceptable in our system of logic, but it is tempting to think they might mark a flaw in Sentential. The derivation above, using negation elimination on a sentence that is completely irrelevant to the inconsistent sentences, can seem like a mere trick. This is especially so if you are tempted by the thought that logic and reasoning have some con‑ nection (recall §§2.3,26.1). Even the weakest version of a proposed connection between logic and reasoning will tell you that if you accept the premises of a valid argument for good reason, then competent deduction of the conclusion provides a reason to accept the conclusion. Once we acknowledge that we are not logically perfect, and that sometimes we have reasons to accept inconsistent claims, then this feature of Sentential means that we have a reason to accept any arbitrary claim ℬ. And this is absurd: that we’ve made a mistake in accepting some inconsistent claims 𝒜,¬𝒜is no reason at all to accept ℬ. So explosion as a derivation rule seems to involve a form of ‘natural deduction’ that no human reasoner would ever follow. It may be denied that we ever have reasons to accept inconsistent claims; but this doesn’t stand up to scrutiny. We all rely on a diversity of sources that contribute to the bodies of information on which we rely, and there is no guarantee that the information delivered from one source will be entirely consistent with the information arriving from another. These gathered pieces of information may not even come together in our thought; we may quarantine them in different fragments (they may appear to us to concern different topics) and may not even notice the tension between them.1 Some have suggested that a different logic ought to be adopted, perhaps a RELEVANCE LOGIC that aims ‘to make relevance … a necessary condition for the validity of a valid entailment…’.2But such alternative logics have counterintuitive features of their own, including the need to intro‑ duce further truth values beyond truth and falsity! Perhaps better to again diminish the bearing of logic as a norm for reasoning: the existence of certain valid implications and acceptance nat‑ ural deduction rules is no grounds for us to reason or infer in accordance with those rules. (I return to the topic of relevance logic in §39.1.) 1For a discussion of this phenomenon, see David Lewis (1982) ‘Logic for Equivocators’, Noûs 16, pp. 431–41, at p. 436. 2Robert K. Meyer (1971) ‘Entailment’, Journal of Philosophy 68, pp. 808–18, at p. 809. 300 NATURAL DEDUCTION FOR SENTENTIAL 33.7 De Morgan Laws Our final additional rules are called the DE MORGAN LAWS. (These are named after the nineteenth century logician Augustus De Morgan.) The first two De Morgan laws show the provable equi‑ valence of a negated conjunction and a disjunction of negations. 𝑚 ¬(𝒜∧ℬ) (¬𝒜∨¬ℬ) DeM, 𝑚 𝑚 (¬𝒜∨¬ℬ) ¬(𝒜∧ℬ) DeM, 𝑚 The second pair of De Morgan laws are dual to the first pair: they show the provable equivalence of a negated disjunction and a conjunction of negations. 𝑚 ¬(𝒜∨ℬ) (¬𝒜∧¬ℬ) DeM, 𝑚 𝑚 (¬𝒜∧¬ℬ) ¬(𝒜∨ℬ) DeM, 𝑚 §33. DERIVED RULES FOR Sentential 301 The De Morgan laws are no genuine addition to the power of our original natural deduction system. Here is a demonstration of how we could derive the first De Morgan rule: 𝑘 ¬(𝒜∧ℬ) 𝑚 ¬(¬𝒜∨¬ℬ) 𝑚+1 ¬𝒜 𝑚+2 ¬𝒜∨¬ℬ ∨I, 𝑚+1 𝑚+3 𝒜 ¬E, 𝑚+1–𝑚+2,𝑚+1–𝑚 𝑚+4 ¬ℬ 𝑚+5 ¬𝒜∨¬ℬ ∨I, 𝑚+4 𝑚+6 ℬ ¬E, 𝑚+4–𝑚+5,𝑚+4–𝑚 𝑚+7 𝒜∧ℬ ∧I, 𝑚+3,𝑚+6 𝑚+8 ¬𝒜∨¬ℬ ¬E, 𝑚–𝑚+7,𝑚–𝑘 Here is a demonstration of how we could derive the second De Morgan rule: 𝑘 ¬𝒜∨¬ℬ 𝑚 ¬𝒜 𝑚+1 𝒜∧ℬ 𝑚+2 𝒜 ∧E, 𝑚+1 𝑚+3 ¬(𝒜∧ℬ) ¬I, 𝑚+1–𝑚+2,𝑚+1–𝑚 𝑚+4 ¬ℬ 𝑚+5 𝒜∧ℬ 𝑚+6 ℬ ∧E, 𝑚+5 𝑚+7 ¬(𝒜∧ℬ) ¬I, 𝑚+5–𝑚+6,𝑚+5–𝑚+4 𝑚+8 ¬(𝒜∧ℬ) ∨E, 𝑘,𝑚–𝑚+3,𝑚+4–𝑚+7 Similar demonstrations can be offered explaining how we could derive the third and fourth De Morgan rules. These are left as exercises. Those mentioned above are all of the additional rules of our proof system for Sentential. 302 NATURAL DEDUCTION FOR SENTENTIAL Key Ideas in §33 ›Our official system of rules can be augmented by additional rules that are strictly speaking unneccessary – nothing is provable with them that couldn’t have been proved without them – but that can nevertheless be used sometimes to speed up proofs. ›Only make use of derived rules when you are told you may do so. ›Some derived rules – such as the rule of double negation elimination – can even be used in place of a rule of our original system, given a different system but with the same things being provable. Practice exercises A. The following proofs are missing their commentaries (rule and line numbers). Add them wherever they are required: you may use any of the original or derived rules, as appropriate. 1 𝑍→(𝐶∧¬𝑁) 2 ¬𝑍→(𝑁∧¬𝐶) 3 ¬(𝑁∨𝐶) 4 ¬𝑁∧¬𝐶 5 ¬𝑁 6 ¬𝐶 7 𝑍 8 𝐶∧¬𝑁 9 𝐶 10 ¬𝐶 11 ¬𝑍 12 𝑁∧¬𝐶 13 𝑁 14 ¬¬(𝑁∨𝐶) 15 𝑁∨𝐶 1 𝑊 →¬𝐵 2 𝐴∧𝑊 3 𝐵∨(𝐽∧𝐾) 4 𝑊 5 ¬𝐵 6 𝐽∧𝐾 7 𝐾 1 𝐿↔¬𝑂 2 𝐿∨¬𝑂 3 ¬𝐿 4 ¬𝑂 5 𝐿 6 ¬𝐿 7 ¬¬𝐿 8 𝐿 §33. DERIVED RULES FOR Sentential 303 B. Give a proof representing each of these arguments; you may use any of the original or derived rules, as appropriate: 1. 𝐸∨𝐹,𝐹∨𝐺,¬𝐹 ∴𝐸∧𝐺 2. 𝑀∨(𝑁 →𝑀)∴¬𝑀 →¬𝑁 3. (𝑀∨𝑁)∧(𝑂∨𝑃),𝑁 →𝑃,¬𝑃 ∴𝑀∧𝑂 4. (𝑋∧𝑌)∨(𝑋∧𝑍),¬(𝑋∧𝐷),𝐷∨𝑀∴𝑀 C. Provide proof schemes that justify the addition of the third and fourth De Morgan rules as derived rules. D. The proofs you offered in response to question Aabove used derived rules. Replace the use of derived rules, in such proofs, with only basic rules. You will find some ‘repetition’ in the resulting proofs; in such cases, offer a streamlined proof using only basic rules. (This will give you a sense, both of the power of derived rules, and of how all the rules interact.) 35 Basic Rules for Quantifier 35.1 Proof‑Theoretic Concepts in Quantifier Quantifier makes use of all of the connectives of Sentential. Helpfully, our natural deduction proof system for Quantifier will simply import all of the basic rules from chapter 6. (Obviously we will get all of the derived rules for free by doing this, but we won’t make use of the derived rules.) We will define a correctly formed natural deduction proof for Quantifier to be a structured sequence of sentences of Quantifier such that each sentence is either an assumption or follows from the previous sentences by any of the Sentential rules or by any of the new rules governing the quantifiers and identity that we will introduce in this chapter.1 The notion of provability of 𝒜from undischarged assumptions 𝒞1,…,𝒞𝑛was introduced for Sentential in §31. The rules for Quantifier are different, but the earlier definitions go through unchanged, once we remember that the notion of proof in Quantifier involves a sequence of Quantifier sentences justified by the rules for Sentential and those for Quantifier. So in what follows I will make use of the single turnstile ‘⊢’ to mean that there is a correctly formed proof using only the rules of Sentential and Quantifier. (And once we introduce the identity rules in §37, I will tacitly assume that proofs can make use of those rules too in justifying a claim using ‘⊢’.) Likewise, the notions of theoremhood, provable equivalence, and joint contrariness all carry over their definitions, because they are defined in terms of the single turnstile. Some proofs in Quantifier don’t need any new rules. Consider this: ¬(∀𝑥𝑃𝑥∨∃𝑦𝑃𝑦)∴¬∀𝑥𝑃𝑥. 1Though there is a category ‘formulae which are not sentences’ in Quantifier, no member of this class will ever appear in any correctly formed proof. 310 §35. BASIC RULES FOR Quantifier 311 1 ¬(∀𝑥𝑃𝑥∨∃𝑦𝑃𝑦) 2 ∀𝑥𝑃𝑥 3 (∀𝑥𝑃𝑥∨∃𝑦𝑃𝑦) ∨I, 2 4 ¬(∀𝑥𝑃𝑥∨∃𝑦𝑃𝑦) R, 1 5 ¬∀𝑥𝑃𝑥 ¬I, 2–3,2–4 The sentences on each line are Quantifier sentences that are not sentences of Sentential, but the main connectives involved are just those governed by the rules we already introduced to handle Sentential proofs. However, not every Quantifier sentence has a Sentential connective as its main connective. So we will also need some new basic rules to govern the quantifiers, and to govern the identity sign, to deal with those sentences where the main connective is a quantifier and where the sentence is an identity predication. 35.2 Universal Elimination Holding fixed the claim that everything is F, you can conclude that any particular thing is F. You name it; it’s F. The same is true for many‑place predicates: if every human is shorter than 3km tall, then Amy is shorter than 3km tall, and Bob is, and Jonquil is, and everyone else you can name. Accordingly, the following reasoning should be fine for the corresponding symbolisations in Quantifier: 1 ∀𝑥𝑅𝑥𝑥𝑑 2 𝑅𝑎𝑎𝑑 ∀E, 1 We obtained line 2 by dropping the universal quantifier and replacing every instance of ‘𝑥’ with ‘𝑎’. Equally, the following should be allowed: 1 ∀𝑥𝑅𝑥𝑥𝑑 2 𝑅𝑑𝑑𝑑 ∀E, 1 We obtained line 2 here by dropping the universal quantifier and replacing every instance of ‘𝑥’ with ‘𝑑’. We could have done the same with any other name we wanted. This motivates the UNIVERSAL ELIMINATION rule (∀E), using the notion for uniform substitution we introduced in §22.4: 312 NATURAL DEDUCTION FOR QUANTIFIER 𝑚 ∀𝓍𝒜 ⋮ 𝒜|𝒸↷𝓍 ∀E, 𝑚 Where 𝒸can be any name. The intent of the rule is that you can obtain any substitution instance of a universally quantified formula: replace every occurrence of the free variable 𝓍in 𝒜with any chosen name. (If there are any – the rule is also good when 𝒜has no free variable, because then the quantifier ∀𝓍 is redundant.) Remember here that the expression ‘𝒸’ is a metalanguage variable over names: you are not required to replace the variable 𝓍by the Quantifier name ‘𝑐’, but you can select any name you like! I should emphasise that (as with every elimination rule) you can only apply the ∀E rule when the universal quantifier is the main connective. Thus the following is outright banned: 1 (∀𝑥𝐵𝑥→𝐵𝑘) 2 (𝐵𝑏→𝐵𝑘) naughtily attempting to invoke ∀E, 1 This is illegitimate, since ‘∀𝑥’ is not the main connective in line 1. (If you need a reminder as to why this sort of inference should be banned, reread §16.) Here is an example of the rule in action. Suppose we wanted to show that ∀𝑥∀𝑦(𝑅𝑥𝑥 → 𝑅𝑥𝑦),𝑅𝑎𝑎∴𝑅𝑎𝑏is provable. The proof might go like this: 1 ∀𝑥∀𝑦(𝑅𝑥𝑥→𝑅𝑥𝑦) 2 𝑅𝑎𝑎 3 ∀𝑦(𝑅𝑎𝑎→𝑅𝑎𝑦) ∀E, 1 4 𝑅𝑎𝑎→𝑅𝑎𝑏 ∀E, 3 5 𝑅𝑎𝑏 →E, 4,2 Here on line 3 we substitute the previously used name ‘𝑎’ for the variable ‘𝑥’ in ‘∀𝑦(𝑅𝑥𝑥→𝑅𝑥𝑦)’; and then on line 4 we substitute the new name ‘𝑏’ for the variable ‘𝑦’ in ‘𝑅𝑎𝑎 → 𝑅𝑎𝑦’. The rule of universal elimination doesn’t discriminate between new and old names. 35.3 Existential Introduction Given the assumption that some specific named thing is an F, you can conclude that something is an F: ‘Sylvester reads, so someone reads’ seems like a conclusive argument. So we ought to allow the inference from a claim about some particular thing being F, to a general claim that something or other is F: §35. BASIC RULES FOR Quantifier 313 1 𝑅𝑎𝑎𝑑 2 ∃𝑥𝑅𝑎𝑎𝑥 ∃I, 1 Here, we have replaced the name ‘𝑑’ with a variable ‘𝑥’, and then existentially quantified over it. Equally, we would have allowed: 1 𝑅𝑎𝑎𝑑 2 ∃𝑥𝑅𝑥𝑥𝑑 ∃I, 1 Here we have replaced both instances of the name ‘𝑎’ with a variable, and then existentially generalised. There are some pitfalls with this description of what we have done. The following argument is invalid: ‘Someone loves Alice; so someone is such that someone loves themselves’. So we ought not to be able to conclude ‘∃𝑥∃𝑥𝑅𝑥𝑥’ from ‘∃𝑥𝑅𝑥𝑎’. Accordingly, our rule cannot be replace a name by a variable, and stick a corresponding quantifier out the front – since that would would permit the proof of the invalid argument. We take our cue from the ∀E rule. This rule says: take a sentence ∀𝓍𝒜, then we can remove the quantifier and substitute an arbitrary name for some free variable in the formula 𝒜(assuming there is one). The ∃I rule is in some sense a mirror image of this rule: it allows us to move from a sentence with an arbitrary name – that might be thought of as the result of substituting a name for a free variable in some formula 𝒜– to a quantified sentence ∃𝓍𝒜. So here is how we formulate our rule of EXISTENTIAL INTRODUCTION: 𝑚 𝒜|𝒸↷𝓍 ⋮ ∃𝓍𝒜 ∃I, 𝑚 So really we should think that the proof just above should be thought of as concluding ∃𝑥𝑅𝑥𝑥𝑑 from ‘𝑅𝑥𝑥𝑑’|𝑎↷𝑥 (i.e., ‘𝑅𝑎𝑎𝑑’). If we have this rule, we cannot provide a proof of the invalid argument. For ‘∃𝑥𝑅𝑥𝑎’ is not a substitution instance of ‘∃𝑥∃𝑥𝑅𝑥𝑥’ – both instances of ‘𝑥’ in ‘𝑅𝑥𝑥’ are bound by the second existential quantifier, so neither is free to be substituted. So the premise is not of the right form for the rule of ∃I to apply. On the other hand, this proof is correct: 1 𝑅𝑎𝑎 2 ∃𝑥𝑅𝑎𝑥 ∃I, 1 314 NATURAL DEDUCTION FOR QUANTIFIER Why? Because the assumption ‘𝑅𝑎𝑎’ is in fact not only a substitution instance of ∃𝑥𝑅𝑥𝑥, but also a substitution instance of ‘∃𝑥𝑅𝑎𝑥’, since ‘𝑅𝑎𝑥’|𝑎↷𝑥 is just ‘𝑅𝑎𝑎’ too. So we can vindicate the intuitively correct argument ‘Narcissus loves himself, so there is someone who loves Narcissus’. As we just saw, applying this rule requires some skill in being able to recognise substitution instances. Thus the following is allowed: 1 𝑅𝑎𝑎𝑑 2 ∃𝑥𝑅𝑥𝑎𝑑 ∃I, 1 3 ∃𝑦∃𝑥𝑅𝑥𝑦𝑑 ∃I, 2 This is okay, because ‘𝑅𝑎𝑎𝑑’ can arise from substitition of ‘𝑎’ for ‘𝑥’ in ‘𝑅𝑥𝑎𝑑’, and ‘∃𝑥𝑅𝑥𝑎𝑑’ can arise from substitition of ‘𝑎’ for ‘𝑦’ in ‘∃𝑥𝑅𝑥𝑦𝑑’. But this is banned: 1 𝑅𝑎𝑎𝑑 2 ∃𝑥𝑅𝑥𝑎𝑑 ∃I, 1 3 ∃𝑥∃𝑥𝑅𝑥𝑥𝑑 naughtily attempting to invoke ∃I, 2 This is because ‘∃𝑥𝑅𝑥𝑎𝑑’ is not a substitution instance of ‘∃𝑥∃𝑥𝑅𝑥𝑥𝑑’, since (again) both occur‑ rences of ‘𝑥’ in ‘𝑅𝑥𝑥𝑑’ are already bound and so not available for free substitution. Here is an example which shows our two proof rules in action, a proof showing that ∀𝑥∀𝑦(𝑅𝑥𝑦∧𝑅𝑦𝑥)⊢∃𝑥𝑅𝑥𝑥: 1 ∀𝑥∀𝑦(𝑅𝑥𝑦∧𝑅𝑦𝑥) 2 ∀𝑦(𝑅𝑎𝑦∧𝑅𝑦𝑎) ∀E, 1 3 (𝑅𝑎𝑎∧𝑅𝑎𝑎) ∀E, 2 4 𝑅𝑎𝑎 ∧E, 3 5 ∃𝑥𝑅𝑥𝑥 ∃I, 4 For another example, consider this proof of ‘∃𝑥(𝑃𝑥∨¬𝑃𝑥)’ from no assumptions: §35. BASIC RULES FOR Quantifier 315 1 ¬(𝑃𝑑∨¬𝑃𝑑) 2 ¬𝑃𝑑 3 (𝑃𝑑∨¬𝑃𝑑) ∨I, 2 4 ¬(𝑃𝑑∨¬𝑃𝑑) R, 1 5 𝑃𝑑 ¬E, 2–3,2–4 6 (𝑃𝑑∨¬𝑃𝑑) ∨I, 5 7 ¬(𝑃𝑑∨¬𝑃𝑑) R, 1 8 (𝑃𝑑∨¬𝑃𝑑) ¬E, 1–6,1–7 9 ∃𝑥(𝑃𝑥∨¬𝑃𝑥) ∃I, 8 One final example, a proof that ∀𝑥∀𝑦(𝑅𝑥𝑦→𝑅𝑦𝑥)⊢∀𝑥(∀𝑦𝑅𝑥𝑦→∃𝑦𝑅𝑦𝑥): 1 ∀𝑥∀𝑦(𝑅𝑥𝑦→𝑅𝑦𝑥) 2 ∀𝑦(𝑅𝑎𝑦→𝑅𝑦𝑎) ∀E, 1 3 (𝑅𝑎𝑏→𝑅𝑏𝑎) ∀E, 2 4 ∀𝑦𝑅𝑎𝑦 5 𝑅𝑎𝑏 ∀E, 4 6 𝑅𝑏𝑎 →E, 3,5 7 ∃𝑦𝑅𝑦𝑎 ∃I, 6 8 (∀𝑦𝑅𝑎𝑦→∃𝑦𝑅𝑦𝑎) →I, 4–7 9 ∀𝑥(∀𝑦𝑅𝑥𝑦→∃𝑦𝑅𝑦𝑥) ∀I, 8 35.4 Empty Domains The following proof combines our two new rules for quantifiers: 1 ∀𝑥𝐹𝑥 2 𝐹𝑎 ∀E, 1 3 ∃𝑥𝐹𝑥 ∃I, 2 Could this be a bad proof? If anything exists at all, then certainly we can infer that something is F, from the fact that everything is F. But what if nothing exists at all? Then it is surely vacuously true that everything is F; however, it ought not follow that something is F, for there is nothing to be F. So if we claim that, as a matter of logic alone, ‘∃𝑥𝐹𝑥’ follows from ‘∀𝑥𝐹𝑥’, then we are 316 NATURAL DEDUCTION FOR QUANTIFIER claiming that, as a matter of logic alone, there is something rather than nothing. This might strike us as a bit odd. Actually, we are already committed to this oddity. In §15, we stipulated that domains in Quantifier must have at least one member. We then defined a logical truth (of Quantifier) as a sentence which is true in every interpretation. Since ‘∃𝑥𝑥 = 𝑥’ will be true in every interpretation, this also had the effect of stipulating that it is a matter of logic that there is something rather than nothing. Since it is far from clear that logic should tell us that there must be something rather than nothing, we might well be cheating a bit here. If we refuse to cheat, though, then we pay a high cost. Here are three things that we want to hold on to: ›∀𝑥𝐹𝑥⊢𝐹𝑎: after all, that was ∀E. ›𝐹𝑎⊢∃𝑥𝐹𝑥: after all, that was ∃I. ›the ability to copy‑and‑paste proofs together: after all, reasoning works by putting lots of little steps together into rather big chains. If we get what we want on all three counts, then we have to countenance that ∀𝑥𝐹𝑥⊢∃𝑥𝐹𝑥. So, if we get what we want on all three counts, the proof system alone tells us that there is something rather than nothing. And if we refuse to accept that, then we have to surrender one of the three things that we want to hold on to! In fact the choice is even starker. Consider this proof: 1 𝐹𝑎 2 𝐹𝑎 R, 1 3 (𝐹𝑎→𝐹𝑎) →I, 1–2 4 ∃𝑥(𝐹𝑥→𝐹𝑥) ∃I, 3 This proof uses only the obvious rule of conditional introduction, and our existential introduction rule. It terminates in a claim that a certain thing exists: a thing that is 𝐹if it is 𝐹, and has no undischarged assumptions. Again the existence of something is a theorem of our logic. The real source of the existential commitment here seems to be the use of the name ‘𝑎’, because our rules implicitly assume that every name has a referent, and hence as soon as you use a name you assume that there is something in the domain for the name to latch on to. Before we start thinking about which to surrender,2we might want to ask how much of a cheat this is. Granted, it may make it harder to engage in theological debates about why there is something 2In light of the second proof, many will opt for restricting ∃I. If we permit an empty domain, we will also need ‘empty names’ – names without a referent. When the name 𝒸is empty, it seems problematic to conclude from ‘𝒸is F’ that there is something which is F. (Does ‘Santa Claus drives a flying sleigh’ entail ‘Someone drives a flying sleigh’?) But empty names are not cost‑free; understanding how a name that doesn’t name anything can have any meaning at all has vexed many philosophers and linguists. §35. BASIC RULES FOR Quantifier 317 rather than nothing. But the rest of the time, we will get along just fine. So maybe we should just regard our proof system (and Quantifier, more generally) as having a very slightly limited purview. If we ever want to allow for the possibility of nothing, then we shall have to cast around for a more complicated proof system. But for as long as we are content to ignore that possibility, our proof system is perfectly in order. (As, similarly, is the stipulation that every domain must contain at least one object.) 35.5 Universal Introduction Suppose you had shown of each particular thing that it is F (and that there are no other things to consider). Then you would be justified in claiming that everything is F. This would motivate the following proof rule. If you had established each and every single substitution instance of ‘∀𝑥𝐹𝑥’, then you can infer ‘∀𝑥𝐹𝑥’. Unfortunately, that rule would be utterly unusable. To establish each and every single substitu‑ tion instance would require proving ‘𝐹𝑎’, ‘𝐹𝑏’, …, ‘𝐹𝑗2’, …, ‘𝐹𝑟79002’, …, and so on. Indeed, since there are infinitely many names in Quantifier, this process would never come to an end. So we could never apply that rule. We need to be a bit more cunning in coming up with our rule for introducing universal quantification. Our cunning thought will be inspired by considering: ∀𝑥𝐹𝑥∴ ∀𝑦𝐹𝑦 This argument should obviously be valid. After all, alphabetical variation in choice of variables ought to be a matter of taste, and of no logical consequence. But how might our proof system reflect this? Suppose we begin a proof thus: 1 ∀𝑥𝐹𝑥 2 𝐹𝑎 ∀E, 1 We have proved ‘𝐹𝑎’. And, of course, nothing stops us from using the same justification to prove ‘𝐹𝑏’, ‘𝐹𝑐’, …, ‘𝐹𝑗2’, …, ‘𝐹𝑟79002,…, and so on until we run out of space, time, or patience. But reflecting on this, we see that this is a way to prove 𝐹𝒸, for any name 𝒸. And if we can do it for any thing, we should surely be able to say that ‘𝐹’ is true of everything. This therefore justifies us in inferring ‘∀𝑦𝐹𝑦’, thus: 1 ∀𝑥𝐹𝑥 2 𝐹𝑎 ∀E, 1 3 ∀𝑦𝐹𝑦 ∀I, 2 The crucial thought here is that ‘𝑎’ was just some arbitrary name. There was nothing special about it – we might have chosen any other name – and still the proof would be fine. And this crucial thought motivates the universal introduction rule (∀I): 318 NATURAL DEDUCTION FOR QUANTIFIER 𝑚 𝒜|𝒸↷𝓍 ⋮ ∀𝓍𝒜 ∀I, 𝑚 𝒸must not occur in any undischarged assumption, or elsewhere in 𝒜 A crucial aspect of this rule, though, is bound up in the accompanying constraint. In English, a name like ‘Sylvester’ can play two roles: it can be introduced as a name for a specific thing (‘let me dub thee Sylvester’!), or as an arbitrary name, introduced by this sort of stipulation: let ‘Sylvester’ name some arbitrarily chosen man. The name doesn’t tell us, when it subsequently appears, whether it was introduced in one way or the other. But if it was introduced as an arbitrary name, then any conclusions we draw about this Sylvester aren’t really dependent on the particular arbitrarily chosen referent – they all depend rather on the stipulation used in introducing the name, and so (specifically) they will all be consequences of the only fact we know for sure about this Sylvester, that he is male. If all men are mortal, then an arbitrarily chosen man, whom we temporarily call ‘Sylvester’, is mortal. If Sylvester is mortal, then there is a date he will die. But since he was selected arbitrarily, without reference to any further particulars of his life, then for any man, there exists a date he will die. And that is appropriate reasoning from a universal generalisation, to another generalisation, via claims about a specific but arbitrarily chosen person.3 An informal example of this sort of reasoning from arbitrary names is this: Consider an arbitrary somebody who travelled from London to Munich in 2016. Call them J Doe. ›If J Doe took the train, then they had to go via Paris, and that leg of the journey alone takes 3 hours. ›If J Doe flew, then they would have spent at least an hour in airport transfers at each end, even setting aside the flight time itself. ›The other options – driving, walking, etc., – are all even slower. So J Doe’s journey took over two hours in every possible case. Therefore – since J Doe is an arbitrary person – every traveller’s journey from London to Munich in 2016 took over two hours. We don’t have stipulations like the above to introduce a name as an arbitrary name in Quantifier. But we do have a way of ensuring that the name has no prior associations other than those linked to a prior universal generalisation, if we insist that, when the name is about to be eliminated from the proof, no assumption about what that name denotes is being relied on. That way, we can know that however it was introduced to the proof, it was not done in a way that involved making specific assumptions about whatever the name arbitrarily picks out. 3The details about how this sort of arbitrary reference works are interesting. A controversial but nevertheless attract‑ ive view of how it might work is Wylie Breckenridge and Ofra Magidor (2012) ‘Arbitrary Reference’, Philosophical Studies 158, pp. 377–400. §35. BASIC RULES FOR Quantifier 319 If you can conclude something about a named object that doesn’t involve mak‑ ing any assumptions about it other than assumptions which we are making more generally, then you can conclude that same something about everything. The simplest way for ensure that a name is not subject to any specific assumptions is if the name was introduced by an application of ∀E, as an arbitrary name in the standard sense. But there are other ways too. In general what we need is that the name not occur in the range of any assump‑ tion which uses the name. If the name has been introduced without making any assumptions about what it denotes, then we are not relying on any special features of what the name happens to denote when we conclude that if this arbitrary thing is F, then everything is F. Consider the following proof to see how this works in action. 1 ∀𝑥(𝐴𝑥∧𝐵𝑥) 2 𝐴𝑎∧𝐵𝑎 ∀E, 1 3 𝐴𝑎 ∧E, 2 4 ∀𝑥𝐴𝑥 ∀I, 3 The crucial step is applying the ∀I rule to the name ‘𝑎’ on the last line. While the name ‘𝑎’ does appear on lines 2 and 3, it doesn’t occur in the assumption – it was introduced on line 2 as an arbitrary instance of the universal assuption. This constraint ensures that we are always reasoning at a sufficiently general level. To see the importance of the constraint in action, consider this terrible argument: Everyone loves Kylie Minogue; therefore everyone loves themselves. We might symbolise this obviously invalid inference pattern as: ∀𝑥𝐿𝑥𝑘∴∀𝑥𝐿𝑥𝑥 Now, suppose we tried to offer a proof that vindicates this argument: 1 ∀𝑥𝐿𝑥𝑘 2 𝐿𝑘𝑘 ∀E, 1 3 ∀𝑥𝐿𝑥𝑥 naughtily attempting to invoke ∀I, 2 This is not allowed, because ‘𝑘’ occurred already in an undischarged assumption, namely, on line 1. The crucial point is that, if we have made any assumptions about the object we are working with (including assumptions embedded in 𝒜itself), then we are not reasoning generally enough to license the use of ∀I. Although the name may not occur in any undischarged assumption, it may occur as a discharged assumption. That is, it may occur in a subproof that we have already closed. For example: 326 NATURAL DEDUCTION FOR QUANTIFIER 1 ∃𝑥(𝐵𝑥∧∀𝑦(¬𝑆𝑦𝑦↔𝑆𝑥𝑦)) 2 (𝐵𝑎∧∀𝑦(¬𝑆𝑦𝑦↔𝑆𝑎𝑦)) 3 ∀𝑦(¬𝑆𝑦𝑦↔𝑆𝑥𝑦) ∧E, 2 4 (¬𝑆𝑎𝑎↔𝑆𝑎𝑎) ∀E, 3 5 ¬(𝑃∧¬𝑃) 6 ¬𝑆𝑎𝑎 7 𝑆𝑎𝑎 ↔E, 4,6 8 ¬𝑆𝑎𝑎 R, 6 9 𝑆𝑎𝑎 ¬E, 6–7,6–8 10 ¬𝑆𝑎𝑎 ↔E, 4,9 11 (𝑃∧¬𝑃) ¬E, 5–9,5–10 12 (𝑃∧¬𝑃) ∃E, 1,2–11 13 𝑃 ∧E, 12 14 ¬𝑃 ∧E, 12 15 ¬∃𝑥(𝐵𝑥∧∀𝑦(¬𝑆𝑦𝑦↔𝑆𝑥𝑦)) ¬I, 1–13,1–14 One trick to this proof is to be sure to instantiate the universally quantified claim at line 3 by using the same name ‘𝑎’ as was already used in line 2. This is because, intuitively, the problem case for this supposed barber arises when you think about whether they shaves themselves or not. But themselves trickiest part of this proof occurs at lines 5–11. By line 4, we’ve already derived a contradictory biconditional. But if we just use it to derive ‘𝑆𝑎𝑎’ and ‘¬𝑆𝑎𝑎’, the contradictory claims we obtain would end up involving the name ‘𝑎’. That would mean we couldn’t apply the ∃E rule, since the final line of the subproof would contain the chosen name, so we couldn’t get our logical falsehood out of the subproof beginning on line 2, and hence could perform the desired reductio on line 1 via ¬I. So our trick is to suppose the negation of an unrelated logical falsehood on line 5, derive the logical falsehood from line 4 in the range of that assumption, and hence use ¬E to derive the logical falsehood ‘𝑃∧¬𝑃’ on line 11. This doesn’t contain the name ‘𝑎’, and hence can be extracted from the subproof to show that line 1 by itself suffices to derive a logical falsehood, and that shows the supposition that there is such a barber is a logical falsehood. 35.8 Justification of these Quantifier Rules Above, I offered informal arguments for each of our quantifier rules that seem to exemplify the pattern of argument in the rule, and to be intuitively valid. But we can also offer justifications for our rules in terms of interpretations of the sentences involved, and the principles governing truth of quantified sentences introduced in §22.4. For example, consider any interpretation which makes ∀𝓍𝒜 true. In any such interpretation, there will be a nonempty domain, and every name will denote some member of this domain. §35. BASIC RULES FOR Quantifier 327 ∀𝓍𝒜is true just in case for any name we like, it will denote something of which 𝒜|𝒸↷𝓍 is true. So in any such interpretation, for each name in the language 𝒸,𝒜|𝒸↷𝓍 will also be true. So the proof rule of ∀E corresponds to a valid argument form. For ∃E, the case is only a little more involved. Suppose ∃𝓍𝒜is true in an interpretation. Then there is some interpretation, otherwise just like the original one, in which some new name 𝒸is assigned to some object in the domain, and where 𝒜|𝒸↷𝓍 is true. Suppose that, in fact, every inter‑ pretation which makes 𝒜|𝒸↷𝓍true also makes ℬtrue, where the new name 𝒸does not appear in ℬ. Could ℬbe false in our original interpretation? No – for everything that appears in ℬis already interpreted in the original interpretation, with the same interpretation as in the interpretation which makes it true. So it must be true in our original interpretation too. So 𝒞1,…,𝒞𝑛,∃𝓍𝒜⊨ℬ (when the name 𝒸makes no appearance in any sentence in this argument), and the proof rule of ∃E corresponds to a valid argument form. You may also offer arguments from intepretations to the effect that our other quantifier proof rules correspond to valid arguments in Quantifier: ›𝒜|𝒸↷𝓍 ⊨ ∃𝓍𝒜– if 𝒸is used in the proof, it must have an interpretation as something in the domain, and so something in the domain satisfies 𝒜; ›If this entailment holds: 𝒞1,…,𝒞𝑛⊨𝒜|𝒸↷𝓍, where the name 𝒸occurs nowhere among 𝒞𝑖or elsewhere in 𝒜, then this entailment also holds: 𝒞1,…,𝒞𝑛⊨∀𝓍𝒜. For we could have substituted any other name for 𝒸and the original entailment would still have succeeded, since it could not have depended on the specific name chosen. So it doesn’t matter what the interpretation of 𝒸happens to be, and if that doesn’t matter, it must be because everything is 𝒜. So we are again comforted: our proof rules can never lead us from true assumptions to false claims, if correctly applied. 328 NATURAL DEDUCTION FOR QUANTIFIER Key Ideas in §35 ›We augment our natural deduction proof system for Sentential by allow‑ ing Quantifier sentences to occur in proofs, and adding rules governing quantifiers to go partway towards a natural deduction system for Quantifier. ›The notation ‘⊢’ for provability carries over from its earlier use in Sentential unchanged, once we understand that a proof can now use the new rules for the new logical connectives in Quantifier. ›The rules for ∀E and ∃I are straightforward and can be applied regardless of which names we deploy. ›But the other quantifier rules ∀I and ∃E contain some important restric‑ tions on which names we can use. These restrictions are motivated by considerations about arbitrary reference which inform us when we can introduce ‘dummy names’ in the course of our proofs and what we can do with them. ›Our proof rules match the interpretation of Quantifier we have given – they will not permit us to say that some claim is provable from some assumptions when that claim isn’t entailed by those assumptions. Practice exercises A. The following three ‘proofs’ are incorrect. Explain why they are incorrect. If the argument ‘proved’ is invalid, provide an interpretation which shows that the assumptions involved do not entail the conclusion: 1. 1 ∀𝑥𝑅𝑥𝑥 2 𝑅𝑎𝑎 ∀E, 1 3 ∀𝑦𝑅𝑎𝑦 ∀I, 2 4 ∀𝑥∀𝑦𝑅𝑥𝑦 ∀I, 3 2. 1 ∀𝑥∃𝑦𝑅𝑥𝑦 2 ∃𝑦𝑅𝑎𝑦 ∀E, 1 3 𝑅𝑎𝑎 4 ∃𝑥𝑅𝑥𝑥 ∃I, 3 5 ∃𝑥𝑅𝑥𝑥 ∃E, 2,3–4 §35. BASIC RULES FOR Quantifier 329 3. 1 ∃𝑦¬(𝑇𝑦∨¬𝑇𝑦) 2 ¬(𝑇𝑑∨¬𝑇𝑑) 3 𝑇𝑑 4 𝑇𝑑∨¬𝑇𝑑 ∨I, 3 5 ¬(𝑇𝑑∨¬𝑇𝑑) R, 2 6 ¬𝑇𝑑 ¬E, 3–4,3–5 7 (𝑇𝑑∨¬𝑇𝑑) ∨I, 6 8 ¬(𝑇𝑑∨¬𝑇𝑑) R, 2 9 ((𝑇𝑑∨¬𝑇𝑑)∧¬(𝑇𝑑∨¬𝑇𝑑)) ∧I, 7,8 10 ((𝑇𝑑∨¬𝑇𝑑)∧¬(𝑇𝑑∨¬𝑇𝑑)) ∃E, 1,2–9 11 ¬∃𝑦¬(𝑇𝑦∨¬𝑇𝑦) ¬I, 1–10 B. The following three proofs are missing their commentaries (rule and line numbers). Add them, to turn them into bona fide proofs. 1 ∀𝑥∃𝑦(𝑅𝑥𝑦∨𝑅𝑦𝑥) 2 ∀𝑥¬𝑅𝑚𝑥 3 ∃𝑦(𝑅𝑚𝑦∨𝑅𝑦𝑚) 4 𝑅𝑚𝑎∨𝑅𝑎𝑚 5 𝑅𝑎𝑚 6 𝑅𝑚𝑎 7 ¬𝑅𝑎𝑚 8 ¬𝑅𝑚𝑎 9 𝑅𝑎𝑚 10 𝑅𝑎𝑚 11 ∃𝑥𝑅𝑥𝑚 12 ∃𝑥𝑅𝑥𝑚 1 ∀𝑥(∃𝑦𝐿𝑥𝑦→∀𝑧𝐿𝑧𝑥) 2 𝐿𝑎𝑏 3 ∃𝑦𝐿𝑎𝑦→∀𝑧𝐿𝑧𝑎 4 ∃𝑦𝐿𝑎𝑦 5 ∀𝑧𝐿𝑧𝑎 6 𝐿𝑐𝑎 7 ∃𝑦𝐿𝑐𝑦→∀𝑧𝐿𝑧𝑐 8 ∃𝑦𝐿𝑐𝑦 9 ∀𝑧𝐿𝑧𝑐 10 𝐿𝑐𝑐 11 ∀𝑥𝐿𝑥𝑥 330 NATURAL DEDUCTION FOR QUANTIFIER 1 ∀𝑥(𝐽𝑥→𝐾𝑥) 2 ∃𝑥∀𝑦𝐿𝑥𝑦 3 ∀𝑥𝐽𝑥 4 ∀𝑦𝐿𝑎𝑦 5 𝐿𝑎𝑎 6 𝐽𝑎 7 𝐽𝑎→𝐾𝑎 8 𝐾𝑎 9 𝐾𝑎∧𝐿𝑎𝑎 10 ∃𝑥(𝐾𝑥∧𝐿𝑥𝑥) 11 ∃𝑥(𝐾𝑥∧𝐿𝑥𝑥) C. In §16 problem part A, we considered fifteen syllogistic figures of Aristotelian logic. Provide proofs for each of the argument forms. NB: You will find it much easier if you symbolise (for example) ‘No F is G’ as ‘∀𝑥(𝐹𝑥→¬𝐺𝑥)’. D. Aristotle and his successors identified other syllogistic forms which depended upon ‘existen‑ tial import’. Symbolise each of the following argument forms in Quantifier and offer proofs. ›Barbari. Something is H. All G are F. All H are G. So: Some H is F ›Celaront. Something is H. No G are F. All H are G. So: Some H is not F ›Cesaro. Something is H. No F are G. All H are G. So: Some H is not F. ›Camestros. Something is H. All F are G. No H are G. So: Some H is not F. ›Felapton. Something is G. No G are F. All G are H. So: Some H is not F. ›Darapti. Something is G. All G are F. All G are H. So: Some H is F. ›Calemos. Something is H. All F are G. No G are H. So: Some H is not F. ›Fesapo. Something is G. No F is G. All G are H. So: Some H is not F. ›Bamalip. Something is F. All F are G. All G are H. So: Some H are F. §35. BASIC RULES FOR Quantifier 331 E. Provide a proof of each claim. 1. ⊢∀𝑥𝐹𝑥∨¬∀𝑥𝐹𝑥 2. ⊢∀𝑧(𝑃𝑧∨¬𝑃𝑧) 3. ∀𝑥(𝐴𝑥→𝐵𝑥),∃𝑥𝐴𝑥⊢∃𝑥𝐵𝑥 4. ∀𝑥(𝑀𝑥↔𝑁𝑥),𝑀𝑎∧∃𝑥𝑅𝑥𝑎⊢∃𝑥𝑁𝑥 5. ∀𝑥∀𝑦𝐺𝑥𝑦⊢∃𝑥𝐺𝑥𝑥 6. ⊢∀𝑥𝑅𝑥𝑥→∃𝑥∃𝑦𝑅𝑥𝑦 7. ⊢∀𝑦∃𝑥(𝑄𝑦→𝑄𝑥) 8. 𝑁𝑎→∀𝑥(𝑀𝑥↔𝑀𝑎),𝑀𝑎,¬𝑀𝑏⊢¬𝑁𝑎 9. ∀𝑥∀𝑦(𝐺𝑥𝑦→𝐺𝑦𝑥)⊢∀𝑥∀𝑦(𝐺𝑥𝑦↔𝐺𝑦𝑥) 10. ∀𝑥(¬𝑀𝑥∨𝐿𝑗𝑥),∀𝑥(𝐵𝑥→𝐿𝑗𝑥),∀𝑥(𝑀𝑥∨𝐵𝑥)⊢∀𝑥𝐿𝑗𝑥 F. Write a symbolisation key for the following argument, symbolise it, and prove it: There is someone who likes everyone who likes everyone that she likes. Therefore, there is someone who likes herself. G. For each of the following pairs of sentences: If they are provably equivalent, give proofs to show this. If they are not, construct an interpretation to show that they are not logically equival‑ ent. 1. ∀𝑥𝑃𝑥→𝑄𝑐,∀𝑥(𝑃𝑥→𝑄𝑐) 2. ∀𝑥∀𝑦∀𝑧𝐵𝑥𝑦𝑧,∀𝑥𝐵𝑥𝑥𝑥 3. ∀𝑥∀𝑦𝐷𝑥𝑦,∀𝑦∀𝑥𝐷𝑥𝑦 4. ∃𝑥∀𝑦𝐷𝑥𝑦,∀𝑦∃𝑥𝐷𝑥𝑦 5. ∀𝑥(𝑅𝑐𝑎↔𝑅𝑥𝑎),𝑅𝑐𝑎↔∀𝑥𝑅𝑥𝑎 H. For each of the following arguments: If it is valid in Quantifier, give a proof. If it is invalid, construct an interpretation to show that it is invalid. 1. ∃𝑦∀𝑥𝑅𝑥𝑦∴∀𝑥∃𝑦𝑅𝑥𝑦 2. ∃𝑥(𝑃𝑥∧¬𝑄𝑥)∴∀𝑥(𝑃𝑥→¬𝑄𝑥) 3. ∀𝑥(𝑆𝑥→𝑇𝑎),𝑆𝑑∴𝑇𝑎 4. ∀𝑥(𝐴𝑥→𝐵𝑥),∀𝑥(𝐵𝑥→𝐶𝑥)∴∀𝑥(𝐴𝑥→𝐶𝑥) 5. ∃𝑥(𝐷𝑥∨𝐸𝑥),∀𝑥(𝐷𝑥→𝐹𝑥)∴∃𝑥(𝐷𝑥∧𝐹𝑥) 6. ∀𝑥∀𝑦(𝑅𝑥𝑦∨𝑅𝑦𝑥)∴𝑅𝑗𝑗 7. ∃𝑥∃𝑦(𝑅𝑥𝑦∨𝑅𝑦𝑥)∴𝑅𝑗𝑗 8. ∀𝑥𝑃𝑥→∀𝑥𝑄𝑥,∃𝑥¬𝑃𝑥∴∃𝑥¬𝑄𝑥 36 Derived Rules for Quantifier In this section, we shall add some additional rules to the basic rules of the previous section. These govern the interaction of quantifiers and negation. But they are no substantive addition to our basic rules: for each of the proposed additions, it can be shown that their role in any proof can be wholly emulated by some suitable applications of our basic rules from §35. (The point here is as in §33.) 36.1 Conversion of Quantifiers In §15, we noted that ¬∃𝑥𝒜is logically equivalent to ∀𝑥¬𝒜. We shall add some rules to our proof system that govern this. In particular, we add two rules, one for each direction of the equivalence: 𝑚 ∀𝓍¬𝒜 ¬∃𝓍𝒜 CQ∀/¬∃,𝑚 𝑚 ¬∃𝓍𝒜 ∀𝓍¬𝒜 CQ¬∃/∀,𝑚 Here is a schematic proof corresponding to our first conversion of quantifiers rule, CQ∀/¬∃: 1 ∀𝓍¬𝒜 2 ∃𝓍𝒜 3 𝒜|𝒸↷𝓍 4 ¬(ℬ∧¬ℬ) 5 ¬𝒜|𝒸↷𝓍 ∀E, 1 6 ℬ∧¬ℬ ¬I, 4–5,4–3 7 ℬ∧¬ℬ ∃E, 2,3–6 8 ¬∃𝓍𝒜 ¬I, 2–7 332 §36. DERIVED RULES FOR Quantifier 333 A couple of things to note about this proof. 1. I was hasty at line 9 – officially I ought to have applied ∧E to line 8, obtaining the con‑ tradictory conjuncts in the subproof, and then applied ¬I to the assumption opening that subproof. (But then the proof would have gone over the page.) 2. Note that we had to introduce the new name 𝒸at line 3. Once we did so, there was no obstacle to applying ∀E on that newly introduced name in line 5. But if we had done things the other way around, applying ∀E first to some new name 𝒸, we would have had to open the subproof with yet another new name 𝒹. 3. The sentence ℬcannot contain the name 𝒸if the application of ∃E at line 8 is to be correct. We introduce this arbitrary logical falsehood precisely so we can show that the contradict‑ oriness of our initial assumptions does not depend on the particular choice of name. The alternative would have been to show that the assumption 𝒜|𝒸↷𝓍 leads to logical falsehood, and then applied ¬I – but that would have left the name 𝒸outside the scope of a subproof and would not have allowed us to apply ∃E. A similar schematic proof could be offered for the second conversion rule, CQ¬∃/∀. Equally, we might add rules corresponding to the equivalence of ∃𝓍¬𝒜and ¬∀𝓍𝒜: 𝑚 ∃𝓍¬𝒜 ¬∀𝓍𝒜 CQ∃/¬∀,𝑚 𝑚 ¬∀𝓍𝒜 ∃𝓍¬𝒜 CQ¬∀/∃,𝑚 Here is a schematic basic proof showing that the third conversion of quantifiers rule just intro‑ duced, CQ∃/¬∀, can be emulated just using the standard quantifier rules in combination with the other rules of our system, in which some of the same issues arise as in the earlier schematic proof: 1 ∃𝓍¬𝒜 2 ∀𝓍𝒜 3 ¬𝒜|𝒸↷𝓍 4 ¬(ℬ∧¬ℬ) 5 𝒜|𝒸↷𝓍 ∀E, 2 6 ¬𝒜|𝒸↷𝓍 R, 3 7 ℬ∧¬ℬ ¬E, 4–5,4–6 8 ℬ∧¬ℬ ∃E, 1,3–7 9 ¬∀𝓍𝒜 ¬I, 2–8 A similar schematic proof can be offered for the final CQ rule. 334 NATURAL DEDUCTION FOR QUANTIFIER 36.2 Alternative Proof Systems for Quantifier We saw in §34 that it is possible to formulate alternative proof systems that can nevertheless establish the same arguments are provable in Sentential. The same is true for Quantifier. The idea is to get rid of the rules for one quantifier, retaining the rules governing the other quantifier, but then to take the conversion of quantifier rules as basic. So, for example, we could consider the system which has ∃I and ∃E, and also has CQ∃/¬∀ and CQ¬∀/∃. With these rules, we can emulate ∀E and ∀I. A schematic proof showing how to emulate ∀E using our other basic rules is this: 1 ∀𝓍𝒜 2 ¬𝒜|𝒸↷𝓍 3 ∃𝓍¬𝒜 ∃I, 2 4 ¬∀𝓍𝒜 CQ∃/¬∀,3 5 ∀𝓍𝒜 R, 1 6 𝒜|𝒸↷𝓍 ¬E, 2–5,2–4 A schematic proof emulating ∀I using our other basic rules is trickier. Here it is: 1 𝛤 ⋮ 𝑚 ¬∀𝓍𝒜 𝑚+1 ∃𝓍¬𝒜 CQ¬∀/∃,𝑚 𝑚+2 ¬𝒜|𝒸↷𝓍 𝑚+3 ¬(ℬ∧¬ℬ) ⋮ 𝑛 𝒜|𝒸↷𝓍 Original proof from, 1 𝑛+1 ¬𝒜|𝒸↷𝓍 R, 𝑚+2 𝑛+2 ℬ∧¬ℬ ¬E, 𝑚+3–𝑛,𝑚+3–𝑛+1 𝑛+3 ℬ∧¬ℬ ∃E, 𝑚+1,𝑚+2–𝑛+2 𝑛+4 ∀𝓍𝒜 ¬E, 𝑚–𝑛+3 To understand this schematic proof, what we need to remember is that, in order for the original ∀I rule to apply, we must already have a proof of 𝒜|𝒸↷𝓍 which relies on assumptions 𝛤that do not mention 𝒸at all. The trick is to make use of that proof inside an assumption about an existential §36. DERIVED RULES FOR Quantifier 335 witness. We don’t try to perform that proof to derive 𝒜|𝒸↷𝓍 and then attempt to manipulate ¬∀𝓍𝒜to generate a logical falsehood. Rather, we first assume ¬∀𝓍𝒜, apply quantifier conversion to obtain ∃𝓍¬𝒜, assume that 𝒸witnesses that existential claim so that ¬𝒜|𝒸↷𝓍, and then use our original proof to derive 𝒜|𝒸↷𝓍 at line 𝑛. To avoid problems with the name appearing at the bottom of the existential witness subproof, we perform the same trick of assuming the falsehood of an arbitrary logical falsehood (so long as ℬdoesn’t include 𝒸), and then we manage to derive from 𝛤what we had hoped to: that ∀𝓍𝒜. Key Ideas in §36 ›The derived rules for Quantifier concern the interaction of quantifiers with negation. ›Do not make use of these derived rules unless you are explicitly told you may do so. Practice exercises A. Show that the following are jointly contrary: 1. 𝑆𝑎→𝑇𝑚,𝑇𝑚→𝑆𝑎,𝑇𝑚∧¬𝑆𝑎 2. ¬∃𝑥𝑅𝑥𝑎,∀𝑥∀𝑦𝑅𝑦𝑥 3. ¬∃𝑥∃𝑦𝐿𝑥𝑦,𝐿𝑎𝑎 4. ∀𝑥(𝑃𝑥→𝑄𝑥),∀𝑧(𝑃𝑧→𝑅𝑧),∀𝑦𝑃𝑦,¬𝑄𝑎∧¬𝑅𝑏 B. Show that each pair of sentences is provably equivalent: 1. ∀𝑥(𝐴𝑥→¬𝐵𝑥),¬∃𝑥(𝐴𝑥∧𝐵𝑥) 2. ∀𝑥(¬𝐴𝑥→𝐵𝑑),∀𝑥𝐴𝑥∨𝐵𝑑 C. In §16, I considered what happens when we move quantifiers ‘across’ various connectives. Show that each pair of sentences is provably equivalent: 1. ∀𝑥(𝐹𝑥∧𝐺𝑎),∀𝑥𝐹𝑥∧𝐺𝑎 2. ∃𝑥(𝐹𝑥∨𝐺𝑎),∃𝑥𝐹𝑥∨𝐺𝑎 3. ∀𝑥(𝐺𝑎→𝐹𝑥),𝐺𝑎→∀𝑥𝐹𝑥 4. ∀𝑥(𝐹𝑥→𝐺𝑎),∃𝑥𝐹𝑥→𝐺𝑎 5. ∃𝑥(𝐺𝑎→𝐹𝑥),𝐺𝑎→∃𝑥𝐹𝑥 6. ∃𝑥(𝐹𝑥→𝐺𝑎),∀𝑥𝐹𝑥→𝐺𝑎 NB: the variable ‘𝑥’ does not occur in ‘𝐺𝑎’. When all the quantifiers occur at the beginning of a sentence, that sentence is said to be in prenex normal form. These equivalences are sometimes called prenexing rules, since they give us a means for putting any sentence into prenex normal form. D. Offer proofs which justify the addition of the other CQ rules as derived rules. 342 NATURAL DEDUCTION FOR QUANTIFIER Practice exercises A. Provide a proof of each of the following. 1. 𝑃𝑎∨𝑄𝑏,𝑄𝑏→𝑏=𝑐,¬𝑃𝑎⊢𝑄𝑐 2. 𝑚=𝑛∨𝑛=𝑜,𝐴𝑛⊢𝐴𝑚∨𝐴𝑜 3. ∀𝑥𝑥=𝑚,𝑅𝑚𝑎⊢∃𝑥𝑅𝑥𝑥 4. ∀𝑥∀𝑦(𝑅𝑥𝑦→𝑥=𝑦)⊢𝑅𝑎𝑏→𝑅𝑏𝑎 5. ¬∃𝑥¬𝑥=𝑚⊢∀𝑥∀𝑦(𝑃𝑥→𝑃𝑦) 6. ∃𝑥𝐽𝑥,∃𝑥¬𝐽𝑥⊢∃𝑥∃𝑦¬𝑥=𝑦 7. ∀𝑥(𝑥=𝑛↔𝑀𝑥),∀𝑥(𝑂𝑥∨¬𝑀𝑥)⊢𝑂𝑛 8. ∃𝑥𝐷𝑥,∀𝑥(𝑥=𝑝↔𝐷𝑥)⊢𝐷𝑝 9. ∃𝑥((𝐾𝑥∧∀𝑦(𝐾𝑦→𝑥=𝑦))∧𝐵𝑥),𝐾𝑑⊢𝐵𝑑 10. ⊢𝑃𝑎→∀𝑥(𝑃𝑥∨¬𝑥=𝑎) B. Show that the following are provably equivalent: ›∃𝑥((𝐹𝑥∧∀𝑦(𝐹𝑦→𝑥=𝑦))∧𝑥=𝑛) ›𝐹𝑛∧∀𝑦(𝐹𝑦→𝑛=𝑦) And hence that both have a decent claim to symbolise the English sentence ‘Nick is the F’. C. In §18, I claimed that the following are logically equivalent symbolisations of the English sentence ‘there is exactly one F’: ›∃𝑥𝐹𝑥∧∀𝑥∀𝑦((𝐹𝑥∧𝐹𝑦)→𝑥=𝑦) ›∃𝑥(𝐹𝑥∧∀𝑦(𝐹𝑦→𝑥=𝑦)) ›∃𝑥∀𝑦(𝐹𝑦↔𝑥=𝑦) Show that they are all provably equivalent. (Hint: to show that three claims are provably equi‑ valent, it suffices to show that the first proves the second, the second proves the third and the third proves the first; think about why.) D. Symbolise the following argument There is exactly one F. There is exactly one G. Nothing is both F and G. So: there are exactly two things that are either F or G. And offer a proof of it. E. What condition on the directed graph of a relation corresponds to that relation being an equi‑ valence relation? [Document text truncated for crawler view.]