One Function, Two Roles: A Model-Counting Measure That Also Induces Consequence

CP=PC problem Adam Olszewski August 18, 2025 Abstract This paper revisits the CP=PC problem: the identification of the probability of a conditional with the corresponding conditional probability. Working with a fixed classical probability space and a minimal implicational backbone (rules (p1)–(p4) plus Modus Ponens), we prove a collapse theorem: if a global CP=PC identity is imposed for a single implicational connective and the logic validates a natural symmetry ( A−B⊢B−A ), then the probability assignment degenerates on the positive-probability fragment, trivializing the symmetric extension. We further show that moving to Łukasiewicz logic does not repair the identification: under truth-averaging semantics, the probability of A⇒B is a nonlinear expectation that generally diverges from Pr ( B|A ). The upshot is methodological: CP=PC should not be treated as a law of an implicational connective. Instead, we propose a division of roles: keep the implicational connective for inference, and introduce a dedicated conditional ( B|A )whose semantics guarantees that its probability equals Pr ( B|A ). This reframes the Stalnaker program by separating inferential structure from probabilistic conditionalization and locating CP=PC in the semantics of an internal conditional rather than in the implicational fragment. 1 Background and Problem Setting The long–standing question known as the conditional probability versus the probability of conditionals (abbreviated CP=PC) asks whether, and in which logical settings, one can identify the probability of a conditional with the corresponding conditional probability. In its classical guise (often called Stalnaker’s Thesis), one writes, for formulas A, B with Pr ( A ) > 0, Pr(A→B) = Pr(B|A) = Pr(A∧B) Pr(A). This identity is immediately attractive—linking the inferential reading of a conditional to the calculus of uncertainty—but it notoriously fails for the material conditional and gives rise to well–known “triviality”/collapse phenomena under natural strengthening assumptions.This distinction between the probability assigned to formulas and the dynamics of conditionalization aligns with analyses of probabilistic readings of logical sentences in recent work, e.g., [Ols22] Our starting point is a simple diagnosis: CP=PC mixes two semantically different roles. The connective “ → ” (or a single binary connective “ − ”) is meant to govern inference inside 1 a propositional logic, while Pr ( B|A )is a clause about conditionalization of a measure. Treating these on a par—by imposing a global identity Pr ( A−B ) = Pr ( B|A )—quietly conflates inferential laws with probabilistic update and all but guarantees pathology. We make this tension precise in a minimal and robust setting. Throughout, we fix a classical probability space (Ω ,F,Pr ), interpret formulas as events via two–valued semantics, and read the left–hand side of CP=PC as the same Pr applied to the classical event E ( A−B ). Within a very weak implicational backbone (the logic of order, i.e., rules (p1)–(p4) and Modus Ponens), we prove a collapse theorem: if one also validates a natural global symmetry for the conditional ( A−B⊢B−A ), then CP=PC forces Pr ( E ( A )) = Pr ( E ( B )) for all positive–probability A, B . In short, the probability assignment degenerates on the positive fragment, and the symmetric extension trivializes. This shows that the program “find a minimal logic (via a single connective −) that validates CP=PC” is ill–posed. The positive lesson is equally clear. CP=PC is not a law of an implicational connective; it is a semantic clause for conditionalization. To give CP=PC substantive and non–degenerate content, one must separate roles: keep − as an inferential implication (e.g., the Łukasiewicz residuum) to support reasoning (rules (p1)–(p4) and MP), and introduce a dedicated conditional operator ( B|A )whose semantics is calibrated so that its probability matches Pr ( B|A ) in the intended probabilistic reading. Minimality then concerns the smallest semantic package for (·|·), not a “minimal implicational logic” via −. We also explain why appealing to richer truth–value spaces does not fix the problem. In Łukasiewicz logic L∞ with truth–averaging, the probability of A⇒B becomes a nonlinear expectation Zmin(1,1−v(A) + v(B)) dµ, which in general does not equal Pr ( B|A ). Thus both classically (with 0/1 events) and many–valuedly (with averaged truth), identifying the probability of an implicational connective with conditional probability fails without additional, nontrivial constraints. In summary, our contributions are: • a precise collapse theorem showing that global CP=PC for a single implicational connective (under a fixed classical Pr and symmetry) forces degeneracy; • a no–go corollary: there is no nontrivial “minimal logic via − ” validating CP=PC in the natural lattice [Co, S]; • a constructive reorientation: CP=PC belongs to a dedicated conditional ( ·|· )with minimal semantic clauses, while −remains an inferential implication. These results recast the Stalnaker problem: the right target is not a minimal implicational logic equating two different notions, but a clean separation of inference and conditionalization with a calibrated internal conditional. Definition 1 (Fixed classical interpretation for CP=PC).Let (Ω ,F,Pr )be a fixed probability space. For each formula A in Form− (single binary connective − ), let E ( A ) ∈ F denote its event under classical, two-valued semantics. For Pr(E(A)) >0, Pr(B|A) := Pr(E(A)∩E(B)) Pr(E(A)) ,Pr(A−B) := Pr(E(A−B)). 2 finitely additive probability measure. In this formulation, often referred to as Stalnaker’s Thesis (ST), the connective → is taken as a logically respectable conditional, and Pr (ů) is an ordinary finitely additive probability measure. Van Fraassen (1976) asked what is the smallest logic of conditionals in which CP=PC could hold nontrivially. Van Fraassen (1976) raised the question of what is the smallest logic of conditionals in which CP=PC holds nontrivially. Example 1.1 (Failure of CP = PC for material implication).Consider the conditional A→Bunderstood as material implication, i.e. A→B≡ ¬A∨B. Then Pr(A→B)=1−Pr(A∧ ¬B)=1−Pr(A) + Pr(A∧B). If Pr(A)>0, the conditional probability is Pr(B|A) = Pr(A∧B) Pr(A). Since Pr(A∧B) = Pr(A) Pr(B|A), we obtain Pr(A→B)−Pr(B|A) = (1 −Pr(A))(1 −Pr(B|A)). Hence CP=PC holds only if Pr ( A ) = 1 or Pr ( B|A ) = 1. For example, if Pr ( A ) = 0 . 5and Pr(B|A)=0.5, then Pr(A→B)=0.75 while Pr(B|A)=0.5, so the equality fails. Motivation: van Fraassen’s program and the search space. Building on van Fraassen’s suggestion, we seek the weakest conditional that could sustain a nontrivial CP=PC. Using general methods due to Czelakowski and Olszewski, we conduct the search abstractly over implicational systems: we consider the lattice of consequence operations above the logic of order Co (generated by (p1)–(p4) and Modus Ponens) and ask whether there exists a minimal implicational logic in this lattice that satisfies CP=PC under a fixed classical reading of probability. The logic Cois given by the following structural schemata: (p1) ⊢A−A (p2) A−B, B −C⊢A−C (p3) A, A −B⊢B (p4) A−B, B −A, C −D, D −C⊢(A−C)−(B−D) Definition 2. Alogic of order is any structural consequence operation in [ Co, S ]satisfying (p1)–(p4). Definition 3 (after [CO22], p. 1427).Let ( S, C )be a logic of order. We call C alogic of implication if the logic C+(S) obtained from C by adjoining the symmetry rule ( S )(i.e., A−B⊢B−A ) is inconsistent. If C is a logic of implication, then the connective − is called an implication in C. 3 The Role of Symmetry It is easy to see that one may add to Cothe symmetry rule (A−B)⊢(B−A) obtaining a consistent strengthening of Co . This naturally raises a key question: is such a symmetry admissible if we also insist on CP=PC? The following lemma shows that the answer is negative. Theorem 4 (Collapse under CP=PC and global symmetry).Assume: 1. Cis a logic of order over Form−: it validates (p1)–(p4) and Modus Ponens; 2. the fixed classical interpretation above; 3. for all A, B with Pr(E(A)) >0, Pr(E(A−B)) = Pr(B|A); 4. the symmetric extension C+(S) validates A−B⊢B−Afor all A, B. Then for all A, B with Pr ( E ( A )) > 0one has PrE ( A )  = PrE ( B )  . Thus Pr collapses on the positive-probability fragment. Proof. From symmetry, E(A−B) = E(B−A). Hence Pr(B|A) = PrE(A−B)= PrE(B−A)= Pr(A|B). By Bayes’ theorem, Pr(B|A) = Pr(A|B) Pr(E(B)) Pr(E(A)) . If Pr ( A|B ) > 0, cancellation yields PrE ( A )  = PrE ( B )  . If Pr ( A|B ) = 0, then also Pr ( B|A ) = 0; varying A, B with positive measure propagates equalities across the positive-probability fragment. Hence collapse. Remark: Theorem 4 shows that global symmetry of the conditional is incompatible with Stalnaker’s Thesis in its probabilistic version. Consequently, no logic in the interval [ C_o, S ] that contains symmetry can serve as a minimal logic supporting CP=PC in a nontrivial sense. Corollary 5 (No nontrivial “minimal logic via − ” for CP=PC).Under the hypotheses of Theorem 4, if the symmetric extension C+(S) is consistent, the collapse obtains. In a logic of order, ( p 4) and MP propagate equivalences through implicational contexts; thus C+(S) trivializes. Therefore, no nontrivial logic in [Co, S]validates the global identity Pr(E(A−B)) = Pr(B|A) (Pr(E(A)) >0) by identifying the implicational connective − with a probabilistic conditional. The search for a “minimal logic (via −) satisfying CP=PC” is ill-posed. 4 Remark (Resolving Stalnaker’s problem by separating roles).CP=PC is a semantic clause for conditionalization, not a law of an implicational connective. To retain content and avoid collapse, keep − as an inferential implication (e.g., Łukasiewicz residuum satisfying ( p 1)–( p 4) and MP), and introduce a dedicated conditional ( B|A )with semantics calibrated so that, in the intended probabilistic reading, P  ( B|A )  = Pr ( B|A ). Minimality then concerns the smallest semantic package for (· | ·), not a minimal implicational logic via −. Łukasiewicz Implication and Truth-Averaging Definition 6 (Łukasiewicz Implication).In a many-valued setting with truth values in [0 , 1], the Łukasiewicz implication is defined pointwise by v(A⇒B) := min(1,1−v(A) + v(B)). In the classical 2-valued case ( v∈ { 0 , 1 } ), this coincides extensionally with material implication. Łukasiewicz Family and the Implicational Core Across all Łukasiewicz logics (finite Ł n with n≥ 2and the standard Ł ∞ ), ⇒ is the residuum of the Łukasiewicz t-norm with truth-function v ( A⇒B ) = min (1 , 1 −v ( A ) + v ( B )). This definition is uniform in n (the value sets differ: Γ n vs. [0 , 1]). In the pure implicational language, validity in Ł ∞ is minimal by inclusion: every implicational tautology of Ł∞is valid in each Łn, and Th⇒(Ł∞) = \ n≥2 Th⇒(Łn). Consequently, any negative result for CP=PC formulated via ⇒ in Ł ∞ transfers to every finite Łn. Notation: Pvs. Pr Let ( W, Σ , µ )be a probability space, and let v : {A, B, C, . . . }×W→ [0,1] be a many-valued evaluation with the standard clauses: v(¬A, w)=1−v(A, w), v(A⇒B, w) = min(1,1−v(A, w) + v(B, w)). •P(A)denotes the truth-average (expected truth degree): P(A) = ZWv(A, w)dµ(w). •Pr ( E )denotes the classical probability of an event E∈ Σ. When a formula A is interpreted as the crisp event EA⊆W (i.e., v ( A, · ) ∈ { 0 , 1 } ), we write Pr ( A ) := µ ( EA ) and Pr(B|A) := Pr(A∧B)/Pr(A)for Pr(A)>0. If v(·, w)∈ {0,1}, then P(A) = ZWv(A, w)dµ(w) = µ(EA) = Pr(A). In general, however, Pand Pr capture different notions: the former averages degrees of truth, the latter measures the size of crisp events. Related discussions of assigning probabilities to logical sentences, and the pitfalls of conflating different probabilistic notions, appear in [Ols22] 5 Łukasiewicz Logic Ł ∞ The language has ¬,⇒ with Modus Ponens and substitution. The standard [0,1]-semantics is v(¬A) = 1 −v(A), v(A⇒B) = min(1,1−v(A) + v(B)). An equivalent Hilbert system uses the axiom schemata: (L1) (A⇒B)⇒((B⇒C)⇒(A⇒C)), (L2) ((A⇒B)⇒B)⇒((B⇒A)⇒A), (L3) (¬A⇒ ¬B)⇒(B⇒A). This system is complete for the standard [0,1]-semantics. Proposition 7 (Truth-averaging does not repair CP=PC for ⇒ ).In many-valued, truthaveraging semantics, there is no general identity P(A⇒B) = Pr(B|A) that holds for all models unless additional, nontrivial constraints are imposed (see [Háj98] for the standard Łukasiewicz semantics and discussions of probabilistic interpretations). The Łukasiewicz truth-function v(A⇒B) = min(1,1−v(A) + v(B)) is nonlinear in (v(A), v(B)), so the truth-average P(A⇒B) = ZWmin1,1−v(A, w) + v(B, w)dµ(w) cannot, in general, collapse to the ratio form that defines classical conditional probability Pr(B|A) = Pr(E(A)∩E(B)) Pr(E(A)) (Pr(E(A)) >0), which depends only on the A -region and normalizes by Pr ( E ( A )). Truth-averaging mixes all worlds (including partial v ( A, w ) ∈ (0 , 1)) without this normalization, hence P( A⇒B )  = Pr(B|A)in general. Example 1.2 (Two-world counterexample).Let W = {w1, w2} with µ ( {w1} ) = µ ( {w2} ) = 1/2. Define v(A, w1)=1, v(B, w1)=0.4; v(A, w2)=0.2, v(B, w2) = 0.2. Then v(A⇒B, w1) = min(1,1−1+0.4) = 0.4, v(A⇒B, w2) = min(1,1−0.2+0.2) = 1, so P(A⇒B) = 1 2·0.4 + 1 2·1=0.7. 6 Under the classical reading with E(A) = {w:v(A, w)=1}={w1}, E(B) = {w:v(B, w)=1}=∅, we have Pr(B|A) = Pr(E(A)∩E(B)) Pr(E(A)) = 0, hence P( A⇒B )=0 . 7  =0= Pr ( B|A ). The mismatch is driven by nonlinearity and the lack of A-conditional normalization in the truth-average. Definition 8 (Internal conditional operator).Extend the language by a binary operator ( · | · ), read “the conditional.” An interpretation ( W, µ, v )internalizes conditional probability if, for all A, B with Pr(A)>0, P((B|A)) = Pr(B|A), and (·|·)satisfies intended sanity laws. Theorem 9 (How to obtain CP=PC).Let − be an implicational connective (e.g., Łukasiewicz implication). If the language is enriched by an internal conditional ( · | · )and the semantics enforces P((B|A)) = Pr(B|A)for all A, B with Pr(A)>0, then CP=PC holds for the internal conditional. Proof. Immediate from the semantic clause for ( · | · ). The implicational connective − supplies the inferential backbone (rules and MP), while ( · | · )internalizes conditionalization: P((B|A)) = Pr(B|A). Corollary 10 (Division of roles).Łukasiewicz implication can serve as the implicational connective, but CP=PC should be formulated for the dedicated conditional ( · | · )whose semantics guarantees P(( B|A )) = Pr ( B|A ). Replacing ( B|A )by A⇒B invalidates CP=PC. 2 Main Result We work in the lattice interval [ Co, S ]of implicational logics over the language Form− with a single binary connective − and Modus Ponens. CP=PC is interpreted on a fixed classical probability space (Ω,F,Pr): for all A, B, Pr(A−B) := Pr(E(A−B)),Pr(B|A) := Pr(E(A)∩E(B)) Pr(E(A)) if Pr(E(A)) >0. 7 Negative result: collapse Theorem 11 (Collapse under CP=PC and global symmetry).Assume: 1. Cvalidates (p1)–(p4) and MP; 2. the fixed classical interpretation above; 3. for all A, B with Pr(E(A)) >0,Pr(E(A−B)) = Pr(B|A); 4. the symmetric extension C+(S) validates A−B⊢B−Afor all A, B. Then for all A, B with Pr(E(A)) >0, Pr(E(A)) = Pr(E(B)), i.e., the probability assignment collapses on the positive-probability fragment. Corollary 12 (No nontrivial “minimal logic via − ” for CP=PC).Under the hypotheses of Theorem 11, if C+(S) is consistent, the collapse obtains. In a logic of order, (p4) and MP propagate equivalences through implicational contexts; hence C+(S) trivializes. Therefore, there is no nontrivial logic in [Co, S]validating Pr(E(A−B)) = Pr(B|A) by identifying − with a probabilistic conditional. The search for a “minimal logic (via − ) satisfying CP=PC” is ill-posed. Positive result: division of roles Theorem 13 (Division of roles and how to obtain CP=PC).Let − be an implicational connective (e.g., the Łukasiewicz residuum) validating (p1)–(p4) and MP. Extend the language by a conditional operator (·|·), and let (W, Σ, µ, v)be a semantics with P(A) = ZWv(A, w)dµ(w). If, for all A, B with Pr(E(A)) >0, P((B|A)) = Pr(B|A), and ( · | · )satisfies the intended sanity laws (certainty/zero cases), then CP=PC holds for ( · | · ). In particular, CP=PC is achieved not by identifying A−B (e.g., A⇒B ) with conditionalization, but by calibrating (B|A)so that its probability matches Pr(B|A). Scope and qualified cases where CP=PC may hold. Our no-go result targets global identifications of the form Pr ( E ( A−B )) = Pr ( B|A )across the full language and arbitrary models, under a fixed classical measure and natural structural principles. It does not preclude qualified, local uses of CP=PC under additional constraints. Typical safe zones include: (i) restricting scope to non-nested occurrences of the conditional and to antecedents/events within a fixed algebra where the semantic reading of A−B is externally calibrated to conditionalization; (ii) conditional logics or probabilistic frameworks that limit the domain of 8 application (e.g., only for a designated class of conditionals or almost-everywhere clauses) so that ratio-style normalization is preserved; and (iii) systems that introduce a dedicated conditional ( B|A )with an explicit semantic clause ensuring P(( B|A )) = Pr ( B|A ), while keeping − purely inferential. In short, CP=PC can function reliably in controlled fragments or with a purpose-built conditional, but not as a global law tying an implicational connective to classical conditional probability. Related Work The classical program of identifying the probability of a conditional with the corresponding conditional probability—often labeled Stalnaker’s Thesis (ST)—goes back to early discussions of conditionals in formal epistemology and philosophical logic (e.g., [Sta70]). While the thesis is prima facie compelling, it clashes with standard truth-functional treatments: for the material conditional, one obtains Pr ( A→B )=1 −Pr ( A ) + Pr ( A∧B ), which in general differs from Pr ( B|A )unless degenerate cases obtain, as is well known in the probability-of-conditionals literature. A substantial body of work documents “triviality” and collapse phenomena that arise when one attempts to validate global identifications of this kind. Van Fraassen’s influential analysis [Fra76] explicitly raised the question of a smallest (nontrivial) logic in which ST could hold and showed severe constraints on any such attempt. By seeking CP=PC via a single conditional connective, this program conflates an inferential operator with probabilistic conditionalization; our collapse theorem shows that, under a fixed classical measure and global symmetry, this identification alone forces degeneracy over the minimal logic-of-order backbone. Subsequent impossibility and triviality results—under diverse technical assumptions about the conditional, background algebra, and admissible probability assignments—reinforced the message that unconditional, global identifications force degeneracy or require nonclassical probabilistic frameworks (e.g., Popper functions) or restricted scopes (e.g., limited nesting, almost-everywhere clauses) [Lew76; Spo83; Edg95]. From the perspective of conditional logics, various systems accommodate fragments of ST only under significant restrictions (e.g., on the form of antecedents, on admissible updates, or via distinct semantic tiers for conditionals versus events). Contemporary surveys emphasize that mixing the inferential role of a conditional with the semantics of conditionalization is the key methodological pitfall: the two belong to different layers of theory and should not be identified without an explicit bridge principle [Edg25; CHN11; Fra24]. The present paper contributes a streamlined no-go result in a minimal implicational setting: within the logic-of-order backbone (rules (p1)–(p4) and MP) and a fixed classical reading of Pr on both sides of CP=PC, adding global symmetry triggers a probabilistic collapse, and the symmetric extension trivializes. This sharpens the negative conclusions of the classical literature by isolating the exact structural features that force degeneracy in the implicational language. On the positive side, our “division of roles” reframes ST as a semantic clause for a dedicated conditional operator ( · | · )—rather than a law of an implicational connective—aligning with approaches that separate inference from probabilistic update and calibrate the internal conditional directly to Pr(B|A). 9