Logical Localism in he Con ex
o Combining Logics
Ca los Beni o-Monsal o
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Tesi doc o al
Logical Localism in he
Con ex o Combining Logics
Au o /a:
Ca los Beni o-Monsal o
Di ec o /a: D . José Ma ínez Fe nández
Di ec o /a: D . Elia Za dini
Tu o /a: D . Manuel Ga cía-Ca pin e o
P og ama de Doc o a : Ciencia Cogni i a y Lenguaje
Facul a de Filoso ia
Ene o de 2023
Logical Localism in he Con ex
o Combining Logics
Ca los Beni o-Monsal o
A hesis submi ed in ul ilmen o he
equi emen s o he deg ee o Doc o o
Philosophy
Supe iso s:
D . José Ma ínez Fe nández
D . Elia Za dini
PhD p og am: Cogni i e Science and Language
Uni e si y o Ba celona
Philosophy Depa men
Ca los Beni o-Monsal o: Logical Localism in he Con ex o Combining
Logics, 2023
SUPERVISORS: D . José Ma ínez Fe nández and D . Elia Za dini
TUTOR: D . Manuel Ga cía-Ca pin e o
This disse a ion has been possible hanks o he FI-DGR-EMC/2199/2017
g an o he Gene ali a de Ca alunya; he FPU(18/06283) g an o he
Spanish Minis y o Uni e si ies; he mobili y g an (EST21/00505) o
bene icia ies o he T aining P og amme o Academic S a (FPU); he
p ojec on Localism and Globalism in Logics and Seman ics
(FFI2015-70707-P) om he Spanish Minis y o Economy and
Compe i i eness and he p ojec ‘Wo lds and T u h Values: Challenges o
Fo mal Seman ics (MUNVAL)’(2019PIDPID-107667GB-I00) om he
Spanish Minis y o Science and Inno a ion.

Abs ac
Logical localism is a claim in he philosophy o logic s a ing ha di e en
logics a e co ec in di e en domains. The e a e di e en ways in which
his hesis can be mo i a ed and I will explo e he mos impo an ones.
Howe e , localism has an ob ious and majo challenge which is known as
‘ he p oblem o mixed in e ences’. The main goal o his disse a ion is o
sol e his challenge and o ex end he solu ion o he ela ed p oblem o
mixed compounds o ale hic plu alism. My app oach in o de o o e a
solu ion is one ha has no been conside ed in he li e a u e as a as I am
awa e. I will s udy di e en me hods o combining logics, concen a ing on
he me hod o jux aposi ion, by Joshua Schech e , and I will y o sol e he
p oblem o mixed in e ences by making a ine ansla ion o he a gumen s
and using combina ion mechanisms as he c i e ion o alidi y. One o he
mos in iguing aspec s o he disse a ion is he syne gy ha is c ea ed be-
ween he philosophical deba e and he echnical me hods wi h he p oblem
o mixed in e ences a he cen e o ha syne gy. I hope o show ha no
only he philosophical deba e bene i s om he me hods o combining logics,
bu also ha hese me hods can be de eloped in new and in e es ing ways
mo i a ed by he philosophical p oblem o mixed in e ences. The p oblem
sugges s ha he e a e ele an in e ac ions be ween connec i es, jus i ied
by he philosophical conside a ions o concep ualising di e en logic sys-
ems, ha he me hods o combining logics should allow o eme ge. The
ecogni ion o his ac is wha d i es he imp o emen s on he me hod o
jux aposi ion ha I de elop. Tha is, in o de o allow o he eme gence o
desi able in e ac ion p inciples, I will p opose al e na i e ways o combining
logic sys ems -speci ically classical and in ui ionis ic logics- ha go beyond
he s anda d o combina ions, which is based on minimali y condi ions so
as o a oid he so-called collapse heo ems.
i
Acknowledgemen s
I am qui e a ac ed o he idea ha all in elligence is collec i e in elligence,
in he sense ha he e is no such hing as an indi isible uni o in elligence
ha we can pinpoin . So, al hough he neu ons o my b ain ha e been
he ones s uggling o i e in he igh pa hs in o de o ul ima ely p oduce
his disse a ion, o he neu ons om o he b ains ha e ce ainly been o an
in aluable help o he co ec i ings o occu . I am, he e o e, hank ul o
e e yone who, in some way o ano he , in e ac ed wi h me.
Among hose in e ac ions, he i s pe son ha showed me a ue passion
o logic and philosophy was Jesus Ma i La azabal. He was my logic eache
and my bachelo ’s hesis supe iso , bu , mos o all, he was a men o and
a iend. I canno exp ess how indeb ed I am o him. I jus would like o
lea e, o whom i may conce n, a glimpse o his mind: ‘Philosophy is an
in ellec ual ac i i y ha will o en lead you o loneliness, bu his loneliness
is necessa y i you wan o hink ca e ully, eely, and no be a ec ed by all
he mess. Howe e , you mus know ha you a e no alone and ha o he
people also deeply ca e abou he same hings as you do. Ul ima ely, he owl
o Mine a always sp eads i s wings wi h he alling o he dusk o b ing o de
o chaos’. Hope ully, his and o he u u e wo ks will be a wo hy es amen
o his in ellec ual hones y and passion.
A clea p oo o he in luence ha Jesus Ma i had on me and my ca ee
is ha he in oduced me o my supe iso s, José Ma ínez and Elia Za dini.
I me hem in a wo kshop ha Jesus Ma i was o ganizing a Donos ia. I
was my las yea as an unde g adua e and I was in he p ocess o deciding
wha o do nex . Since Jesus Ma i knew hem bo h and highly app ecia ed
and espec ed hem, he encou aged me o go and alk o hem and seek hei
ad ice. I ended up s udying he mas e in Analy ic Philosophy, whe e Pepe
augh me he cou se on Philosophical Logic and supe ised my Mas e ’s
Thesis, wi h an eye al eady on a PhD. He in oduced me o e e y opic ha
ii
his disse a ion is conce ned wi h and has guided me h ough e e y h ead
o ge a whe e I am, while gi ing me enough eedom. I am deeply hank ul
o all his. Also, o he ime, e o and igou ha he has dedica ed o
imp o ing his disse a ion and o gi ing me he con idence on mysel and
my wo k when I mos needed i . I eel lucky o ha e had he oppo uni y
o wo king oge he wi h someone whom I ha e admi ed since he was my
eache .
Elia became my co-supe iso igh on ime, bu I ce ainly wish he would
ha e joined us be o e. Since he i s alk ha I wi nessed a Donos ia, I
ha e hough ha he is one o he sha pes and mos b illian philosophe s
ha I know. Ve y ew o he imes I ha e me someone who is able o o e
you some help ul and deep insigh s in almos any opic, om cuisine o
philosophical logic. When Pepe in o med me abou he possibili y o ha ing
Elia as a supe iso I was eally exci ed, bu also a li le bi in imida ed. In
he end, he has been an indispensable pa o he las pe iod o my wo k,
when mos o he cooles ideas ha e o igina ed. No only has he gi en me
ou s anding insigh s o imp o e my esea ch, bu also has had he mos
kind and suppo i e wo ds o inspi e me a he lowes poin s o he p ocess.
Coming om him, hey mean he wo ld o me.
The e a e many o he people om he academia ha ha e been impo an
in all hese yea s. I would like o hank e e y membe o Logos o being so
close and kind. I has been an honou o be pa o he bes esea ch g oup
in analy ic philosophy and be su ounded by he bes minds in he ield. I
especially hank And ea Ri adulla, wi h whom I s a ed he p ocess o he
PhD a e we inished he mas e . I know she has a b illian u u e ahead as
a esea che i she wan s o.
I would also like o men ion and hank he people ha ha e pa icipa ed
in he Philosophical Logic eading g oup o ganized by Pepe. Ob iously, many
o he pape s ha we ead and discussed we e ela ed o my esea ch, so I
ha e g ea ly bene i ed om ou discussions. I am pa icula ly g a e ul o
Se gi Oms, S en Rosenk anz, Niccolò Rossi and Pila Te és.
Du ing my PhD I comple ed a esea ch s ay a he Buenos Ai es Logic
G oup. Gi en he si ua ion wi h COVID-19, i was qui e di icul o make
he s ay possible and I had o pos pone i almos wo yea s. Bu , inally,
a he end o Feb ua y 2022 I could a el o A gen ina. The e we e many
easons o choosing his pa icula esea ch g oup. Fi s o all, hey a e an
ou s anding eam, o med by e y alen ed and b illian esea che s who a e
able o sys ema ically publish in he op le el in e na ional jou nals like a
iii
well-oiled machine. Second, I was lucky enough o know some o he membe s
o he g oup be o e my s ay. I me Damian Szmuc a he ‘PhDs in Logic’
celeb a ed in P ague and had him a e wa ds in he iny gues oom o my
la a Ba celona (so y o ha ). I also knew Lucas Rosenbla , who was
a Logos when I joined and le sho ly a e o e u n home, and Edua do
Ba io, whom I had me in a wo kshop ha Pepe o ganized. Damian and
Edua do we e e y suppo i e and help ul wi h all he pape wo k p e ious o
he s ay and, mos impo an ly, keeping he hope ha I could inally a el.
I am deeply g a e ul o ha . Las ly, le me be comple ely hones , I wan ed
o a el o A gen ina o climb and ek a Pa agonia, which I de ini ely did.
Those mon hs wi h he Buenos Ai es Logic G oup we e a gi and com-
ple ely su passed my expec a ions. I wan o hank all he membe s o he
g oup o making me eel a home and o all he g ea discussions which
we e a ue inspi a ion o my wo k. Special hanks go o Edua do, Damian,
Lucas, Na alia Buaca and Paula Teijei o. I canno hank Ma iela Rubin
(Ma u) enough o e e y hing she did o me. You a e my a ou i e pe son
in he Ame ican con inen .
I ha e had he chance o p esen ing pa s o my disse a ion in a ious
enues. I would like o hank he people ha ha e a ended and helped me
wi h ques ions o commen s. Among all, le me men ion Pablo Cob e os,
who migh no know ha he has always gi en me wise ad ise whene e we
ha e encoun e ed and An onio Yus e-Ginel who has become a iend wi h
whom I can sha e philosophy, logic, climbing and bee s.
O cou se, he e is li e ou side he academia oo and mos o he people
ha a e essen ial o me and o wha I am a e ou side s. I am deeply g a e ul
o many o my iends: Adu , Raúl, Fynn, Ál a o, And ea, Gonzalo, Lau a
and Guille mo. Also o he iends o my pa ne who ha e welcomed me and
become iends o mine oo: Alejand o, Ma ía, Ma io, Sand a, Mau o, Lau a
and Sa a. And o he ones who know me since I was a 5 yea s old kid: Iñaki
A bide, Julen Ga mendia and Julen A u i. The o he day I ealized ha
you ha e ne e asked me wha my hesis was abou , and I am uly g a e ul
o ha . You a e in many o my dea es and happies memo ies.
I canno o ge o hank my amily oo: Julio, Ca men, Jo ge and Ma ia,
hanks o all he wa m h and suppo . I know ha my g and a he , Eugenio
Monsal o, will also be p oud and shed some ea s when eading his name,
despi e no knowing wha o say when asked wha his g andson does o a
li ing. I am lucky o ha e a big amily and he e a e many o he membe s o
i o whom I eel deeply g a e ul. I you a e eading his, I hope you know who
i
In oduc ion
The i s p emiss and he conclusion a e mo al claims and, by assump ion,
a e go e ned by logic L1. The second p emiss, on he o he hand, is a
claim abou he obse able ac o someone (ac ing as an execu o o he
powe o he U.S. go e nmen ) pou ing wa e o e a clo h co e ing someone
else’s ace. So, i belongs o he domain o middle-sized objec s o e en s
o malized by L2. Ne e heless, he a gumen seems o be in ui i ely alid.
Bu , he ques ions a e, which is he logic ha accoun s o he alidi y o
he a gumen ? And which a e he p inciples o easoning ha allow us o
eason ac oss domains? These a e no mino issues o a philosophical hesis
claiming ha he co ec applica ion o logic sys ems is local, since easoning
ac oss domains is some hing qui e pe asi e.
Thus, he philosophical hesis ha I am going o analyse is logical localism
and he challenge o localism ha I will y o add ess, he main challenge o
localism, indeed, is he p oblem o mixed in e ences. Logical localism is o en
conside ed as a o m o logical plu alism. Tha is, as a hesis claiming ha
he e is mo e han one legi ima e ela ion o logical consequence. These ypes
o philosophical posi ions ega ding logic we e no a se ious op ion be o e he
appea ance and consolida ion o non-classical logics a he beginning o he
20 h cen u y. These logics ejec some o he classical p inciples, he eby
allowing o ques ion he gene al alidi y o uniqueness o classical logic.
The eme gence o non-classical logics -no ably, ele an , in ui ionis ic and
many- alued logics- ga e ise o some o he mos impo an ques ions in he
philosophy o logic: a e al e na i e logics eally ‘logic’? Do hey dese e
he same s a us as classical logic? Is he e jus one co ec logic o a e
he e many? Some o he p oponen s o al e na i e logics con inued de end-
ing monis heses, now agains classical logic and in a ou o one o he
al e na i es. Michael Dumme , o example, a gued o he co ec ness o
in ui ionis ic logic, claiming e en ha classical connec i es had no meaning
a all.
Howe e , he ac o ha ing a plu ali y o sys ems and o being able
o check ou ha some o hem appea ed o ha e applica ions whe e hey
excelled he mos , made plu alis ic p oposals mo e and mo e plausible. These
p oposals ha e had a ious o ms and I will p esen he mos ele an ones
o my pu poses la e on.
The p oblem o mixed in e ences is he challenge a ound which he philo-
sophical and he echnical aspec s o he disse a ion e ol e. A e p esen -
ing he philosophical amewo k I will in oduce he challenge o localism
and I will mos ly ocus on he e sion ha Chase W enn o e s, which is, o
3

In oduc ion
my mind, he bes and mos de ailed p esen a ion o he p oblem o mixed
in e ences o logical localism.
My app oach in o de o o e a solu ion o he challenge is one ha has
no been explo ed, o conside ed, in he li e a u e as a as I am awa e. The
s a egy will be o explo e di e en me hods o combining logics, concen-
a ing specially on he me hod o jux aposi ion, by Joshua Schech e , and
ying o sol e he p oblem o mixed in e ences by making a ine ansla ion
o he a gumen s and using combina ion mechanisms as he c i e ion o a-
lidi y. Bu combina ions o logics b ing abou o he po en ially p oblema ic
issues ega ding in e ac ions be ween connec i es and, in he limi case, hey
b ing abou wha a e known as collapse heo ems. These p oblems will be
analysed and I will de elop he combina ion mechanisms ha ing in mind ha
he collapse has o be a oided.
One o he mos in iguing aspec s o he disse a ion is he syne gy ha
is c ea ed be ween he philosophical deba e and he echnical me hods wi h
he p oblem o mixed in e ences a he cen e o ha syne gy. I hope o
show ha no only he philosophical deba e bene i s om he me hods o
combining logics, bu also ha hese me hods can be de eloped in new and
in e es ing ways mo i a ed by he philosophical p oblem o mixed in e ences.
The p oblem sugges s ha he e a e ele an in e ac ions be ween connec-
i es, jus i ied by he philosophical conside a ions o concep ualising di -
e en logic sys ems, ha he me hods o combining logics should allow o
eme ge. The ecogni ion o his ac is wha d i es he imp o emen s on he
me hod o jux aposi ion ha I de elop. Tha is, in o de o allow o he
eme gence o desi able in e ac ion p inciples, I will p opose al e na i e ways
o combining logic sys ems, speci ically classical and in ui ionis ic logics,
ha go beyond he s anda d o combina ions, which is based on minimali y
condi ions so as o a oid collapse heo ems.
Besides he echnical de elopmen s, he new combina ion mechanisms
in oduce u he sub le ies wi hin he philosophical deba e a ound localism
and allow o a mo e ine-g ained analysis o al e na i e kinds o localisms and
e en o domains. Tha is, he analysis sugges s ha he e a e some p ope ies
o he domains ha a e no cap u ed only by he logic ha co esponds o
a gi en domain. Ins ead, hose p ope ies a e e ealed when he domain
in e ac s wi h ano he , i.e. when easoning ac oss domains. Thus, i migh
be he case ha easoning ac oss he domain o middle-sized objec s and he
ma hema ical domain equi es di e en in e ac ions om hose equi ed by
easoning ac oss he domains o middle-sized objec s and e hics, e en i one
4
In oduc ion
belie es ha he logic o he ma hema ical and he e hical domain is he
same, say, in ui ionis ic logic. Thus, I hink ha combining logics can b ing
us close o a solu ion o he p oblem o mixed in e ences and, mo eo e , help
us disce n wi h mo e accu acy he in icacies o he philosophical deba e.
The s uc u e in which I will un a el hese ideas is he ollowing: in
chap e 2, I p esen he concep ual amewo k in which logical localism is
going o be cha ac e ized. In his concep ual amewo k, he opics o logical
and ale hic plu alisms play a c ucial ole, so i will be ele an o cla i y wha
hey a e and how hey ela e o localism. Then, I will conclude he chap e
by p esen ing he main challenge o logical localism, namely, he p oblem o
mixed in e ences and I will go h ough some o he mos no able a emp s
o sol e i . The las pa will be de o ed o Chase W enn’s e sion o he
p oblem, which is he mos elabo a e e sion o he p oblem in he li e a u e.
Chap e 3 is mean o es ablish he connec ion be ween he p oblem o
mixed in e ences and he ield o combining logics. In o de o jus i y ha
b idge, I s a by ocusing on some logical concep ions abou mixed easoning
and I in oduce he no ions o ‘in e ac ion p inciples’, ‘b idge p inciples’ and
‘collapse heo ems’, exis en in he li e a u e, and elabo a e hem. Then, I
p esen some popula me hods o combining logics, paying special a en ion
o he one upon which I am going o build my own mechanisms, namely,
jux aposi ion.
In chap e 4 I de elop a solu ion o he p oblem o mixed in e ences.
Fi s , I app oach he p oblem by applying he o iginal me hod o jux a-
posi ion and discuss i s i ues and po en ial sho comings. Based on he
limi a ions o he me hod, I a gue ha he combina ion mechanisms should
allow o mo e in e ac ion be ween he logics being combined, in o de o ge
a mo e encompassing solu ion o he p oblem o mixed in e ences. The e-
o e, I p opose some new al e na i e mechanisms in which he desi ed b idge
p inciples can na u ally eme ge in he combina ion p ocess.
Finally, chap e 5 concludes he disse a ion by looking in o some p omis-
ing u u e wo k. As I will y o show, he analysis sugges s ha he e a e
many in e es ing philosophical and logical/algeb aic issues o be so ed ou
in he icini y.
5
Chap e 2
Logical Localism
2.1 Wha is Logical Localism? A concep ual
amewo k
In his sec ion I will y o se he amewo k o he discussion ha conce ns
me o he disse a ion. As I al eady ad anced in he in oduc ion, he opic
o localism is e y closely ela ed o, i no included in, he deba e a ound
logical plu alism, which is a cen al deba e wi hin he philosophy o logic.
Gi en i s cen ali y, he e a e many o he issues in he ield ha a ec he
discussion, such as he deba e a ound which he chie aim o logic is, he
no ma i e s a us o logic, he sense in which logic is o mal, whe he logical
consequence should be spelled ou model- heo e ically, p oo - heo e ically o
kep p imi i e, and so on and so o h.
Those opics a e huge and each one o hem dese es mo e han a disse a-
ion, as shown, o ins ance, by J. G. MacFa lane wi h his b illian hesis on
o mali y (MacFa lane (2000)). Howe e , i is beyond he scope o my hesis
o deepen on hose opics and I eckon ha i would no e en be help ul o
ul illing my mo e modes and speci ic aim, namely, o wo k ou me hods o
combining logics in o de o ind possible solu ions o he p oblem o mixed
in e ences ha challenges he philosophical posi ion o localism.
Ne e heless, in cha ac e izing logical localism hose c ucial issues will
ine i ably come up, since, as I said, hey a ec how one hinks abou he
na u e o logic1. Thus, I will y o make explici , whe e applicable, wha I
1No ice ha he e a e, a leas , h ee senses o ‘logic’: a consequence ela ion, a pa -
icula logic sys em o he discipline. I belie e ha he e will be no con usion h oughou
6
Logical Localism
am assuming o how hose issues migh a ec he cha ac e iza ion o localism
ha I will be de eloping.
Le me, hen, be o e going in o he de ails o localism, s a by laying
down a gene al assump ion I will make conce ning he chie aim o logic.
Following he mains eam adi ion and, mo e conc e ely, G aham P ies
(P ies (2006)) and J. C. Beall & G. Res all (Beall and Res all (2000,2001,
2006)), which a e a guably he mos impo an and in luen ial de ende s
o monism and plu alism, espec i ely, I will assume ha he chie subjec
ma e o logic is logical consequence. Beall and Res all e y nicely pu i a
he beginning o hei book Logical Plu alism:
Logical consequence is he hea o logic; i is also a he cen e o phi-
losophy and many heo e ical and p ac ical pu sui s besides. Logical
consequence is a ela ion among claims (sen ences, s a emen s, p opo-
si ions) exp essed in a language. An accoun o logical consequence
is an accoun o wha ollows om wha -o wha claims ollow om
wha claims (in a gi en language, whe he i is o mal o na u al). An
accoun o logical consequence yields a way o e alua ing he connec-
ions be ween a se ies o claims- o , mo e speci ically, o e alua ing
a gumen s. (Beall and Res all,2006, p. 3)
And also in Beall and Res all (2000):
The chie aim o logic is o accoun o consequence, o say, accu a ely
and sys ema ically, wha consequence amoun s o, which is no mally
done by speci ying which a gumen s (in a gi en language) a e alid.
All o his, a leas oday, is common g ound.
(Beall and Res all,2000, p. 475)
Simila ly, P ies w i es:
Wha is logic? Uncon o e sially, logic is he s udy o easoning.[...]The
s udy o easoning, in he sense in which logic is in e es ed, conce ns
he issue o wha ollows om wha . Less c yp ically, some hings
-call hem p emises- p o ide easons o o he s -call hem conclu-
sions.[...]The ela ionship be ween p emise and conclusion in each case
is, colloquially, an a gumen , implica ion, o in e ence. Logic is he
in es iga ion o ha ela ionship. A good in e ence may be called a
alid one. Hence, logic is, in a nu shell, he s udy o alidi y.
(P ies ,2006, p. 176)
he ex since I usually use hose mo e speci ic wo ds ins ead o he mo e gene al ‘logic’.
In any case, he con ex should su ice in o de o disambigua e he gene ic uses.
7
Logical Localism
Thus, I will adhe e o his widely accep ed adi ion. Logic is he sys em-
a ic s udy o wha ollows om wha ; o which p emises s and in he logical
consequence ela ion o which conclusions. Tha is, he aim o logic is o
accoun o he alidi y o a gumen s.
This is no , howe e , he only exis ing posi ion ega ding wha logic is
abou . J. an Ben hem, o ins ance, has a mo e ‘libe al’ concep ion o logic
and a gues ha he iew o logic as being abou consequence ela ions may
ha e had some sense when i was hough o p o ide he ounda ions o
ma hema ics. Bu , since he 1930s he ield has changed and b oadened i s
scope. Logic is now, an Ben hem claims, abou de inabili y, compu a ion
and mo e ( an Ben hem,2008, p. 183). Indeed, an Ben hem de ends ha
he main issue o logic is “ he a ie y o in o ma ional asks pe o med by
in elligen in e ac ing agen s, o which in e ence is only one among many,
in ol ing obse a ion, memo y, ques ions and answe s, dialogue, o gene al
communica ion” ( an Ben hem,2008, p. 182).
I do no ha e any pa icula conce n wi h his concep ion and I belie e
ha he discussion on whe he logic is X o Y is no e y ui ul. Howe e ,
i does a ec he plausibili y o plu alism and localism how b oad he domain
o applica ion o logic is. To pu i simply, i logic is abou so many hings
beyond logical consequence, as an Ben hem claims, i will be mo e p obable
ha he e is mo e han one ‘co ec logic’ and i will be less likely ha one
logic does all he job.
When I use ‘co ec logic’ I mean, oughly, he logic ha is mos ui ul,
mos adequa e o he da a, o e all simples , e c. Thus, in his case, I do no
aim o imply any me aphysical iew on whe he he e is, o no , an objec i e
eali y ha logic seeks o cap u e. Thus, i is a sense o ‘co ec ness’ a ailable
bo h o a ealis and an ins umen alis (in he sense o Haack (1978)).The
di e ence be ween he ealis and he ins umen alis a ises, hough, wi h
espec o which he T ue logic is. Since o he ins umen alis he e is no
ex a-sys emic alidi y, bu jus alid-in-L, he e is no T ue logic. Fo he
ealis on he o he hand he in a-sys emic no ion o alidi y is ying o
cap u e ‘ eal’ ex e nal alidi y. Bu , hen, i is logically possible o concei e
a wo ld in which he co ec logic, a e weighing he heo e ical i ues, is
no he T ue logic. Imagine, o ins ance, ha he T ue logic is one in which
e e y in e ence ule has some coun e examples. S ill, i could be he case
ha he mos ui ul and adequa e logic o be applied, i.e., he co ec logic,
was one wi h uni e sal in e ence ules.
On his no e, and in o de o con inue laying down some assump ions, i
8

Logical Localism
is ele an o he discussion ha we cla i y a bi mo e he no ion o ‘applica-
ion’. I is la gely uncon en ious, nowadays, ha he pu e/applied dis inc ion
holds when speaking abou logics. No ably, P ies (2003,2006) jus i ies he
dis inc ion by d awing an analogy wi h geome y and a i hme ic (an idea due
o Łukasiewicz). The analogy ies o es ablish ha , in he same way ha
he e a e many pu e geome ies (Euclidean, Riemannian, Lobache skian,
e c.), he e a e many pu e logics (classical, in ui ionis ic, pa aconsis en ,
connexi is , e c.). These a e “well-de ined ma hema ical s uc u e[s] wi h
a p oo - heo y, model heo y, e c.” (P ies ,2006, p. 195). We de ine hem,
we s udy hei p ope ies, p o e esul s abou hem, ela ions be ween hem,
and so on. Wi h espec o pu e logics, much like pu e geome ies, he e is no
doub abou plu alism. I is an uncon en ious ac ha he e is a plu ali y
o pu e logics.
The e a e, howe e , o he aspec s o doing logic ha ha e o do wi h
he applica ion o pu e logics o di e en domains and p oblems. This is a
common p ac ice wi hin philosophical logic, o ins ance, whe e pu e logics
a e o en applied in o de o deal wi h pa adoxes, o sys ema ically accoun
o easoning abou knowledge, necessi y, obliga ion, mo ali y, e c. Bu , i
is also he case, as R. Cook poin s ou , ha “logics ha e been cen al o
he s udy o a numbe o phenomena, including many ha ha e, a bes ,
an indi ec connec ion o human easoning such as elec onic ci cui design,
da abase managemen , and in e ne secu i y”(Cook,2010, p. 494).
Despi e he a ie y o applica ions ha logics ha e been used o , hough,
some people a gue ha he e is a p i ileged applica ion o logic, a canonical
applica ion, which is he analysis o easoning. Quo ing P ies ,
he mos impo an and adi ional applica ion o a pu e logic [is]
he canonical applica ion: he applica ion o a logic in he analysis
o easoning [...]. The cen al pu pose o an analysis o easoning is
o de e mine wha ollows om wha -wha p emises suppo wha
conclusions- and why. An a gumen whe e his is, in ac , he case is
alid. (P ies ,2006, p. 196)
Thus, he mo e in e es ing plu alis hesis would be wi h espec o i s
canonical applica ion. I is no enough he me e exis ence o a a ie y o pu e
logics, no e en he ac ha he e migh be a ious i al logics compe ing o
being he bes codi ica ion o easoning (wha P ies (2006) calls heo e ical
plu alism). As Cook s esses,
9
Logical Localism
i logical plu alism is o be a subs an ial and con o e sial hesis, some-
hing mo e mus be in ended. Tha some hing mo e is he no ion o
logical consequence - ha is, a logic is ‘co ec ’, o ‘accep able’, e c., i
and only i i is a co ec (o accep able, e c.) codi ica ion o logical
consequence. The idea ha he philosophically p ima y (bu ob iously
no only) goal o logical heo izing is o p o ide a o mal codi ica ion
o logical consequence in na u al language aces back (a leas ) o he
wo k o Al ed Ta ski. (Cook,2010, p. 495)
Le us, he e o e, assume ha he chie subjec ma e o logic is logical
consequence and ha logic’s canonical applica ion is he analysis o eason-
ing. One could hink ha a legi ima e way o a guing o plu alism would
be o emphasize ha he e a e di e en ways o accoun ing o logical con-
sequence, e. g. model- heo e ically, p oo - heo e ically o ega ding i as a
p imi i e no ion. I ha e ied o a gue o he model- heo e ic app oach in
Beni o-Monsal o (2022) and I belie e ha he deba e is philosophically sub-
s an ial, bu his will no be ele an o ou discussion he e, since i is no he
sou ce o plu ali y ha is in e es ing o p esen pu poses. We a e in e es ed
in he hesis ha he e a e di e en logics which cap u e di e en legi i-
ma e ela ions o logical consequence when canonically applied ( ega dless o
whe he he logic is p esen ed model- heo e ically o p oo - heo e ically).
Logical localism is one o hose plu alis heses, bu logical plu alism,
unde s ood à la Beall and Res all, o ins ance, is also a posi ion in a ou o
ha kind o plu ali y. Thus, wha I wan o do, now, is o s a singling ou
and delimi ing he logical localis posi ion. This is a delica e ask, because
logical plu alism is no an unequi ocal hesis and encompasses many posi ions
unde he same naming. Bu I am less in e es ed in he exege ical wo k
han in p o iding a concep ual map o he a ailable heo e ical posi ions
and add essing whe e localism s ands in ha map. This is why I will be
aking some au ho s almos like a che ypical igu es o he di e en posi ions.
Le me p oceed, hen, o cla i y how I unde s and logical plu alism, bes
ep esen ed by Beall and Res all, in opposi ion o logical monism, which has
P ies as one o i s mos popula de ende s. This opposi ion is also in e es ing
because bo h sides ag ee on he chie subjec ma e o logic, namely, logical
consequence.
10
Logical Localism
2.1.1 Logical Plu alism
I sounds like a uism, bu i is always nice o emembe ha in o de
o e en concei e logical plu alism we equi e o know o he exis ence o
di e en , al e na i e logic sys ems. Howe e , i is also always amusing o
emembe Kan ’s hesis on New onian physics and A is o elian logic, i. e.
syllogis ic. Some au ho s e e now o Hugh McColl as he i s p oposal
o wha could be conside ed a logical plu alis philosophy o logic. I is no
coincidence ha he was also a pionee in he de elopmen o non-classical
logics, including, many- alued, p obabili y, ele an and connexi e logics (see
Rahman and Redmond (2008)). His plu alism, hough, seems o be close
o wha I call ‘localism’ han o Beall and Res all’s plu alism, and i would
ce ainly all unde wha Cook, in Cook (2010), calls ‘ ela i ism’. This is
wha Rahman and Redmon commen on his espec :
MacColl’s philosophy is a kind o ins umen alism in logic which led
him o se he basis o wha migh be conside ed o be he i s plu-
alism in logic. The poin condensed in he epig aph amoun s o he
ollowing: i could well be ha in some con ex s o easoning he exis -
ing a gumen a ion demands a ype o logic which is no applicable in
o he s. When cons uc ing a symbolic sys em o a pa icula ype o
logic, he co esponding exp essions in use should he e o e be aken
in o ca e ul conside a ion.
(Rahman and Redmond,2008, pp. 540–541)
So, we can say ha McColl’s plu alism is one ha claims ha he ap-
plica ion o logic is ela i e o ‘con ex s o easoning’ and, he e o e, ha
he e migh be di e en co ec logics o di e en con ex s. In his sense,
he applica ion o logic is local, despi e logic being a heo y o easoning,
because his a ies om con ex o con ex . In o he wo ds: he canonical
applica ion o logic is he analysis o easoning, bu he e a e, so o speak,
‘sub-canonical applica ions’.
A second miles one in he his o y o logical plu alism is Rudol Ca nap
and his p inciple o ole ance. In The Logical Syn ax o Language (1937)
Ca nap says:
In logic he e a e no mo als. E e yone is a libe y o build his own
logic, i.e. his own language, as he wishes. All ha is equi ed o him
is ha , i he wishes o discuss i , he mus s a e his me hods clea ly,
and gi e syn ac ical ules ins ead o philosophical a gumen s.
(Ca nap,1937, §17)
11
Logical Localism
A c ucial poin ha esul s om his libe y o building ou own logic
om ou own language, is ha he e is no ex e nal logical eali y ha o ces
a pa icula One T ue Logic. The esul is a kind o con en ionalism, simila
o ha o McColl, by which one ge s di e en co ec 2logics by a ying
he linguis ic amewo k. Cook summa ises Ca nap’s plu alism in a e y
illumina ing way:
Ca nap’s iew ce ainly amoun s o a o m o logical plu alism. [...]
Bu his is a dependen plu alism, esul ing om an unde lying el-
a i ism – ha is di e en logics esul om a ying he language in
ques ion (i is wo h no ing ha Ca nap does no ad oca e plu alism
wi hin a amewo k – di e en linguis ic amewo ks migh be go -
e ned by di e en logics, bu wi hin a pa icula amewo k he e is
a single logic ha co ec ly codi ies he (in e nal) logical consequence
ela ion o ha amewo k). Thus, Ca nap’s ole ance amoun s o
a e sion o logical plu alism, bu no a e sion o SLP [Subs an ial
Logical Plu alism]. (Cook,2010, pp. 497–498)
Wha Cook means by ‘subs an ial logical plu alism’, is a plu alism such
ha he language is kep ix, he dema ca ion o he logical/nonlogical o-
cabula y is also ixed and, ye , he e a e (a leas ) wo logical consequence
ela ions ha cap u e wo legi ima e di e en senses o ‘ ollows om’. This
is, in ac , wha Beall and Res all wan o achie e. So, le me p esen he
undamen als o hei p oposal.
2.1.2 Beall and Res all’s Logical Plu alism
As we said be o e, Beall and Res all adhe e o he mains eam adi ion
o aking logic o be abou he consequence ela ion. Abou sys ema ically
de e mining wha ollows om wha . The accoun o logical consequence
ha Beall and Res all (2000,2001,2006) deploy is a gene aliza ion o he
adi ional seman ic one, i.e. o Ta ski’s accoun o logical consequence. They
call i Gene alised Ta ski Thesis (GTT):
•(GTT) An a gumen is alidxi and only i , in e e y casexin which he
p emises a e ue, so is he conclusion.
2I would say ha ‘co ec ’ o Ca nap means jus he logic (o he logics) ha be e
ul ils some p agma ic goal se by he subjec .
12
Logical Localism
me con as localism wi h globalism o he sake o making localism mo e
clea and dis inc .
Localism agains Globalism
The opposi e hesis o localism, as I will unde s and he no ions, is no
monism bu globalism. Globalism is he hesis acco ding o which he appli-
ca ion o logic is global, i.e. independen o he subjec -ma e o he domain
o easoning o which i applies. The e o e, globalism s ays wi hin he o ho-
doxy o logic in ha i e ains he alleged opic-neu ali y o logic and seems
o suppo he adi ional idea o he uni e sali y o eason.
I ag ee wi h some o he in ui ions ha sus ain he idea o uni e sali y o
eason, bu I hink i is an o e simpli ica ion o in e om ha deside a um
ha , since logic is a heo y o co ec easoning and easoning is uni e sal,
hen logic has o be applied globally in o de o i o be a candida e o
being co ec . One could push back by a guing ha eason can be uni e sal,
in he sense o being a human acul y ha we can employ i espec i e o
subjec -ma e , in any domain o inqui y, despi e ac ual easoning aking
a ious o ms and p inciples in di e en domains. To gi e an analogy, he e
is some hing uni e sal o e e yone ge ing a gold medal a he Olympics,
namely, hey did be e and won o e hei compe i o s. Bu , a he same
ime, winning has many o ms and i ma e ializes in di e en ways, whe he
you win a gold medal in climbing o in pole aul , o ins ance.
The e is an impo an empi ical a gumen on he side o he globalis
hough. We do seem o eason ac oss domains, om mo al and ac ual
p emises o mo al conclusions, om ma hema ical and physical o physical,
easoning abou he in e play o mac o and mic o-objec s, and so on and so
o h. This is, o my mind, he mos impo an empi ical ac ha localism
has o accoun o . I is, indeed, he undamen al ac ha sus ains he
p oblem o mixed in e ences. Thus, e en i he applica ion o logic is local,
we mus be able o gi e an explana ion o how hose domains migh in e ac .
Mo eo e , I eckon ha he e is also impo an empi ical e idence in
a ou o localism and challenging globalism, namely, he amoun o logical
sys ems ha a e used and a e cons an ly being de eloped, no jus o any
applica ion, bu o canonical applica ions, o sub-canonical applica ions, like
o malising easoning abou ague phenomena, easoning wi h inconsis en
in o ma ion, easoning abou u h, quan um-mechanical phenomena, e c.
E en wi hin he domain o ma hema ics, ha one could ega d as a single
19

Logical Localism
homogeneous domain, he e a e p oposals o employing di e en logics in
di e en b anches, like pa aconsis en , cons uc i e o classical ma hema ics
(see, o ins ance, P ies (2019); Shapi o (2014a,b)). In ac , he e is no
eason o suppose ha we ha e eached he ul ima e s age o a ie ies o
easoning and ha , he e o e, no mo e new domains o easoning will appea ,
which, po en ially, could equi e e en new logical sys ems o be sys ema ised.
Thus, I belie e ha globalism aces an impo an challenge oo, simila
o he scope p oblem ha adi ional heo ies o u h ha e o ace, namely,
ha ‘ he plausibili y o each in la ionis ’s candida e o he [ u h] p ope y
Fdi e s ac oss di e en egions o discou se’ Pede sen and W igh (2018).
Simila ly, globalism has o ace he empi ical ac ha he plausibili y o any
logic sys em, L, di e s ac oss di e en egions o discou se o domains o
easoning.
Localism as a o m o Rela i ism
Following wi h he cha ac e iza ion o localism, I belie e i is in e es ing o
place localism wi hin he excellen axonomy o ‘plu alis ’ heo ies ha Cook
(2010) p o ides. Cook’s iew on ela i ism is such ha ,
One is a ela i is abou a pa icula phenomenon X i and only i
one hinks ha he co ec accoun o X is a unc ion o some dis inc
se o ac s Y. Thus, ela i ism abou X amoun s o accep ance o he
ollowing schema:
The co ec accoun o X is ela i e o Y.
(Cook,2010, pp. 492–493)
In his sense, i is clea ha localism is a o m o ela i ism, namely, one
holding ha he co ec accoun o ‘ ollows om’ o ‘consequence’ o ‘ alid’,
is ela i e o domains (lea ing open whe he domains a e indi idua ed by
subjec -ma e , by he on ological p ope ies o he objec s wi hin ha do-
main, o wha no ).
Howe e , localism is no a ype o Subs an ial Logical Plu alism, unde
Cook’s ca ego iza ion, since i is di icul o hold ha , i he logic o ea-
soning abou u h is a pa aconsis en logic, he logic o e alua i e discou se
is in ui ionis ic and he logic o easoning abou middle-size phenomena
is classical logic, o ins ance, hose logics’ co esponding connec i es, say,
nega ions, a e going o ha e he same meaning. Whe he one hinks ha he
20
Logical Localism
meaning o he connec i es is de e mined by he ules o in e ence o by hei
u h-condi ions o sa is ac ion condi ions, i is easonable o assume ha
he meanings will change7. The e o e, since Cook’s concep ion o Subs an ial
Logical Plu alism implies ha he plu alism a ises wi hin a ixed language
and a ixed in e p e a ion o he logical/non-logical di ide, localism canno
be a Subs an ial Logical Plu alism.
This should no be in e p e ed as localism no being a subs an ial o
in e es ing hesis. Cook makes his clea , bu I guess ha he li e a u e
has had a endency o ocus mo e on he ype o plu alism claiming ha
‘ he e is no genuine deba e be ween ad oca es o di e en all-pu pose logics’
(Field,2009, p. 344). The eason o his bias migh be ha he mos popula ,
de ailed and bes de eloped accoun o logical plu alism is Beall and Res all’s
and hey explici ly say ha hei in ended plu alism is no a ela i ism.
The plu ali y o logics, acco ding o hem, a e applicable o any domain o
subjec -ma e .
Nei he should we conclude ha e e y o m o ela i ism is a localism. A
popula ela i is iew is de ended by Achille Va zi in Va zi (2002). Acco d-
ing o him, which conclusions a e in a logical consequence ela ion o which
p emises depends on how one concei es he logical/non-logical di ide. Fo
ins ance, whe he o no one ega ds iden i y as a logical symbol a ec s he
possible models ha one will accep . I one akes i as a logical symbol, she
will accep only he models ha do jus ice o he in ended in e p e a ion o
he p edica e. O he wise, one could accep u he models.
Rega dless o whe he we ag ee wi h Va zi o no , i seems clea o me ha
his ela i ism is no a o m o localism. I is no ha he plu ali y o accoun s
o he logical consequence ela ion depends on a domain o easoning, a
subjec -ma e , some on ological p ope y, e c. His ela i ism does no ha e
any hing o do wi h domains, bu wi h he speci ic se o logical cons an s
ha one adop s.
To conclude his sec ion on he cha ac e iza ion o logical localism, le
me nex e e o some localis p oposals in he li e a u e. We ha e al eady
said ha McColl’s and Ca nap’s plu alisms can be aken as localis accoun s.
Now, I will mo e on o explain how localism has been concep ualized by o he
au ho s.
7Howe e , I am going o discuss mo e his issue once my imp o emen s on jux aposi ion
ha e been p esen ed, since I belie e ha he me hod migh allow o some in e p e a ions
compa ible wi h localism wi hou meaning- a iance o connec i es.
21
Logical Localism
Some Localis p oposals in he li e a u e
As I de ined i abo e, logical localism is he hesis s a ing ha di e en logic
sys ems a e equi ed in o de o sys ema ically accoun o co ec easoning
in di e en domains. The main di e ence be ween he al e na i e p oposals
esides in how one indi idua es he domains.
In he case o McColl, he domains a e con ex s o easoning and his
con en ionalism seems o lea e qui e some eedom wi h espec o wha
de e mines a con ex o easoning. I migh be a pa icula p oblem, a opic,
e c. The e is no speci ic ea u e ha o ces a new con ex o easoning, simply
some con ex s equi e di e en ools in o de o accoun o wha ollows om
wha in ha con ex . Le me gi e some o he examples o localism now.
New on da Cos a’s localism8
New on da Cos a’s localism is also based on he obse a ion ha he e a e
a plu ali y o logics because di e en domains ask o di e en logic sys ems.
Mo eo e , in da Cos a’s iew, he e is a conc e e eason o his domain
a ia ion, namely, ha easoning abou di e en kinds o objec s equi es
di e en logics. The e o e, domains o easoning a e de e mined by he on-
ological p ope ies o he objec s belonging o he domain. An example o
hese kinds o objec s and he a ia ion o logics ha hey equi e is ha o
mac o-objec s e sus quan um objec s:
[...] I is clea ha o common objec s, such as a book o a pe -
son, [...] [∀x(x=x)] applies appa en ly wi hou a single impo an
di icul y. Any pe son wha e e , say A, e en hough hey unde go
mul iple modi ica ions in he cou se o hei li e, emains in a ce ain
sense iden ical wi h hemsel : A = A . Tha appea s e en mo e clea ly
as conce ns abs ac objec s: o example, he equali y 1 = 1 seems
e iden and indispu able [...]. Howe e , hings a e no as simple as
nai e ealism would lead one o belie e. In quan um physics, elemen-
a y pa icles, acco ding o all appea ances, ansg ess he p inciple
o iden i y. Thus Sch ödinge a i med ha he ela ion o iden i y
be ween pa icles was de oid o sense: “i is no a p oblem ha de-
pends on ou capaci y o p o ing he iden i y in ce ain cases and
ou incapaci y o p o ing i in o he cases. I is ce ain ha he issue
8I ha e no been able o ind he ele an ma e ial in English and, since my Po uguese
and F ench a e no good enough, I will ely on P ies ’s in e p e a ion o da Cos a and in
some ansla ions o English made by himsel .
22
Logical Localism
o ‘iden i y’ is, eally and uly, de oid o sense”. I could be ha
he posi ion o Sch ödinge is accep able only empo a ily and ha
he u u e will show us ha he is mis aken. None heless he ac is
ha quan um physics shows he possibili y o dialec ising he idea o
iden i y, and consequen ly, he e y law ha co esponds o i .
(da Cos a,1997, p. 120) as ci ed om (P ies ,2000, p. 440)
On da Cos a’s iew, hen, he on ological di e ences be ween objec s
en o ce di e en logical p ope ies oo. Bu he accoun goes e en u he ,
as P ies obse es:
Da Cos a’s plu alism is mo e adical han I ha e so a indica ed,
hough. He en isages no only ha objec s o di e en kinds may
ha e di e en logical p ope ies, bu ha di e en logical ope a o s
may also need o be used in easoning abou di e en kinds o objec s.
Thus, o example, classical nega ion is app op ia e o dealing wi h
pla onic objec s, and in ui ionis nega ion is app op ia e o dealing
wi h men al cons uc ions. (P ies ,2006, p. 198)
Then, da Cos a’s localism could imply9 ha he e is meaning- a iance
bo h a connec i e le el and also wi h espec o logical consequence and
alidi y. Tha is, di e en kinds o objec s cons i u e di e en domains and
he easoning wi hin hose domains a ies because he di e se on ological
p ope ies o he objec s ca y o e di e en logical p ope ies. The e o e,
di e en logical connec i es and consequence ela ions a e equi ed in each
domain in o de o p o ide heo ies o co ec easoning wi hin hose domains.
We will soon see ha a coun e a gumen ha P ies gi es agains da
Cos a’s localism cons i u es one o he majo challenges ha I wan o ad-
d ess wi h he aid o he me hods o combining logics. Bu , he e is an issue
ha P ies does no men ion and ha o me seems p oblema ic o , a leas ,
dubious. This is he domain indi idua ion jus in i ue o he on ological
p ope ies o objec s. I hink he e a e good easons, gi en by D. Edwa ds in
Edwa ds (2018), o ins ance, o a gue agains ha c i e ion. To illus a e,
ake he s a emen s ‘ he numbe πis i a ional’ and ‘ he numbe πis beau-
i ul’. Bo h s a emen s a e abou he same objec , namely, he ma hema ical
objec π. Howe e , while we would ce ainly assign he i s p oposi ion o
9I ake he p ecau ion o using ‘could’ because I ha e no ound ou whe he da Cos a
de ends ha he e is an ex a-sys emic no ion o alidi y. I he e is one, hen i could be
a gued ha he meaning o ‘ alid’ does no eally change.
23
Logical Localism
he ma hema ical domain, we will mos likely no assign he second one o
he domain o ma hema ics, bu o he aes he ical domain. As Edwa ds
sugges s,
i is no wha a sen ence is abou ha we should be conside ing o
domain membe ship, i is a he how he hing he sen ence is abou
is ep esen ed, by he use o a p edica e o a ibu e a p ope y.
(Edwa ds,2018, p. 96)
Tha is, domain membe ship o a p oposi ion seems o ha e o do wi h how
he objec is ep esen ed, i.e. wi h he p edica e, a he han wi h he ob-
jec . Ano he example migh come om ague phenomena. I is well known
ha agueness has been a e ile ield o p oposing al e na i e non-classical
logics, bu i seems ha he objec s o which we migh p edica e ague p op-
e ies will appea in classical domains oo. We migh say, o ins ance, ‘Pu in
is bald’, bu also ‘Pu in is he p esiden o Russia’. Howe e , i da Cos a’s
example ega ding quan um objec s and iden i y is accep ed, ha would con-
s i u e an example in which he kind o objec would be enough o indi idua e
a domain. So, maybe he igh c i e ion o domain indi idua ion has o go
beyond da Cos a’s and Edwa ds’ accoun s and make oom o bo h. Domain
indi idua ion and he en o cemen o a di e en logic sys em wi hin a domain
could be a ma e o kinds o objec s and he way objec s a e ep esen ed.
Dide ik Ba ens’ localism
In Ba ens (1990), D. Ba ens p esen s a se ies o a gumen s agains he idea
o global pa aconsis ency, i.e. he idea ha he co ec logic is pa aconsis en
and i s applica ion is global because inconsis encies a e inhe en o human
easoning. His iew, agains wha he akes o be a dogma ic a i ude owa ds
pa aconsis ency, is ha ‘each logic [has] a pa icula se o domains in which
i is adequa e’ (Ba ens,1990, p. 209).
The way in which we should concei e o hese domains, ollowing P ies ’s
e minology in P ies (2006), is as ‘p oblem-sol ing si ua ions’. One o hose
si ua ions ha Ba ens uses as example is a me a- heo e ical si ua ion. Ac-
co ding o Ba ens, despi e he e being domains in which a pa aconsis en
logic is equi ed, e. g. inconsis en domains,
classical desc ip ions o many logical sys ems, including pa aconsis-
en ones, a e possible, and [...] whene e his is so, pa aconsis en
desc ip ions a e oo poo o be adequa e. The same applies o o he
24

Logical Localism
domains: whene e a domain is consis en , a pa aconsis en desc ip-
ion is incomple e. (Ba ens,1990, p. 227)
The eason why pa aconsis en logics a e oo poo o ce ain si ua ions
o domains, is because hey lack he exp essi e s eng h ha classical logic
has. Fo ins ance, acco ding o Ba ens, he e is a sense o ‘ ejec ion’ ha
he classical nega ions exp ess ha canno be cap u ed by a pa aconsis en
nega ion ha allows bo h Aand ¬pA o be ue. In his sense, domain
indi idua ion seems o be some hing mo e p agma ic, ha ing o do wi h he
u ili y o a logic o a gi en p oblem, simila ly o McColl’s posi ion, han
he mo e subs an ial c i e ion o wha cons i u es a domain ha da Cos a
de ends.
S ewa Shapi o’s ela i ism as localism
We ha e b ie ly men ioned abo e S. Shapi o’s localism. I hink i is
wo h commen ing on i he e because i ep esen s an in e es ing case. So,
p ima acie, one would hink ha , o mos o he possible sound accoun s
ega ding domain indi idua ion, ma hema ics would be a domain. Whe he
you cha ac e ize domains by he kinds o objec s, by he p edica es used
o ep esen ing he objec s, by he ype o easoning o p oblem-sol ing
si ua ions, i seems ha ma hema ics is a good candida e o being a domain.
In spi e o all ha , Shapi o a gues o localism wi hin ma hema ics.
As wi h he p e ious p oposals, he e a e many de ails ha I am no
in e es ed in add essing he e. My pu pose is o illus a e di e en exis ing
app oaches o localism depending on how one unde s ands domains. So, wi h
Shapi o, he way in which he sepa a es he domains o applica ion o logics is
by aking domains as s uc u es, whe e a s uc u e is a legi ima e b anch o
ma hema ics. Hence, acco ding o Shapi o, ‘logical consequence and alidi y
a e ela i e o s uc u e. Tha is, one canno say wha he p ope logic is un il
one says which s uc u e is being discussed’ (Shapi o,2014a, p. 321). And,
he poin is ha Shapi o a gues o a a ie y o in e es ing and applicable
ma hema ical b anches, i.e. s uc u es, ha equi e di e en logics.
Fo ins ance, in ui ionis ic analysis is inconsis en i one has classical logic
as i s backg ound logical heo y. Bu , he e is no eason o dismiss such
heo y. I is a legi ima e b anch o ma hema ics wi h po en ial applica ions
and, ye , i equi es us o d op some classical p inciples (mos amously,
excluded middle) on pain o inconsis ency.
25
Logical Localism
Tha is no he only example ha Shapi o poin s ou . Simila si ua ions
a e ound wi h espec o o he in ui ionis ic heo ies and also pa aconsis en
ones. Thus, Shapi o’s localism is a localism al eady wi hin ma hema ics ha ,
po en ially, could be ex ended i hose ma hema ical s uc u es a e applied,
say, in some physical heo y. Jus wi hin ma hema ics, hen, we ha e legi -
ima e s uc u es some o which equi e classical logic, o he s in ui ionis ic
logic and some o he s pa aconsis en logic.
Pede sen and Lynch: om ale hic plu alism o localism
Nikolaj J. L. L. Pede sen and Michael P. Lynch a gue o a o m o local-
ism ha Lynch names ‘domain-speci ic logical plu alism’ (DLP), in Pede sen
(2014) and Lynch (2008), o ins ance. I g oup hem oge he because o how
simila ly mo i a ed hey a e, namely, hey y o connec ale hic plu alism
wi h domain-speci ic logical plu alism. Tha is, he idea ha u h a ies
ac oss domains ( he p ope y o being ue o he way p oposi ions a e ue,
o ins ance) wi h he idea ha his a ia ion o u h o ces a a ia ion on
logic. He e is how Pede sen ames i :
ea u es o he u h p ope y o a domain play a c ucial ole in de-
e mining he logic o he domain. In pa icula , he u h p ope ies
o some domains ha e he ea u e o being epis emically cons ained
and go hand in hand wi h cases ha deli e in ui ionis ic logic, while
he u h p ope ies o o he domains ha e he ea u e o being epis-
emically uncons ained and go hand in hand wi h cases ha deli e
classical logic. In sho , ale hic plu alism yields logical plu alism.
(Pede sen,2014, p. 262)
Lynch, simila ly, a gues ha , al hough he e is no di ec una oidable a -
gumen om u h plu alism o domain-speci ic logical plu alism, one can
indi ec ly make he connec ion like his:
I he e is mo e han one way o mani es u h, and some o he
mani es ing p ope ies a e epis emically de ined p ope ies like supe -
wa an , and some no , hen di e en domains will admi o di e en
mani es a ions o he consequence ela ion. And his means, among
o he hings, ha a gumen o ms ha a e alid in some domains may
no be so in o he s.All his o cou se, assumes ha he e is mo e han
one way o play he u h- ole. I he e is no , hen he e may s ill
be mo e han one consequence ela ion, bu his will p esumably be
mo i a ed by o he hings han a iew abou he na u e o u h.
(Lynch,2008, pp. 134–135)
26
Logical Localism
So, acco ding o Pede sen and Lynch, he hing ha cha ac e izes a domain
and dis inguishes i om o he domains is how he u h p ope y is mani-
es ed wi hin ha domain o which u h p ope y a domain has. This, in
u n, migh be he ac o ha de e mines co ec easoning wi hin a domain,
making i possible ha co ec easoning and, he e o e, logic, a ies om
one domain o ano he .
Since he a gumen om ale hic plu alism o localism has been one o
he mos popula in he li e a u e, especially in he li e a u e ha ing o do
wi h heo ies abou u h, we will di e in o he de ails gi en by Pede sen and
Lynch la e on in sec ion 2.1.4. Le me inish, now, by adding ha his way o
indi idua ing domains, by looking a he u h p ope y, could be de eloped
mo e by linking u h p ope ies wi h on ological/me aphysical p ope ies,
as Pede sen (2014) does, o ins ance. I we ollow his pa h, we migh end
up in some hing no so a om da Cos a’s idea o domains indi idua ed by
kinds o objec s. Thus, Pede sen a gues ha
i one g an s ha he e is bo h a co espondence domain and a su-
pe wa an domain [i.e., a domain wi h an epis emically cons ained
u h p ope y], one should also g an ha he e is a domain wi h e-
spec o which one is commi ed o me aphysical ealism and a domain
o which one is no hus commi ed. Bu , i one is no commi ed
o me aphysical ealism, one mus be commi ed o some o he me a-
physical iew on he en i ies in he ele an domain. All in all, his
amoun s o a o m o me aphysical plu alism—o a he e y leas , i
seems o be e y congenial o a o m o me aphysical plu alism.
(Pede sen,2014, p. 271)
The e o e, domains would be ul ima ely indi idua ed by he on ologi-
cal/me aphysical p ope ies, which would hen make u h mani es in di -
e en ways and, inally, his di e en u h mani es a ions would impose
di e en consequence ela ions.
One can see ha he e a e impo an localis p oposals in he li e a u e.
Despi e he ac ha he philosophy o logic has no been e y in e es ed in
logical localism (maybe, as I said, because he dominan posi ion has been
Beall and Res all’s logical plu alism and o he au ho s ha e ied o challenge
i ) we can ind, al eady om he beginning o plu alis ic p oposals abou
logic, his o ical and ele an con ibu ions wi h a localis spi i . Mo eo e ,
he li e a u e on heo ies o u h and, in pa icula , on ale hic plu alism, has
clea ly had a endency o a ou localis implica ions wi h espec o logic,
27
Logical Localism
as he na u al philosophical posi ion o an ale hic plu alis . Howe e , one
could be a localis abou logic wi hou commi ing o plu alism abou u h.
In he nex sec ion, I wan o p opose a ede ini ion o he heo e ical op-
ions and he concep ual map ega ding he deba e a ound logical plu alism,
b oadly unde s ood.
Rede ining he concep ual map
Wha I ha e p esen ed so a yields a pic u e o he deba e a ound logical
plu alism ha allows, I belie e, o a new concep ualiza ion. I ha e ied o
ame he localis hesis by con as ing i o globalism and dis inguishing i
om Beall and Res all’s plu alism, which, in u n, I ha e con as ed wi h
monism.
Thus, he localis hesis s a es ha he e is a mul iplici y o domains o
discou se, wi h possibly di e en c i e ia o co ec easoning, ha equi e
adop ing di e en logics. Globalism, on he o he hand, is he posi ion de-
ending ha he applica ion o logic is global, in he sense ha logical laws
and alid a gumen s mus be applicable ega dless o he con en , he subjec -
ma e o he domain o easoning. Unde he assump ion ha he e is a
canonical applica ion o logic and ha his is he applica ion o easoning
(assump ion ha I made ollowing P ies (2006)), his means ha local-
ism implies, con a y o globalism, ha he e a e sub-canonical applica ions
(since di e en domains o easoning equi e di e en logical heo ies).
Now, i we allow plu alism and monism o be heses abou he plu ali y o
uniqueness o legi ima e logics wi h espec o a (sub)canonical applica ion,
we ge a iche concep ual amewo k. As a as I know, Haack (1978) is
he i s (and only?) o concep ualize some hing simila (bu only combining
localism/globalism wi h plu alism and wi h sligh ly di e en senses). In his
case, we ge ou heo e ical posi ions:
•Global Monism: he e is jus one co ec logic and i is neu al wi h
espec o he domain o which i is applied.
•Global Plu alism: he e a e a a ie y o logics ha a e equally co -
ec and hei applica ion is global, i.e. independen o he objec s o
easoning.
•Local Monism: Di e en domains o discou se equi e di e en logics,
bu he e is only one co ec logic o each domain.
28
Logical Localism
can be ue nei he in Ino in any s a e o in o ma ion ex ending I, because
we a e no allowing o inconsis en s a es o in o ma ion. So, ¬¬(p∨ ¬p)
is wa an ed in Iand in any s a e o in o ma ion ex ending I. The e o e, we
ge ha , o ha epis emically cons ained domain, he e is a p oposi ion p
and a s a e o in o ma ion Isuch ha ¬¬(p∨ ¬p)is supe wa an ed, i.e.,
ue, bu p∨ ¬pis no .
The e o e, he logic o ha epis emically cons ained domain should gi e
us 0α∨¬αand ¬¬α0α. This a gumen , de eloped by Lynch and Pede sen,
s ongly sugges s ha he logic ha goes wi h supe wa an is no classical
logic bu in ui ionis ic logic.
Bu , on he con a y, i we also ha e he epis emically uncons ained,
comple e and consis en domain, wi h co espondence u h p ope y and a
s ong o m o he p inciple o bi alence (i.e., e e y sen ence is ei he ue
o alse bu no bo h), hen, we will ha e a domain whose logic is, mos
likely, classical logic. This would show ha he legi imacy o di e en u h
p ope ies o di e en domains jus i ies ha he e a e di e en domains ha
equi e di e en logics. Hence, localism.
We ha e jus conside ed wo kinds o domains wi h di e en u h p op-
e ies bu , a guably, one could make a de ence o pa aconsis ency and he
i ues o adop ing a pa aconsis en logic, by a simila s a egy. Fo ins ance,
one could a gue ha in a domain o discou se like humou , being ue migh
be ela ed o he ac ha some people ake i as ue.Wi h such a low s an-
da d o u h, i is e y likely ha bo h he p oposi ion ha some joke is
unny and i s nega ion a e bo h ue. So, i seems a leas plausible ha
some in e es ing logic sys ems, he ones ha a e usually in oked as being
good candida es o be canonically applied, can be mo i a ed o di e en do-
mains wi h a simila s a egy, namely, in i ue o being he logic ha be e
i s a gi en u h p ope y o a domain.
We will see in he nex sec ion ha his connec ion o ale hic plu alism
and localism wo ks o mo i a ing localism, bu also ca ies o e o he p ob-
lems and challenges o hese philosophical heo ies. Tha is, he e is a s ong
analogy, i no iden i y, be ween he p oblems ha a e usually a ibu ed o
ale hic plu alism and localism. While his migh i sel be p oblema ic, I ake
i as a possible ad an age since I migh kill wo bi ds wi h one s one. He e
I am aiming jus a localism bu , i he o he bi d alls, i will ha e been a
happy acciden .
35

Logical Localism
2.2 Main Challenge: The P oblem o Mixed
In e ences
We ha e jus a gued o he plausibili y o logical localism and ha e gi en
easons in a ou o i . Howe e , he e is a c ucial empi ical ac ha would
seem o coun in a ou o globalism and ha localism has o accoun o ,
namely, ha we do eason ac oss domains. Thus, localis heses, in ui i e as
hey migh be, ha e o ace an impo an challenge; a challenge ha P ies
(2006) aises and ha I will summa ise as ollows:
The P oblem o Mixed In e ences: one migh de end ha he e a e
a a ie y o domains ha equi e di e en logics. Bu he e a e cases
in which one easons abou he in e ac ion o di e en domains, wi h
p emises abou di e en kinds o objec s coming om hose domains.
Wha kind o logic do we use, hen? An unde lying logic o bo h
domains? This would gi e easons o hinking ha he e is a logic
o global applica ion. Maybe a new logic speci ic o ha domain o
in e ac ion? Bu which one? The in e sec ion o he logics in ol ed
in each o he in e ac ing domains migh be oo weak o be o any
u ili y. Mo eo e , i should be aken in o accoun ha i we s a
ying o mix he connec i es o di e en logics some o hem may
collapse. The in ui ionis ic condi ional, o ins ance, collapses in o he
classical condi ional unde he p esence o classical nega ion (P ies ,
2006, p. 199).
This, I belie e, encompasses all he p oblems ha localism has o answe ,
specially, o he echnical app oach o combining logics ha I will be aking
and ha is h ea ened by he collapse o connec i es. The mo e speci ic
p oblems ha we ind by dissec ing his main challenge, a e he p oblems o
mixed compounds and collapse heo ems. Le me cla i y ha he p oblem
o collapse heo ems does no usually appea , e en men ioned, in he mo e
philosophical li e a u e10. Bu , since he app oach ha I will be aking o
answe ing he challenge is ha o he me hods o combining logics and hese
me hods a e h ea ened by collapse heo ems, I include i in cha ac e izing
he p oblem o mixed in e ences.
10In ac , e en he mo e logically o ien ed au ho s, including P ies , despi e men ioning
he collapse heo ems, ake hem as knockdown a gumen s, ob ia ing, o igno ing, he ac
ha he e a e me hods o combining logics designed o a oid he collapse. Hope ully, his
disse a ion is also use ul o closing he gap be ween hose seemingly isola ed wo lds.
36
Logical Localism
Wi h espec o he p oblem o mixed compounds, one could a gue ha
i is a sub-p oblem o mixed in e ences. As we p esen he p oblems i will be
clea e why, bu simply no icing ha mixed in e ences can ha e as p emises
o conclusions mixed compounds makes i clea enough11. Mo eo e , i is
wo h poin ing ou , as Lynch does, ha
[ he p oblem o mixed in e ences] does no a ise solely o hose who
ha e come o DLP [domain-speci ic logical plu alism, a.k.a., localism]
ia u h plu alism. The issue o how o deal wi h mixed in e ence
and compounds is an issue o any logical plu alis who akes i ha
dis inc logics ope a e in di e en domains o discou se.
(Lynch,2008, p. 137)
The i s one who pu o wa d he p oblem o mixed compounds, as a chal-
lenge o ale hic plu alism, was T. Williamson in Williamson (1994), whe e
he e iews W igh ’s T u h and Objec i i y, hough some people also e e o
Tappole (2000) since she aised he same p oblem in a no o ious eply o
Beall (2000). The p oblem o mixed compounds o ale hic plu alism goes
as ollows: a sen ence like ‘Alex killed 13 people and killing o un is w ong’
can well be ue. Howe e , i is no ob ious how he ale hic plu alis could
explain he u h o he conjunc ion. Su ely, she can ake he i s conjunc
o be ue1,T1, (maybe in a co esponden is sense) and he second conjunc
o be ue2,T2, (maybe a no ion o u h g ounded in social ag eemen ). Bu ,
hen, in which sense is he conjunc ion ue? Which is he u h p edica e
ha applies o i ? I seems ha i can be nei he T1no T2, so maybe he e
is ano he u h p edica e ha applies o he conjunc ion. Bu , i is plausible
o hink ha he conjunc ion will be ue in ha u he way i and only i
he conjunc s a e ue in ha e y same way oo. Why do we need hen he
o he u h p edica es T1and T2?
The e sion o he p oblem, as applied o logical localism, is analogous.
Basically, he challenge is o answe which he logic o a compound p opo-
si ion should be. This migh seem innocuous, bu conside a compound
p oposi ion like pc∨ ¬qp, whe e pcis a classical p oposi ion and qpis a pa a-
consis en p oposi ion. I he logic ha go e ns he domain o pcis classical
11Ano he op ion is o conside mixed compounds as a p oblem o ale hic plu alism only,
a guing ha he ques ion abou he co ec logic only makes sense o in e ences, no o
p oposi ions. In any case, gi ing a logic o mixed in e ences (wi h mixed compounds) will,
mos likely, equi e sol ing how he seman ic alue o a mixed compound is de e mined o
explaining which ule is go e ning he in oduc ion o elimina ion o he main connec i e
in a mixed compound.
37
Logical Localism
logic and he logic ha go e ns he domain o qpis P ies ’s logic o pa adox,
LP, which is he logic o he compound? The answe is no i ial a all,
since he way we answe i will a ec , say, whe he we can use disjunc i e
syllogism as a alid in e ence o no (because i is no alid unde LP). This
is he sense in which he p oblem o mixed compounds is a sub-p oblem o
he p oblem o mixed in e ences, in he e sions di ec ed agains localism.
Responding o he p oblem o mixed in e ences equi es ha ing an accoun
o he logic o mixed compounds, o begin wi h12.
The p oblem o mixed in e ences was, indeed, i s pu o wa d by Tap-
pole in Tappole (1997) and i is based on wha she akes o be a cen al
pla i ude abou u h: ‘ u h is wha is p ese ed in alid in e ences’ (Tap-
pole ,1997, p. 210). Bu , hen, she p oceeds o conside he ollowing alid
in e ence: We ca s a e unny
This ca is we
This ca is unny
Tappole ’s objec ion, hen, goes like his:
The alidi y o an in e ence equi es ha he u h o he p emises
necessi a es he u h o he conclusion. Bu how can his in e ence
be alid i we a e o suppose wi h C ispin W igh ha wo di e en
kinds o u h p edica es a e in ol ed in hese p emises? Fo he
conclusion o hold, some unique u h p edica e mus apply o all
h ee sen ences. Bu wha u h p edica e is ha ? And i he e is
such a u h p edica e, why isn’ i he only one we need?
(Tappole ,1997, p. 210)
Thus, Tappole challenges he u h plu alis wi h a ilemma: (a) ei he
one denies ha mixed in e ences a e alid, (b) accep s ha he e is a gene ic
u h p ope y ha all domain-speci ic u h p ope ies ha e in common
(which would make his speci ic ones edundan ) (c) o ejec s he s anda d
accoun o alidi y as necessa y u h p ese a ion.
12Again, one migh p e e o ese e he naming ‘p oblem o mixed compounds’ o he
e sion a ec ing ale hic plu alism. Then, he e sion o he p oblem a ec ing localism
would be jus a sub-p oblem o he p oblem o mixed in e ences. I belie e his is jus
a e minological issue bu he eade should keep in mind ha he p oblem o mixed
compounds is gene ally ega ded as a p oblem o ale hic plu alism.
38
Logical Localism
Many au ho s ha e eplied o Tappole by ying o make he case o
a possible sa is ac o y way ou o he ilemma. Especially in e es ing e-
sponses a e hose o Beall (2000); Co noi (2013); Lynch (2008,2009) and
Yu (2017). Bu , as I said be o e, I am mo e in e es ed in sol ing he p ob-
lem in i s logical localis e sion. Tha is he aim o all he combina ion
mechanisms ha I will be p esen ing la e on. I , as a side e ec , we ge an
in e p e a ion o he solu ion ha is also sa is ac o y o ale hic plu alism, i
would be he icing on he cake.
So, he e sion o he p oblem o mixed in e ences ha is di ec ed agains
localism goes, oughly, as ollows: suppose ha he e a e (a leas ) wo
componen s, wi hin he p emises o conclusion o an a gumen , belonging o
di e en domains whose logics a e L1and L2, espec i ely. Then, which is
he c i e ion o alidi y o he a gumen ? Tha is, which is he logic ha
cap u es co ec easoning o ha mixed domain?
2.2.1 Some a emp s o sol e he p oblem o mixed
in e ences
Al hough no much, he e ha e been some au ho s who ha e a emp ed o
mee he challenge. Howe e , e y ew o hem ha e been sys ema ic enough
in hei e o , wi h he excep ion o (maybe among o he s ha I am no
awa e o ) Co noi (2013); Lynch (2008,2009); W enn (2018) and Yu (2017,
2018). Le me p esen hei accoun s and explain why I ake hem o be
unsa is ac o y.
Co noi ’s algeb aic accoun
The aim o A. Co noi in Co noi (2013) is, i s and o emos , o sol e he
p oblem o mixed in e ences in he e sion di ec ed agains ale hic plu alism
ha Tappole aises. Howe e , a e p o iding an accoun o ha , he goes
on o ex end he idea o make oom o a ia ion in each domain’s logic.
So, Co noi s a s by modelling he idea ha each domain has a dis inc
u h p ope y. In o de o do his, he de ines seman ic alues, V, o be n-
uples, nbeing he numbe o domains and each elemen o he uple being
ei he 1 o 0:
V={ha1, ..., ani| each ai∈ {1,0}}
39
Logical Localism
So, ha ing a 1 in he i- h posi ion means ha he p oposi ion is in he i- h
domain and i is uei. While ha ing a 0 in he i- h posi ion means ei he
ha he p oposi ion is no in he i- h domain o ha i is alsei.
Fu he , assume ha a omic p oposi ions can ha e a mos one u h
p ope y, so he e can only be a 1 in he uple ep esen ing he seman ic
alue o an a om. He, hen goes on o p o ide a classical accoun o connec-
i es, in he sense ha o each componen o he uple, he nega ion in e s
he alues, while conjunc ion and disjunc ion a e minimum and maximum
ope a ions espec i ely13. Tha is,
•Fo all p oposi ion A, i (A) = ha1, ..., ani hen (¬A) = h1−(a1), ..., 1−
(an)i.
•Fo all p oposi ion Aand B, i (A) = ha1, ..., aniand (B) = hb1, ..., bni
hen (A∧B) = hmin(a1, b1), ..., min(an, bn)i.
•Fo all p oposi ion Aand B, i (A) = ha1, ..., aniand (B) = hb1, ..., bni
hen (A∨B) = hmax(a1, b1), ..., max(an, bn)i.
As Yu (2017) igh ly no ices, hese alua ion unc ions al eady yield un-
welcome esul s. Fi s , no ice ha i , say, an a omic p oposi ion is ue,
in he sense speci ic o i s domain, he nega ion o he p oposi ion will be
ue in he o he domains ep esen ed in he uple. Fo ins ance, suppose
we a e conside ing a ma hema ical domain and an e hical domain. Take he
p oposi ion A= ‘1 + 1 = 2’. The alue o his a omic p oposi ion will be
(A) = h1,0iassuming ha he i s elemen o he uple ep esen s he
alue in he ma hema ical domain and he second he alue o he e hical
domain. I is al eady ques ionable ha he e hical u h- alue o ha p opo-
si ion is alse, bu i is e en wo s ha he nega ion o he p oposi ion, i.e.,
(¬A) = h0,1i, is e hically ue. I ma hema ical p oposi ions a e no ap o
he e hical u h p edica e, why should he nega ion o a ue ma hema ical
sen ence be e hically ue? And how could one e en make sense o he ac
ha ‘one plus one does no equal wo’ is e hically ue?
A second unwelcome esul comes om conjunc ion. Again, conside
he ma hema ical p oposi ion A= ‘1 + 1 = 2’ and he e hical p oposi ion
B= ‘Killing babies o un is w ong’ wi h alua ions (A) = h1,0iand
(B) = h0,1i. The conjunc ion o hese p oposi ions, acco ding o Co noi ’s
13He e I am ollowing Yu’s cha ac e iza ion o Co noi ’s accoun , in Yu (2017), since i
is a good summa y o he p oposal.
40

Logical Localism
p oposal, is alse! I is alse in bo h o he ele an senses, i.e., (A∧B) =
hmin(1,0), min(0,1)i=h0,0i. So he alua ion making each o he conjunc s
ue in hei ele an domains makes he conjunc ion alse in e e y sense.
Co noi , p oposes a solu ion o he p oblem wi h nega ion, by in oducing
a hi d u h- alue, 1
2, and mo ing o a non-classical logic14. The idea is ha
a omic p oposi ions ge a mos a 1 in he uple and o e e y o he place
hey ge he alue 1
2, cap u ing he idea o being unde ined o hose domains.
Wi h his co ec ion, he p oblem wi h nega ion is alle ia ed, because i a
p oposi ion has alue 1
2in he i- h place, i will also ge ha alue i we nega e
he p oposi ion. Bu , his epai comes a he expense o ha ing enounced o
classical logic and, in any case, i does no sol e he p oblem wi h conjunc ion.
I we ha e again, A=‘1+1=2’andB= ‘Killing o un is w ong’, now
wi h alua ions (A) = h1,1
2iand (B) = h1
2,1i, he conjunc ion o hese
p oposi ions is unde ined, i.e., (A∧B) = hmin(1,1
2), min(1
2,1)i=h1
2,1
2i.
This is equally coun e in ui i e and unsa is ac o y.
On op o hese p oblems wi h he u h- alue unc ions, he e is a u he
limi a ion in Co noi ’s accoun ha Yu does no iden i y, since i is a limi a-
ion o his accoun oo, as I will explain in a momen . So, wha Co noi ies
o do is o gi e a solu ion o he p oblem o mixed in e ences o ale hic plu-
alism. The eason o swi ching o uples in o de o ep esen u h- alues,
is ha he wan ed o be able o handle di e en u h-p ope ies, each ep e-
sen ed by a di e en posi ion in he uple, wi hin a single logic sys em. Fi s ,
he consequence ela ion ha esul s, is classical logic (see (Co noi ,2013,
pp. 570–571)). Then, he in oduces a hi d u h- alue and ge s a pa acom-
ple e and pa aconsis en sys em. And, inally, he in oduces Hey ing algeb as
in he uples and ge s in ui ionis ic logic (see (Co noi ,2013, pp. 575–577)
o he de ails). In ha way, he claims ha localism is accommoda ed in
he b oade pic u e o ale hic plu alism. Bu his is only hal way ue. He
has a emp ed o handle di e en u h-p ope ies wi h di e en logics, bu
wi h a single logic each ime! he e is no men ion o mixed in e ences and
which he c i e ion o alidi y migh be when we need o combine di e en
logic sys ems.
E en a si ua ion as simple as he ollowing does no ge an answe : sup-
pose pcis a classical p oposi ion and qia p oposi ion belonging o an epis-
emically cons ained domain. Take he p oposi ion ha ¬pc∨qi. Do we
14We do no need o en e in o many de ails. Jus know ha he esul ing logic is he
in e sec ion o in ui ionis ic logic and P ies ’s logic o pa adox.
41
Logical Localism
ha e ha ¬pc∨qi`pc→qi? We ind no c i e ion in Co noi ’s accoun
and, he e o e, no solu ion o he p oblem o mixed in e ences in i s e sion
di ec ed agains localism. In ac , Co noi says ha wi h his p oposal, u h
plu alis s ‘can allow ha he logic o unmixed in e ences can some imes be
domain dependen ’ (Co noi ,2013, p. 577, my emphasis).
Yu’s logic o ale hic and logical plu alis s
We ha e jus seen ha A. Yu, in Yu (2017,2018), igh ly iden i ies some
o he unwelcome esul s in Co noi ’s p oposal, so his solu ion a oids hose
p oblems in a qui e nice way. Yu’s undamen al idea is ha he e is an
isomo phism be ween domains, u h p ope ies and alsi y p ope ies, ha
beha e in he ollowing way:
he e is a one- o-one co espondence be ween domains, domain-speci ic
u h p ope ies, and domain-speci ic alsi y p ope ies. Pu e domains
a e associa ed wi h exac ly one subjec ma e , while impu e domains
a e associa ed wi h wo o mo e subjec ma e s. Whe e domains
a e ei he pu e o impu e, pu e domains gene a e all domains. Each
a omic sen ence is assigned o exac ly one domain. Nega ions a e al-
ways assigned o he same domain as he negand, while conjunc ions
and disjunc ions may o may no be assigned o he same domain
as each ope and, depending on whe he o no he ope ands a e as-
signed o he same domain. Each a omic sen ence is hen assigned a
domain-speci ic u h alue, whe e he ele an domain is he one i is
assigned. The domain-speci ic u h alues o nega ions, conjunc ions,
and disjunc ions a e de e mined by he domain-speci ic u h alues
o he ope ands. Logical consequence necessa ily p ese es domain-
speci ic u h. (Yu,2018, pp. 413–414)
In o de o a oid dis ac ion wi h he de ails o he p oposal,15 le me gi e
a conc e e example o oughly illus a e how i wo ks: suppose we ha e a
p oposi ion abou he physical middle-sized domain and a p oposi ion abou
he e hical domain, say pc=‘ he dog is a home’ and qi=‘ o u ing is w ong’.
Conside ing ha hese a e pu e domains, hey can be combined in o de o
p oduce he impu e domain o ‘physical middle-sized and e hical’. Equally,
15To cla i y, he de ails a e no impo an because he p oposal is no designed o he
p oblem o mixed in e ences ‘localism s yle’ bu o ‘ale hic plu alism s yle’. O he wise,
ob iously, he de ails would be impo an , as happens wi h Lynch’s and W enn’s accoun s.
42
Logical Localism
he u h- alues and alsi y- alues p esen he same s uc u e. The physical
p oposi ion will ake he alues Tco Fc, while he e hical p oposi ion will
be ei he Tio Fi. Howe e , i we make he conjunc ion o bo h p oposi ions
o ge pc∧qi ha compound p oposi ion will ha e he u h- alues ha
co espond o i s impu e domain o ‘physical middle-sized and e hical’. Le
us call hem Tci o Fci.
The e a e a numbe o wo ies wi h his ype o p oposal al eady known
in he li e a u e. In Edwa ds (2008) we ind a simila iew, which also elies
on he in ui ion ha compounds a e ue in a de i a i e sense (i.e., ‘impu ely
ue’, using Yu’s e minology) and, o ins ance, Co noi (2009) makes he
case agains he p oli e a ion o u h-p ope ies ha such a iew equi es.
One could a gue back, ollowing he li e a u e on he me aphysics o unda-
men ali y, ha he e is no p oblem wi h mul iplying he u h p ope ies as
long as hey a e de i a i e, since a spa se on ology is only ele an a he
undamen al le el.
In any case, I eckon ha Yu’s accoun s ill has p oblems. Le me poin
ou a e y ob ious one. Take pc o be he p e ious p oposi ion. By Yu’s
accoun , ¬pcis also om he physical domain and, he e o e, ¬pc∨pc oo.
In ac , since i is a au ology, i s u h- alue is Tc o any in e p e a ion. By
an analogous easoning ¬qi∧qi, which is a con adic ion belonging o he
e hical domain, always ge s alue Fi. Now, make he disjunc ion o hese
wo p oposi ions o ge (¬pc∨pc)∨(¬qi∧qi). Acco ding o Yu, his is a
p oposi ion belonging o he impu e domain o ‘physical middle-sized and
e hical’ and whose u h- alues can only be Tci o Fci. Howe e , his is a
e y odd esul , since he e is no doub ha his impu e p oposi ion is ue
in i ue o he same ac s as ¬pc∨pcand, he e o e, should be ue in exac ly
he same way. In o he wo ds, he e hical p oposi ion does no con ibu e
any hing o he u h o he impu e p oposi ion and, ye , i makes he impu e
p oposi ion ake a di e en u h p ope y o ha o he physical p oposi ion
in i ue o which is ue.
Be ha as i may, his would be a p oblem o his p oposed solu ion o he
p oblem o mixed in e ences as applied o ale hic plu alism. Tha is, i is a
way o accommoda ing a a ie y o u h alues, wi hin a single consequence
ela ion and c i e ion o alidi y. In Yu (2017) he unde lying logic is classical
and in Yu (2018), i is ex ended o allow o a non-classical unde lying logic
oo. Bu , his is a om being a solu ion o mixed in e ences in which
we allow he ac ion o mo e han one unde lying logic wi hin he p emises
h ough he conclusion. Yu has jus gi en a way in which a non-classical
43
Logical Localism
domain can ha e a non-classical logic, bu has no conside ed which is he
logic o an a gumen wi h componen s go e ned by di e en logics. This is
he same p oblem ha Co noi has. They ge non-classical logics o some
domains wi h non-classical u h- alues, bu hey do no o e a me hod o
combining he logics o hose domains in mixed in e ences.
Lynch’s modes y c i e ion
As we said abo e, Lynch is awa e o he connec ion be ween he p oblem
o mixed in e ences in i s ale hic and logical e sions and, mo eo e , no ices
ha i is a p oblem o localism e en i ha localism does no come om
ale hic plu alism. So, once he has sol ed he p oblem o ale hic plu alis s
by means o his unc ionalis accoun , o so he belie es, he mo es on o sol e
he p oblem o localism. The solu ion, Lynch claims, has wo pa s. Fi s ,
he localis ,
being a plu alis a e all, will ake i ha wi hin a domain, wha
quali ies as he go e ning logic will be de e mined by wha mani es s
u h in ha domain. (Lynch,2008, p. 137)
So, o ins ance, ollowing Lynch’s analysis, he logic go e ning he domain
in which supe wa an mani es s u h will be in ui ionis ic logic.
The second pa is he mo e di icul one. Lynch de ends ha in o de
o e alua e he alidi y o a mixed in e ence we ha e o apply a c i e ion o
modes y. To unde s and why, conside he nex a gumen 16:
Nix:I i is no he case ha o ensi e jokes a e unny,
hen snow isn’ whi e
Snow is whi e
O ensi e jokes a e unny
Which we can o malize as:
¬pi→ ¬qc
qc
pi
16I is a modi ied e sion o he a gumen ha Lynch and W enn also call ‘Nix’. The
p oposi ions a e di e en bu he s uc u e is he same. I jus wan ed o use a di e en
exempla o show he ex en o possible ins ances.
44
Logical Localism
modali ies. The collapse heo ems we e also mo i a ed by e lec ing on clas-
sical logic and non-classical ones. Mo e conc e ely, on imagining a dialogue
be ween a classical logician and an in ui ionis , willing o coope a e in o -
de o unde s and wha he o he pa y means. Howe e , mixed in e ences
cons i u e a new philosophical challenge o combina ion mechanisms, in he
sense ha , i logical localism is co ec , hen he e a e si ua ions o mixed
easoning ha migh gene a e in e es ing in e ac ions be ween logic sys ems
ha ha e no been conside ed ye . I will y o show ha his is, in ac , he
case and ha he philosophical p oblem o mixed in e ences migh igge
impo an de elopmen s and esea ch lines o he combina ions o logics.
51

Chap e 3
Mixed Reasoning and
Combining Logics
3.1 In e ac ion P inciples
In si ua ions o combined easoning, such as he ones ep esen ed by he cases
o mixed in e ences, one should expec some in e ac ion be ween he logical
sys ems ha a e being combined. I hose logical sys ems aim a cap u ing
he modes o easoning o gi en domains, i is easonable o expec ha
he combina ion o he logical sys ems will cap u e some in e ac ion o he
combined easoning, o he wise i would no be a case o mixed easoning in
he i s place.
One o he mos common e e ences when dealing wi h in e ac ion p in-
ciples is Da id Hume’s na u alis ic allacy, namely, he hesis ha om a
ac ual s a emen one canno deduce a no ma i e one, ha is, ha om
‘wha is’ one canno de i e ‘wha ough o be’. To pu i in e ms o mode n
modal logic, Hume’s hesis cons i u es an objec ion o in e ac ion p inciples
such as,
p→ p
s a ing ha , i pis he case, hen one ough o p1.
1One could say ha his is a limi case o an in e ac ion p inciple, since he e is no
modal ope a o in he an eceden . We will see in a momen , hough, ha i alls unde
he de ini ion o b idge p inciple, which is a ype o in e ac ion p inciple. In any case, i
is easonable o a gue ha he lack o a modal ope a o is i ele an , since he an eceden
is exp essing a ac ual modali y and his is wha in e ac s wi h he deon ic one.
52
Mixed Reasoning and Combining Logics
Ano he his o ical e e ence on hese ma e s is he ‘ough -implies-can’
hesis, usually a ibu ed o Immanuel Kan . This is he hesis ha i some-
hing is obliga o y hen i mus be possible, o malized as
p→♦p.
And ye ano he in e ac ion p inciple ha is o en imes included in epis emic
logics comes om Pla o’s cha ac e iza ion o knowledge as ‘jus i ied ue
belie ’. Thus, his unde s anding o wha cons i u es ‘knowledge’ mo i a es
ha ing an in e ac ion p inciple cap u ing ha i a subjec xknows ha p,
hen pis he case,
Kxp→p.
These in e ac ion p inciples all unde he subca ego y o ‘b idge p inci-
ples’, which we e unde s ood as p inciples linking ac uali ies o no ms and,
mo e gene ally, as p inciples linking di e en modali ies. Thus, we can de ine
mo e o mally a b idge p inciple as an axiom schema which has a leas one
occu ence o an schema ic le e unde he scope o a modal ope a o , ?,
and a leas one occu ence ou side he scope o ?.
One hing o no ice is ha hese b idge p inciples a y in hei analy ic-
i y, so while some o hem migh be highly desi ed in i ue o hei meaning
and how he in e ac ion is es ablished, o he s migh be mo e p oblema ic
and in need o some philosophical jus i ica ion, o e en ejec ed. The usual
in e de ini ion be ween necessi y and possibili y, namely, α≡ ¬♦¬α, could
be included among he analy ic b idge p inciples. The b idge p inciple, p e-
iously men ioned, connec ing he knowledge o some p oposi ion wi h he
u h o ha p oposi ion ce ainly needs some philosophical jus i ica ion2,
and an axiom s a ing ha i pis he case hen some ini e agen xknows
ha pis clea ly bad and ejec ed by any easonable epis emic logic.
As we will see la e on, some o hese b idge p inciples can be a oided
by some o he me hods o combining logics. This can be seen as a posi i e
consequence o he combina ion mechanisms, as one could add, a e he
combina ion, he b idge p inciples ha one desi es in he combined logic as
addi ional axioms, while a oiding he p oblema ic ones. Howe e , one migh
ake his p ocedu e o be qui e ad hoc and, he e o e, i is an in e es ing
issue o wonde whe he one could combine gi en logical sys ems in such a
2One could a gue ha New on knew some laws o physics, despi e hose laws no being
s ic ly ue.
53
Mixed Reasoning and Combining Logics
way ha he desi ed b idge p inciples a ose in he e y combina ion p ocess
(Schu z,1991, p. 46)3.
Bu , le me now elabo a e and widen mo e he no ion o b idge p inci-
ple. Fo now, we ha e only conside ed b idge p inciples o be in e ac ions
be ween a iables unde he scope o a modal ope a o and a iables ou -
side hei scope. Howe e , I eckon, ollowing Ca nielli and Coniglio (2007),
ha we should hink o b idge p inciples as p inciples es ablishing, mo e
gene ally, connec ions o in e ac ions be ween connec i es. Bu no e e y
in e ac ion be ween connec i es will coun as a b idge p inciple. In a sense,
we wan hese in e ac ions o be “new”, meaning ha we did no ha e hese
in he o iginal logics being combined. Thus, we ake b idge p inciples o be
“any in e ac ions (i.e., de i a ions) among dis inc logic ope a o s which a e
no ins ances o alid de i a ions in he indi idual logics being combined”
(Ca nielli and Coniglio,2007, p. 8).
Take, o ins ance, he ollowing mixed in e ence:
We ca s a e unny
Fi z Roy isn´ he highes moun ain o we ca s a e no unny
Fi z Roy isn´ he highes moun ain
Assume ha we hink CL is he logic o he physical domain and ha IL
is he logic o he domain abou humou . So, le us o malize he a gumen
dis inguishing he connec i es o each logic by he subindexes cand i.
pi
¬cqc∨i¬ipi
¬cqc
We can see ha he e is an in e ac ion be ween connec i es om di e en
logics, bu his is no a b idge p inciple, since i is an ins ance o a alid
de i a ion in IL, namely, Disjunc i e Syllogism. Howe e , had we ansla ed
he a gumen like his,
pi
¬cqc∨c¬ipi
¬cqc
we would ha e deli e ed a b idge p inciple, since his a gumen cap u es an
in e ac ion be ween classical and in ui ionis ic connec i es, bu i is no longe
an ins ance o a alid a gumen in IL, no in CL.
3I will show la e on ha my solu ion o mixed in e ences mee s his deside a um.
54
Mixed Reasoning and Combining Logics
Ano he example in o de o u he illus a e wha cha ac e izes b idge
p inciples among in e ac ion p inciples is gi en by Ca nielli and Coniglio
(2007). Conside he combina ion o he logic o classical conjunc ion, L∧,
and he logic o classical disjunc ion, L∨. In his logic he de i a ion
p∧q`(p∧q)∨
is an in e ac ion p inciple which is no a b idge p inciple, since i is a sub-
s i u ion ins ance o
p`p∨
ha is a alid de i a ion in L∨. Howe e , he dis ibu i e law o conjunc ion
o e disjunc ion,
p∧(q∨ )`(p∧q)∨(p∧ )
is a b idge p inciple o L∧∨, since i is no a subs i u ion ins ance o any alid
de i a ion ei he in L∧o L∨. In some sense, hen, in e ac ion p inciples a e
new de i a ions jus because hey in ol e some ocabula y ha we lacked
be o e he combina ion. Bu , indeed, he in e es ingly new in e ac ions come
om b idge p inciples. These eally a e new de i a ions ha appea in he
combina ion p ocess and ha we e no p esen be o e.
3.1.1 Collapse
Whoe e is amilia wi h he deba e a ound logical plu alism has su ely come
ac oss he no ion o ‘collapse’, as applied o he plu alis p oposals. One o
he i s collapse a gumen s, i no he i s , appea s in Williamson (1988),
and o he ele an e sions can be ound in Read (2006), P ies (2006),
Kee e (2014) and S ei (2020). P ies ´s e y well known a gumen , is di ec ed
agains Beall and Res all’s plu alism:
Le sbe some si ua ion abou which we a e easoning; suppose ha
sis in di e en classes o si ua ions, say, K1and K2. Should one use
he no ion o alidi y app op ia e o K1o o K2? We canno gi e
he answe ’bo h’ he e. Take some in e ence ha is alid K1bu no
K2,α`β, and suppose ha we know (o assume) αholds in s; a e
we, o a e we no en i led o accep ha βdoes? Ei he we a e o we
a e no : he e can be no plu alism abou his. (P ies ,2006, p. 203)
55
Mixed Reasoning and Combining Logics
The essence o he a gumen is ha he e a e wo legi ima e consequence
ela ions ha disag ee wi h espec o a pa icula a gumen , i.e. α`1βbu
α02β, and, a he same ime, he e is a subjec who knows ha αholds and
ha α`1βbu α02β. I would seem, hen, ha he subjec is en i led o
accep ha βholds in s, as P ies de ends igh a e he quo ed pa ag aph.
So, he a ional hing o do o he agen is o accep i and, he e o e, hese
wo logics would collapse o he s onges one.
Ne e heless, his is no he no ion o collapse ha we a e going o ocus
on in his sec ion; o he e is ano he ype o collapse ha has ecei ed
almos no a en ion in he philosophical li e a u e in spi e o being a c ucial
challenge o some plu alis (meaning ’plu alis ’ in a elaxed, almos in o mal,
sense) p oposals such as localism.
Be o e going in o he de ails, le us oughly in oduce his ype o collapse
by saying ha i does no depend on a pa icula a gumen o e which wo
logics disag ee. Nei he does i in ol e any assump ions abou he no ma i -
i y o logic in o de o i o wo k. I is a he a echnical esul , consis ing
o a numbe o heo ems, known as collapse heo ems, which show ha by
eely combining di e en logic sys ems, each (possibly) wi h i s own s ock
o connec i es, he logics collapse o one o hem, because hei di e en
connec i es end up beha ing as me e no a ional a ian s.
Le me no ice ha I will be ollowing Schech e in he e minology and
dis inguish, as he does, collapse and weak collapse. The p ecise no a ion and
concep s will be p esen ed in he nex sec ion. Bu le us, o he momen ,
say ha a logic, L, wi h wo s ocks o logical connec i es, collapses when o
e e y o mula δ,δ0, exac ly alike excep o some o all o hei subsc ip s,
{δ} ` δ0.
Fu he mo e, le us say ha a logic, L,weakly collapses when he e is a
ansla ion, , be ween he se o o mulas wi h connec i es om s ock 1 and
he se o o mulas wi h connec i es om s ock 2, such ha , i Γ`α, hen
(Γ) ` (α).
The i s collapse esul s can be aced back o Ca nap (1943) and Poppe
(1948), al hough he mos common e e ences a e Ha is (1982), in he mo e
philosophically o ien ed li e a u e, and del Ce o and He zig (1996) and Gab-
bay (1996), in he li e a u e e ol ing a ound ib ing. Howe e , despi e hei
di e en ’ adi ions’, all hese sou ces sha e he common ea u e o dealing
uniquely wi h he case o combining classical and in ui ionis ic logics.
56

Mixed Reasoning and Combining Logics
3.1.2 Collapse Theo ems
In his pape ‘Wha ’s So Logical abou he “Logical”Axioms?’, J. H. Ha is
in i es us o imagine an in ui ionis logician and a classical logician willing
o coope a e in o de o unde s and he axioms ha he o he pa y deems
alid, aking hem as syn ac ical meaning pos ula es o how he o he unde -
s ands he connec i es. Assume, hen, ha bo h logicians wan o en e ain
a dialogue in a common logic Lo e a sha ed language L.
Ha is deli e s he ollowing axiom schema a as he ones ha bo h clas-
sical and in ui ionis logicians would accep 4:
1. Deduc ion P ope y (DEDL): A1, ..., Ak, A `LBi A1, ..., Ak`L
A→xB o all L- o mulas A1, ..., Ak, A and B.
2. Modus Ponens (MPL): Γ`LAand Γ`LA→xB, hen Γ`LB.
3. A, B `LA∧xB.
4. (a) `LA∧xB→xA
(b) `LA∧xB→xB
5. (a) `LA→xA∨xB
(b) `LB→xA∨xB
6. A→xC, B →xC`LA∨xB→xC
7. A→xB, A →x¬xB`L¬xA
8. ¬xA, A `LB
On op o hese sha ed axioms, we also bo h accep ha Γ`LAi A∈Γ
(9). Mo eo e , he classical logician will also wan o include he ollowing
axiom:
8c`L¬c¬cA→cA
4Le us use subsc ip x o ei he he in ui ionis (i) e sion o he classical (c) e sion
o he connec i es. Also, no ice ha we will only ocus on he p oposi ional pa .
57
Mixed Reasoning and Combining Logics
Howe e , an in e es ing poin is ha we will no need his las axiom, 8c,
in o de o p o e he collapse heo ems. Taken oge he , hese heo ems a e
mean o es ablish he collapse o he logic, i.e. ha o e e y L- o mula Ax,
he in ui ionis ic and he classical e sions a e in e de i able, `LAi↔xAc.
Le us now show some o he mos in e es ing collapse heo ems.
Theo em 3.1.1.Assume logic Lsa is ies schema 1 - 8 o bo h x∈ {i, c}.
Then o e e y L- o mula A and B and bo h x∈ {i, c}we ha e `LA→i
B↔xA→cB.
P oo . I p o e he igh o le di ec ion, which is ac ually mo e in e es ing.
The o he di ec ion is analogous. We ha e ha A→cB`LA→cBby (9).
Then, A→cB, A `LBby (1) and A→cB`LA→iBagain by (1). 
Theo em 3.1.2.Assume logic Lsa is ies schema 1 - 8 o bo h x∈ {i, c}.
Then o e e y L- o mula A and bo h x∈ {i, c}we ha e `L¬iA↔x¬cA.
P oo . Again om igh o le . ¬cA, A `LBby (8) and ¬cA`LA→iB
by (1). ¬cA, A `L¬iBby (8) and ¬cA`LA→i¬iBby (1). Then apply
wice DEDL o (7) in o de o ge `L(A→iB)→i((A→i¬iB)→i¬iA),
and, inally, by wo applica ions o MPLwe ge ¬cA`L¬iA.
As one migh expec now, he es o he collapse heo ems o ∧and ∨
wo k in a simila way. Thus, all hese heo ems lead o he unse ling esul
ha when I s a e he law o excluded middle using he classical connec i es,
his is logically equi alen o he in ui ionis ic e sion, i.e., `L(A∨i¬iA)↔x
(A∨c¬cA). In ac , one can p o e he ollowing s onge heo em which
s a es he collapse o he logic Lo e he language L.
Theo em 3.1.3.Le Aibe any L- o mula whose connec i es a e in ui ionis ic
and le Acbe he co esponding L- o mula in i s classical e sion. Assume L
sa is ies schema 1 - 8 o bo h x∈ {i, c}. Then `LAi↔xAc.
P oo . By induc ion on he numbe o connec i es (see Ha is (1982), The-
o em 8).
3.1.3 Collapse: he limi ing case o b idge p inciples
We s a ed he sec ion alking abou in e ac ion p inciples and p oceeded
o p esen ing wha collapse is. No ice ha , indeed, he collapse heo ems
a e no hing mo e han some esul s ha p o ide new de i a ions, i.e., b idge
58
Mixed Reasoning and Combining Logics
p inciples, es ablishing connec ions be ween he co esponding logical con-
nec i es o each logic. The p oblem o hese heo ems, howe e , is ha he e
is oo much in e ac ion be ween he logical connec i es. In ac , depending
on he me hod o combining logics, one can gi e su icien condi ions o
which new in e ac ions p o oke he o al collapse. In he case o jux aposi-
ion, o ins ance, one can show ha i we added α→1βa` α→2β5, he
logic would collapse (see Schech e (2011), P op. 7.3.)
As we a e going o see la e on, one o he mos di icul , ye a he same
ime in e es ing, aspec s o combining logics is o calib a e how much in e -
ac ion is oo much in e ac ion and, also, how li le in e ac ion migh be oo
li le. We know ha he collapse o he connec i es is a one ex eme o
in e ac ions, bu he e is an analogous p oblem, i s no iced by Béziau in
Béziau (2004) and la e mo e deeply analysed in Béziau and Coniglio (2011)
a he o he ex eme, namely, he an i-collapse p oblem. The au ho s cha ac-
e ize his p oblem as “ he impossibili y o ob aining, in he logics ob ained
by ib ing, in ended in e ac ion ules which a e jus i ied, o ins ance, by
well-known models o sequen ules” (Coniglio,2007, p. 379). In ac , his
is no a p oblem speci ic o he combina ion me hod o ib ing, bu also o
jux aposi ion and, in gene al, o any combina ion mechanism ha , when
combining he logics L1and L2seeks he logic L12, which is he minimal
conse a i e ex ension6o he logics being combined. This means ha he
combined logic, because o i s minimali y, will no include among i s alid
in e ences any p ope ly new in e ac ions, ha is, b idge p inciples, no , a
o io i, he in ended jus i ied ones.
In Béziau (2004), he an i-collapse p oblem was illus a ed wi h he case
o combining he logic o conjunc ion, L∧, and he logic o disjunc ion, L∨.
We ha e al eady seen ha some o he desi able b idge p inciples o his
combina ion a e he dis ibu i i y laws, bo h o conjunc ion o e disjunc ion
and ice e sa. One can easily check ha i we pu oge he he wo- alued
u h ables o ∧and ∨we ac ually ge ha he dis ibu i i y laws a e
sa is ied. Ne e heless, he s anda d combina ion mechanisms (like ib ing
o jux aposi ion) will no yield hese b idge p inciples in he combined logic
L∧∨, since his logic will be he minimal conse a i e ex ension o L∧and
L∨.
I migh be a ma e o disag eemen whe he he logic o conjunc ion
5As usual, αa` βis an abb e ia ion o α`βand β`α.
6We will gi e he o mal de ini ions la e on.
59
Mixed Reasoning and Combining Logics
and disjunc ion, L∧∨ is dis ibu i e o no 7, bu he e a e o he applica ions
o combining logics ha mo e ob iously equi e he eme gence o b idge
p inciples, o ins ance, he applica ion ha aims a eco e ing a logic om i s
agmen s. This has been one o he mo e ecen ields o de elopmen in he
applica ion o combina ion mechanisms and i has gi en ise o new me hods
o combining logics, gi en how inapp op ia e he s anda d mechanisms a e
in o de o go beyond minimali y.
One o he ew me hods de eloped along hose lines is me a- ib ing, p o-
posed by Coniglio (2007). This me hod eco e s classical logic when com-
bining he logic o (classical) nega ion wi h he logic o (classical) condi-
ional. We will see when showing he applica ions o jux aposi ion ha his
is no possible wi h i , since we do no ge he P inciple o Pseudo-Sco us,
`α→(¬α→β), o ins ance.
Bu he eme gence o some b idge p inciples can also be p oblema ic,
e en i hey do no lead o collapse, jus in i ue o no being well jus i ied
o being philosophically aul y. As Ca nielli and Coniglio (2007) ecall, i one
combines wo no mal modal logics, L1and L2, wi h he me hod o p oduc
o modal logics, each one wi h i s own ope a o s iand ♦i, he ollowing
b idge p inciples pop up a he seman ic le el:
•-commu a i i y: 12α↔21α
•♦-commu a i i y: ♦1♦2α↔♦2♦1α
•(1, 2) Chu ch-Rosse p ope y: ♦12α↔2♦1α
•(2, 1) Chu ch-Rosse p ope y: ♦21α↔1♦2α
Bu i he logic L1is an ale hic modal logic and L2is an epis emic logic, we
would ha e,
♦Kα ↔K♦α
wi h ‘K´s anding o ‘knowledge’, which seems highly implausible. Jus
conside he ac ha i is possible ha an agen knows ha Higgs Boson
exis s, while he agen no knowing whe he i is possible ha Higgs Boson
exis s. This b idge p inciple clea ly has epis emological sho comings.
One should al eady see ha he issue o b idge p inciples and how o
p oduce hem is a e y delica e one. They a e no good o bad on hei
7See Béziau and Coniglio (2011) o an in e es ing analysis.
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Mixed Reasoning and Combining Logics
O he me hods o combining logics ha do no belong o he mains eam
o ca ego ial ib ing a e ecumenism, by P awi z (2015), and chunk and pe -
mea e, by B own and P ies (2004)(see, specially, P ies (2014)). The e a e
some easons why I will no ocus on hese me hods. One o hem is ha
he e is only so much ime one has o doing a hesis and some hings, he
ones ha do no seem so ele an o ce ain pu poses, ha e o be le ou .
Ano he eason, his one speci ic o ecumenism, is ha he philosophical mo-
i a ion o P awi z when de eloping he me hod appea s o be subs an ially
independen o mine. The philosophical mo i a ion o ecumenism is o gi e
an in e en ialis seman ics o classical connec i es. My mo i a ion, hough,
is o p o ide a sys em ha combines logics wi h he aim o ha ing a localis
eading o he mechanism.
This mo i a ion o mine could possibly ma ch one o P ies ’s applica ions
o chunk and pe mea e, indeed. I onic as i migh sound, P ies , oge he
wi h M. B. B own (B own and P ies (2004)) de eloped a s a egy o “han-
dling he applica ion o di e en logics in combina ion” (P ies ,2014, 333).
The p oblem is ha he me hod o chunk and pe mea e is no e y sys ema ic
no gene al and, mainly, depends on how one wan s o design he mechanism
o a speci ic applica ion. On op o his limi a ion, he e is no me a heo-
e ical esul and no p ese a ion heo em ha can help us unde s and mo e
he mechanism o he sake o sys ema izing i .
Maybe i is because o he monopoly ha algeb aic ib ing has on he ield
o combining logics, bu , he u h is ha nei he ecumenism, no chunk and
pe mea e, and no e en jux aposi ion appea men ioned in he en y “Com-
bining Logics” o he S an o d Encyclopedia. I hink his is un o una e,
specially o jux aposi ion, which is a e y well de eloped me hod wi h so
much po en ial as I hope o show.
Be o e mo ing on o he p esen a ion o some o he me hods jus men-
ioned, le me e e o wo di e en app oaches o he combina ion o logics
in o de o cla i y which one o hose I am ocusing on. The app oaches a e
hose o spli ing e sus splicing logics. Wi h espec o spli ing logics,
we may hink abou an analy ic p ocedu e ha pe mi s us o decom-
pose a gi en logic in o simple componen s. [...] A p o o ypical case o
spli ing occu s when one succeeds in desc ibing a gi en logic in e ms
o simple componen s by means o ansla ing he o iginal logic in o
a collec ion o simple , auxilia y logics, using wha is called possible-
ansla ions seman ics. (Ca nielli e .al.,2008, p. 10)
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Mixed Reasoning and Combining Logics
So, spli ing is a op-down analy ic app oach aiming o decompose a logic
in o simple agmen s. Splicing, on he o he hand, is a “p ocess, by which
a bunch o logics is syn hesized o ming a new logic”(Ca nielli e .al.,2008,
p. 10). So, i is a bo om-up, syn he ic app oach, by means o which simple
logics a e combined in o de o ob ain a mo e complex sys em. This is he
app oach ha we a e going o be using. So, e e y me hod ha I am going
o p esen now, and also jux aposi ion, a e cases o splicing logics.
3.2.1 Fib ing by unc ions
The me hod o ib ing, also known as ‘ ib ing by unc ions’, was o iginally
p oposed by D. Gabbay in Gabbay (1996)12. As we said abo e, his mech-
anism only applies o logics wi h K ipke seman ics. S ill, i is a powe ul
me hod o combining such logics. Le me s a by gi ing some de ini ions:
De ini ion 3.2.1 (De ini ion 7 in Coniglio and Fe nández (2005)).Amodal
signa u e is a signa u e Csuch ha C1={¬,},C2={→} and Ck=
∅in any o he case. A K ipke model ( o modal logics) is a iple m=
hWm, Rm, hmisuch ha Wmis a nonemp y se ( he se o possible-wo lds o
m); Rm⊆Wm×Wm( he accessibili y ela ion o m); and hm:V −→ ℘(Wm)
is a mapping ( he m- alua ion). A K ipke seman ics is a class K o K ipke
models.
Le us, hen, deno e he modal logics by he pai L=hCL, K i. Gi en
wo logics, L1and L2we de ine he ib ed language,L(C⊗), which is ob ained
om he ib ed signa u e,C⊗, namely:
C1
⊗={¬,1,2};C2
⊗={→};Ck
⊗=∅in any o he case.
Now, in o de o ge a ib ed logic, we need o pe o m he ib ing o he
K ipke models. The undamen al idea is o ake ib ed K ipke models wi h
dis inguished ac ual wo lds and o connec he wo lds o one model wi h he
wo lds o he o he , in such a way ha i we a e e alua ing, say, a o mula
like 2αin a K ipke model o L1in a wo ld om Wm1, we can mo e o he
co esponding wo ld om Wm2in model m2, in o de o check he alidi y
o 2α. Thus, a ib ed model o K 1and K 2is a iple ( , g, h)such ha :13
12In o de o p esen he me hod o ib ing, I will be ollowing Gabbay (1996), bu also
Ca nielli and Coniglio (2020) and Coniglio and Fe nández (2005), since hese a e e y
clea and concise explana ions o he me hod.
13Udeno es he disjoin union o se s.
68
Mixed Reasoning and Combining Logics
:]
m1∈K 1
Wm1−→ ]
m2∈K 2
Wm2;
g:]
m2∈K 2
Wm2−→ ]
m1∈K 1
Wm1;
h:V −→ ℘(W)
wi h W:= (Um1∈K 1Wm1)](Um2∈K 2Wm2)and and gas ans e map-
pings: om he se o wo lds o he class o models K 1o L1in o he class
o models K 2o L2, and g om he se o wo lds o he class o models K 2
o L2in o he class o models K 1o L1.
The ib ed s uc u e K 1⊗K 2is he class o all he ib ed models o
K 1and K 2. The sa is ac ion o a o mula αby he ib ed model ( , g, h)
in he wo ld w, deno ed ( , g, h)wα, is de ined ecu si ely as usual when
he main connec i e o αis Boolean, i.e., ¬o →, and when he modal
ope a o and he wo ld co espond o he same logic ( ha is, in a si ua ion
o s anda d modal logic). The ele an cases a e hose in which he modal
ope a o and he wo ld o e alua ion ha e di e en o igins. Fo ins ance,
when e alua ing he o mula 1αin he ib ed model ( , g, h)and he wo ld
w2. The sa is ac ion clause goes as ollows: he model ( , g, h)sa is ies 1α
in w2∈Wm2,( , g, h)w21αi ( , g, h)w0
1α o e e y w0
1∈Wm1such
ha g(w2)Rm1w0
1. The o he case, wi h he subindexes o he box and he
wo ld in e changed, wo ks analogously.
Thus, wi h his no ion o sa is ac ion we cha ac e ize logical consequence
in he usual way, as p ese a ion o sa is ac ion om p emises o conclusion.
The ib ed consequence ela ion is, `K 1⊗K 2⊆℘(L(C⊗)) ×L(C⊗)and he
logic L⊗=hC⊗,`K 1⊗K 2iis he ib ing o L1and L2.
Again, he me hod o ib ing is limi ed by i s applicabili y o modal logics
wi h K ipke seman ics, al hough, we will soon see ha plain ib ing is a
na u al way o ex ending he me hod beyond ha limi . In any case, ib ing
is c ucial in he his o y o combining logics, since i inspi ed he de elopmen
o new, mo e gene al, me hods. Among hem, ca ego ial ib ing has been he
one a ound which almos he whole ield o combining logics has o bi ed.
3.2.2 Ca ego ial (o Algeb aic) Fib ing
Since ca ego ial ib ing was p esen ed in Se nadas e .al. (1999), many de el-
opmen s ha e eme ged a ound i . So, his b ie and ough p esen a ion o
69
Mixed Reasoning and Combining Logics
he me hod should no be aken o ep esen o be ai h ul o he whole pic-
u e. In ac , e en he me hod i sel , lea ing aside subsequen imp o emen s
a ound i , poses qui e a challenge o be summa ized in a simple way, since
i in ol es some ideas om ca ego y heo y. This is why I will be ollowing
Schech e (2011), on op o Ca nielli e .al. (2008), and his way o p esen -
ing ca ego ial ib ing; a way ha be e lends i sel o be compa ed wi h
jux aposi ion, which is he main me hod ha I will be using.
Thus, I shall con ine mysel o he seman ics o ib ing. Ano he eason
why ocusing on he seman ics is in e es ing o us is ha , as I poin ed
ou abo e, he collapse o he algeb aic ib ing o CL and IL occu s a he
seman ic le el, no when ib ing hei Hilbe calculi (see Rasga e .al. (2002)).
So, i is in e es ing o my pu poses o show a me hod o combining logics
whose seman ics collapse, since his is some hing ha I wan o a oid when
ying o sol e he p oblem o mixed in e ences appealing o a combina ion
mechanism.
Following Ca nielli e .al. (2008) he seman ic uni o algeb aic ib ing is
an algeb a, hB, Φi, unde s ood as a uple wi h a se , he ca ie se , B, and
a amily o ope a ions, Φ. Bu , “in o de o ensu e he p ese a ion o some
p ope ies by ib ing, i is con enien o conside en iched algeb as, called
in e p e a ion s uc u es” (Ca nielli e .al.,2008, p. 92). An in e p e a ion
s uc u e o e a signa u e C, is a uple hB, ≤,Φ,>i, whe e hB, ≤,>i is a
pa ial o de wi h a op elemen >and hB, Φiis an algeb a o e C. In he
e minology o Schech e (2011), his in e p e a ion s uc u es a e pa ially
o de ed uni al s uc u es, he ‘uni al’ e e ing o he ac ha he >is he
unique designa ed alue. Mo eo e , he ela ion ≤allows us o compa e he
u h alues in B, which makes possible o de ine wo di e en no ions o
en ailmen , al hough we will only ocus on one; global en ailmen . Gi en
a class o pa ially o de ed uni al s uc u es, i.e., a class o in e p e a ion
sys ems, B, we say ha Γglobally en ails αin Bi o e e y in e p e a ion
s uc u e in Band alua ion, i.e., o e e y model, i e e y γ∈Γge s alue
>, hen αge s alue >.
Now, ake an in e p e a ion s uc u e B=hB, ≤,Φ,>i o e a signa u e
C. Suppose ha C0is a subsigna u e o C,C0≤C. We de ine he educ o
B o C0as he uple B|C0=hB, ≤,Φ|C0,>i, whe e Φ|C0is he es ic ion o Φ
o he connec i es in C0. Reduc s play an impo an ole in algeb aic ib ing,
since when doing he algeb aic ib ing o wo classes o pa ially o de ed
uni al s uc u es, B1and B2, o e he signa u es C1and C2, espec i ely,
wha we ge is a class o pa ially o de ed uni al s uc u es and, o each
70
Mixed Reasoning and Combining Logics
s uc u e in he class, we mus ha e ha B|C1∈B1and B|C2∈B2. Tha
is, he algeb aic ib ing o B1and B2is he class o pa ially o de ed uni al
s uc u es, B12 ={B :B|C1∈B1and B|C2∈B2}o e he ib ed signa u e
C1∪C2.
This way o cons uc ing he ib ed seman ics has some limi a ions, hough.
Limi a ions ha a e c ucial o he philosophical p oblem ha I am dealing
wi h. This is because, as I poin ed ou abo e, algeb aic ib ing is no well
sui ed o combining he seman ics o classical and in ui ionis ic logics. The
eason is he ollowing: classical logic is sound and comple e wi h espec o
he class o Boolean algeb as, BA. In ui ionis ic logic is sound and comple e
wi h espec o he class o Hey ing algeb as, HA, which gene alize Boolean
algeb as. Then, he algeb aic ib ing o he seman ics o CL and IL is he
class o pa ially o de ed uni al s uc u es, le us call i Bci, o e he ib ed sig-
na u e Cc∪Ci, wi h Cc={¬c,∧c,∨c,→c,↔c}and Ci={¬i,∧i,∨i,→i,↔i},
such ha Bci ={B|B|Cc∈BA and B|Ci∈HA}. Bu , i one looks a he
de ini ion o educ , one can easily see ha he ca ie se s o B|Ccand B|Ci
coincide. Thus, o e e y in e p e a ion s uc u e in he ib ed class o in-
e p e a ion s uc u es o be a Boolean algeb a and a Hey ing algeb a o
i s ele an educ s, e e y s uc u e has o be a Boolean algeb a, i.e., e -
e y B|Ciis a Hey ing algeb a ha is also Boolean. Hence, he in ui ionis ic
connec i es will beha e exac ly like he classical ones and, so, hey will be in-
e subs i u able. This means, acco ding o he de ini ions gi en abo e, ha
he algeb aic ib ing o CL and IL collapses and, a o io i, weakly collapses.
Recall, hough, ha esponding o W enn’s challenge is an impo an pa
o gi ing a solu ion o he p oblem o mixed in e ences and ha , his chal-
lenge, essen ially in ol es being able o gi e a c i e ion o alidi y o mixed
in e ences wi h componen s coming om domains go e ned by classical and
in ui ionis ic logics. I we a e no able o implemen a combina ion mecha-
nism o hose logics while a oiding he collapse o he connec i es, we will
no be able o mee he challenge. Bu , hen, algeb aic ib ing does no seem
a good candida e in o de o accoun o he p oblem o mixed in e ences.
We know, howe e , ha he e a e o he op ions ha wo k be e .
3.2.3 Di ec Union and Plain Fib ing
Di ec union and plain ib ing a e an ex ension o Gabbay’s o iginal no ion
o ib ing o ano he class o logics, namely, logics cha ac e ized by ma ix
seman ics. These me hods we e p oposed in Coniglio and Fe nández (2005)
71
Mixed Reasoning and Combining Logics
and, al hough hey can s ill be de eloped mo e and u he me alogical esul s
could be ob ained, hey a e in e es ing as a con inua ion o Gabbay’s wo k
and also because o he simila i ies wi h jux aposi ion.
In ac , I should cla i y ha hey a e no wo o ally independen me h-
ods. Di ec union is he me hod ha one applies when we a e in he simple
and smoo h scena io o combining wo logics in “which he domain and des-
igna ed alues o he ma ices in ol ed a e he same. In such cases, he
combined logic can be simply ob ained by pu ing oge he bo h ma ices”
(Coniglio and Fe nández,2005, p. 1596). Wi h plain ib ing, howe e , we
can ace he mo e di icul scena io in which he domains and he alues des-
igna ed a e di e en . The idea, in his case, is o build, wi h app op ia e
unc ions, a bigge ma ix ha encompasses hose o he logics being com-
bined. Once we ha e his gene al ma ix o bo h logics, we do hei di ec
union. In his sense, di ec union is he inal s age o plain ib ing. Le me
begin by laying ou some de ini ions and, hen, we will see he me hods wi h
a li le bi mo e de ail.
De ini ion 3.2.2 (De ini ion 5 in Coniglio and Fe nández (2005)).Gi en a
signa u e C, a C−ma ix is a pai M=hA, Diwhe e A=hA, Diis an
algeb a o e Cand D⊆Ais he se o designa ed alues o M.
De ini ion 3.2.3 (De ini ion 6 in Coniglio and Fe nández (2005)).Le C
be a signa u e and le Kbe a class o C-ma ices. The ma ix seman ics
induced by K(deno ed by `K) is de ined by: Γ`Kαi o e e y C-ma ix
M=hA, Dibelonging o Kand e e y alua ion , i (Γ) ⊆D hen (α)∈
D.
We conside , i s , he simple case in which he domains and he desig-
na ed alues o ma ices a e he same. In his case, he di ec union consis s
o pu ing oge he bo h ma ices in he ollowing way:
De ini ion 3.2.4 (De ini ion 9 in Coniglio and Fe nández (2005)).Le L=
hCi, Mii(wi h i∈ {1,2}) be wo ma ix logics, whe e each Mi=hAi, Diiis a
Ci-ma ix. Assume ha A1=A=A2and D1=D=D2. The di ec union
o L1and L2is he logic L1+L2=hC1]C2,`M1+M2iwhe e `M1+M2is he
consequence ela ion de ined by he C1]C2-ma ix M1+M2=hA, Disuch
ha , i c∈Ck
iand a1, ..., ak∈A, hen cM1+M2(a1, ..., ak) = cMi(a1, ..., ak)
(i∈ {1,2}).
72

Mixed Reasoning and Combining Logics
So, one can see ha hose ypes o combina ions a e p e y s aigh -
o wa d. Since he domains and he designa ed alues a e he same, he
deno a ion o a connec i e om Ciis jus he same in he o iginal ma ix Mi
and in he di ec union o ma ices, M1+M2. Tha is, we will no ge he
mo e di icul case in which he a gumen o a u h- unc ional connec i e is
a seman ic alue o which he u h- unc ion was no de ined.
We now conside he mo e in e es ing case o combining logics cha ac-
e ized by ma ix seman ics wi h di e en domains. This is he me hod o
plain ib ing and he eade will immedia ely see he simila i ies wi h ib-
ing, which is a special case o his mo e gene al app oach. In Coniglio and
Fe nández (2005), hey ea , i s , he case in which he e is no es ic ion
o he ans e mappings, i.e., un es ic ed plain ib ing. Howe e , I di ec ly
conside he si ua ion in which we es ic he mappings in a way speci ied
below.
De ini ion 3.2.5 (De ini ion 14 in Coniglio and Fe nández (2005)).Le
L=hC, Mibe a ma ix logic, whe e each M=hA, Diis a C-ma ix wi h
domain Aand se o designa ed alues D. Le A0and D0be wo se s such
ha D0⊆A0. Suppose, wi hou loss o gene ali y, ha A∩A0=∅. Finally,
le :A0−→ Abe a mapping. The C-ma ix M is de ined as ollows:
i s domain is A]A0; i s se o designa ed alues is D]D0and o c∈Ck
i
and a1, ..., ak∈A]A0,cM (a1, ..., ak) = cM(a1, ..., ak)whe e, o e e y aj
(j= 1, ..., k):
- I aj∈A, hen aj=aj.
- I aj∈A0, hen aj= (aj).14
De ini ion 3.2.6 (De ini ions 13, 15 in Coniglio and Fe nández (2005)).Le
L=hCi, Mii(wi h i= 1, 2) be wo ma ix logics, whe e each M=hAi, Dii
is a Ci-ma ix wi h domain Ai. The ib ed signa u e is gi en by C1]C2and
he ib ed language is L(C1]C2). A ib ed alua ion is a iple ( , g, ), whe e
( , g)∈AA1
2×AA2
1, such ha ( , g)is admissible and ∈(A1]A2)V(Vbeing
he se o p oposi ional a iables). A pai ( , g)∈AA1
2×AA2
1is admissible i
i sa is ies: (x)∈D2i x∈D1, o e e y x∈A1; and g(y)∈D1i y∈D2,
o e e y y∈A2. Gi en φ∈L(C1]C2)and a ib ed alua ion ( , g, ), we
de ine ( , g, )(φ)∈A1]A2by ecu sion on he complexi y o φ:
- I φ∈ V hen ( , g, )(φ) = (φ);
14We will see an illus a ion o his kind o ma ices in Example 3.2.1.
73
Mixed Reasoning and Combining Logics
- I φ=c(β1, ..., βk) hen ( , g, )(φ) = c(( , g, )(β1), ..., ( , g, )(βk))
whe e o e e y o mula βj(j= 1, ..., k):
•I c∈Ck
iand ( , g, )(βj)∈Ai hen ( , g, )(βj) = ( , g, )(βj)
o (i= 1,2);
•I c∈Ck
1and ( , g, )(βj)∈A2 hen ( , g, )(βj) = g(( , g, )(βj));
•I c∈Ck
2and ( , g, )(βj)∈A1 hen ( , g, )(βj) = (( , g, )(βj));
We say ha a ib ed alua ion ( , g, )sa is ies φi ( , g, )(φ)∈D1]D2.
The plain ib ed consequence ela ion `M1M2⊆℘(L(C1]C2))×L(C1]C2)is
de ined as ollows: Γ`M1M2φi , o e e y ib ed alua ion ( , g, )sa is ying
simul aneously all he o mulas o Γ, we ha e ha ( , g, )sa is ies φ. The
plain ib ing o L1and L2is he pai L1 L2=hC1]C2,`M1M2i.
Now, his is al eady a much mo e in e es ing combina ion mechanism,
because he e is a wide a ie y o logics, o agmen s o logics, ha we can
combine. Mo eo e , we know by P oposi ion 10 in Coniglio and Fe nández
(2005), ha he plain ib ing o wo ma ix logics, L1 L2, is a conse a i e
ex ension o bo h L1and L2. Wha his means, is ha C1⊆C1]C2,
C2⊆C1]C2and Li= (L1 L2)|Ci o (i= 1,2). Tha is, when we es ic
he plain ib ing o each o he signa u es, we ge he o iginal logics.
No ice ha his is an impo an ea u e, since i shows al eady ha , i
he se s o alid in e ences o he logics a e di e en , hen he plain ib ing
o hem does no weakly collapse and, he e o e, does no collapse ei he . To
see his, hink, o ins ance, abou he case o doing he plain ib ing o CL
and LP. We know ha in CL we ha e modus ponens bu in LP we don’ .
This means ha in CLLP we will ha e ha p→cq, p `CLLP q, bu
p→pq, p 0CLLP q, which is enough o show ha CLLP does no weakly
collapse, because we ha e ound a alid in e ence in he plain ib ing, whose
ansla ion o he in e ence wi h co esponding connec i es is no longe alid.
Since I wan o a oid excessi e de ail un il we plunge in o he me hod o
jux aposi ion, le me conclude he sec ion by u he illus a ing he me hod
o plain ib ing wi h an example, gi en in Coniglio and Fe nández (2005)
which, I hope, will help o ge a be e pic u e o he me hod.
Example 3.2.1.Take he nega ion agmen o he pa aconsis en ma ix logic
P1.15 Le , hen, L1be he agmen o P1de ined o e he signa u e {¬P1},
15This is a pa aconsis en logic in oduced in Se e (1973).
74
Mixed Reasoning and Combining Logics
gi en by he ma ix M1wi h domain A1={T, T1, F}and se o designa ed
alues D1={T, T1}.
¬P1
T F
T1T
F T
Le also L2be he agmen o CL de ined o e he signa u e {→c}gi en
by he ma ix M2wi h domain A2={1,0}and se o designa ed alues
D2={1}.
→c1 0
1 1 0
0 1 1
Now, we s a ib ing he ma ices by aking he disjoin unions o he
domains and designa ed alues. So, le he domain be A={T, T1, F, 1,0}
and se o designa ed alues D={T, T1,1}and le ( , g)∈AA1
2×AA2
1be
he ollowing admissible alua ion (T) = (T1) = 1, (F) = 0, g(1) = T
and g(0) = F. Then, (M1)gand (M2) a e gi en by he ollowing ma ices,
espec i ely: 16
¬
T F
T1T
1F
F T
0T
→T T11F0
T1 1 1 0 0
T11 1 1 0 0
1 1 1 1 0 0
F1 1 1 1 1
0 1 1 1 1 1
16These ma ices a e he ex ensions o he o iginal ma ices wi h he seman ic alues o
each o he . This is possible, because we ha e he alua ions, i.e., he ans e mappings,
in o de o know how o ansla e he seman ic alues o one ma ix in o he o he , so as
o be able o de e mine he seman ic ma ix o a connec i e when ha ing as a gumen s
possibly new seman ic alues ha we e no in i s o iginal de ini ion.
75
Mixed Reasoning and Combining Logics
Thus, le Lbe he logic o e {¬,→} cha ac e ized by he ma ix M( ,g)=
(M1)g+ (M2) gi en by he wo ables abo e, wi h {T, T1,1}as he se o
designa ed alues. No ice, hough, ha he e is an addi ional admissible pai
( , g0)such ha g0(1) = T1and g0(0) = F. The e o e, he ib ed logic L1L2
is cha ac e ized by he se o ma ices:
M1M2={(M1)g+ (M2) ,(M1)g0+ (M2) }
We can check ha , o ins ance, he mixed o mula (p→q)→ ¬¬(p→q)
is no alid in L1 L2, since o he pai ( , g0)and a alua ion o he
p oposi ional a iables such ha (p) = (q)=1, he ( , g0, )(p→q)=1
and, since g0(1) = T1, by he u h- able o nega ion ¬(T1) = Tand ¬(T) =
F. Bu , gi en ha he pai is admissible, (F) = 0 and, he e o e, wi h
p→qbeing 1 and ¬¬(p→q)being 0, (p→q)→ ¬¬(p→q)is going o be
0. So, he o mula is no alid in L1 L2.
Clea ly, plain ib ing cons i u es a big s ep o wa d wi h espec o Gab-
bay’s ib ing and, also, wi h espec o algeb aic ib ing (a leas conce ning
he collapse heo ems). I, ce ainly, belie e ha plain ib ing is also a good
candida e o success ully be applied in add essing he p oblem o mixed in-
e ences. Howe e , I will be s icking o jux aposi ion since, al hough maybe
being mo e ma ginal han any p oposal in he ib ing adi ion, i is mo e
de eloped han plain ib ing.
Modula ed ib ing and c yp o ib ing whe e designed o a oid he col-
lapses bu also o be applied o an e en la ge a ie y o logic sys ems, which
makes hem, de ini ely, e y a ac i e i he ul ima e goal o combina ion
mechanisms is o a i e a a uni e sal me hodology o combining any gi en
logic sys ems, independen ly o how hey a e p esen ed, e.g., algeb aically,
axioma ically, wi h non-de e minis ic seman ics, e c. The eason o no
ocusing on hem and going wi h jux aposi ion as my p e e ed me hod o
mee ing he challenge, hen, is ha he cases o mixed in e ences ha I could
po en ially add ess wi h jux aposi ion and he me hod i sel we e challeng-
ing enough. Mo eo e , I ha e been able o come up wi h ways in which
jux aposi ion could be imp o ed, in di ec ions ha a e qui e illumina ing o
hinking abou in e ac ion p inciples and combina ions o logics, as I will y
o show.
Hence, I am no claiming ha jux aposi ion, o any o my imp o emen s
he eo , a e going o be he bes solu ion and he one ha encompasses he
g ea e di e si y o logics. Bu , in o de o sol e any di icul p oblem i is
usually a good idea o di ide i o, a some poin , conque i . My modes goal
76
Mixed Reasoning and Combining Logics
P oposi ion 3.3.3 (S ong Conse a i eness).Suppose C1and C2a e dis-
join signa u es. Suppose each o `1and `2is consis en and has no me e
ollowe s. Suppose `12 is he jux aposi ion o `1and `2. Then `12 is a s ong
conse a i e ex ension o each o `1and `2.
P oposi ion 3.3.4 (P ese a ion o Consis ency).Suppose C1and C2a e
disjoin signa u es. Suppose each o `1and `2is consis en and has no
me e ollowe s. Suppose `12 is he jux aposi ion o `1and `2. Then `12 is
consis en . I Γ⊆Sen (Ci,Pi) is consis en wi h espec o `i, hen Γis
consis en wi h espec o `12.
The p ese a ion o consis ency om each consequence ela ion `1and
`2 o he jux aposed one, elies on he ac ha `12 is a s ong conse a i e
ex ension o `1and `2.
Le us now ad ance o he mo e challenging comple eness esul s. Schech e ’s
s a egy is based on a modi ica ion o he Lindenbaum-Ta ski cons uc ion.
The idea is o p o ide di ec p oo s o s ong comple eness ha apply in a
a ie y o cases by inding sui able equi alence ela ions in o de o build he
Lindenbaum-Ta ski models. He e we ocus on he essen ial esul s.
Le ∼be an equi alence ela ion on Sen (C,P). We say ha ∼is a
cong uence o e C−whene e :
•Fo e e y c−∈C−n, α1, ..., αn, β ∈Sen (C,P) and k∈ {1, ..., n}, i
αk∼β, hen c−α1...αk...αn∼c−α1...β...αn.
∼is compa ible wi h `and Γ⊆Sen (C,P) whene e :
•Fo e e y α, β ∈Sen (C,P), i α∼β hen Γ`αjus in case Γ`β.
We say ha ∼is s ongly compa ible wi h `and Γ⊆Sen (C,P) jus in
case ∼is a compa ible wi h `and Γand:
•Fo e e y α, β ∈Sen (C,P), i bo h Γ`αand Γ`β hen α∼β.
83

Mixed Reasoning and Combining Logics
We say ha ∼is sui able o C−,`and Γjus in case ∼is a cong uence
o e C−compa ible wi h `and Γ. We say ha ∼is uni al sui able o C−,
`and Γjus in case ∼is a cong uence o e C−s ongly compa ible wi h `
and Γ. In o de o cons ue he Lindenbaum-Ta ski models, Schech e makes
use o sui able and uni al sui able equi alence ela ions.
Le us gi e some addi ional de ini ions be o e mo ing o he esul s. Sup-
pose Γis a nonemp y subse o Sen (C12,P12) consis en wi h espec o `12.
Fo each i∈ {1,2}, suppose ∼Γ
iis an equi alence ela ion on Sen (C12,P12)
sui able o Ci,`12 and Γ. We de ine:
•|α|Γ
i={β|α∼Γ
iβ}
•BΓ
i={|α|Γ
i|α∈Sen (C12,P12)}
•DΓ
i={|α|Γ
i|Γ`12 α}
•I ci∈Cn
i,ΦΓ
i(ci)(|α1|Γ
i, ..., |αn|Γ
i) = |ciα1...αn|Γ
i
•BΓ
i=hBΓ
i,DΓ
i,ΦΓ
ii
•BΓ
12 =hBΓ
1,BΓ
2i
•I αis an i-a om, VΓ
i(α) = |α|Γ
i
•MΓ
12 =hBΓ
1,VΓ
1,BΓ
2,VΓ
2i
Then, BΓ
12 and MΓ
12 a e he Lindenbaum-Ta ski jux aposed s uc u e and
he Lindenbaum-Ta ski jux aposed model o C1,C2,`12 and Γ, buil wi h
∼Γ
1and ∼Γ
2.
Now, suppose o e e y i∈ {1,2}and nonemp y Γ⊆Sen (C12,P12)
consis en wi h espec o `12,∼Γ
iis an equi alence ela ion on Sen (C12,
P12) sui able o Ci,`12 and Γ. Le us de ine:
B∼
12 ={BΓ
12|Γ⊆Sen (C12,P12) is nonemp y and consis en wi h espec o
`12 and BΓ
12 is buil wi h ∼Γ
1and ∼Γ
2}
B∼
12 is he Lindenbaum-Ta ski class o jux aposed s uc u es o C1,C2
and `12 buil wi h ∼Γ
i.
84
Mixed Reasoning and Combining Logics
Theo em 3.3.1 (S ong Comple eness).Suppose `12 has no me e ollowe s.
Suppose o e e y i∈ {1,2}and nonemp y Γ⊆Sen (C12,P12) consis en
wi h espec o `12,∼Γ
iis an equi alence ela ion on Sen (C12,P12) sui able
o Ci,`12 and Γ. Then `12 is s ongly comple e wi h espec o B∼
12.
Theo em 3.3.2 (S ong Soundness).Suppose `12 is he jux aposi ion o `1
and `2. Suppose o e e y i∈ {1,2}and nonemp y Γ⊆Sen (C12,P12)
consis en wi h espec o `12,∼Γ
iis an equi alence ela ion on Sen (C12,
P12) sui able o Ci,`12 and Γ. Then `12 is s ongly sound wi h espec o
B∼
12.
P oposi ion 3.3.5.Suppose `12 has no me e ollowe s. Suppose o e e y
i∈ {1,2}and nonemp y Γ⊆Sen (C12,P12) consis en wi h espec o `12,
∼Γ
iis an equi alence ela ion on Sen (C12,P12) sui able o Ci,`12 and Γ.
Then i `12 is consis en , he e is a cohe en non i ial jux aposed model
based on B∼
12.
Summa izing he esul s:
Theo em 3.3.3.Suppose `12 has no me e ollowe s. Suppose o e e y
i∈ {1,2}and nonemp y Γ⊆Sen (C12,P12) consis en wi h espec o `12,
∼Γ
iis an equi alence ela ion on Sen (C12,P12) sui able o Ci,`12 and Γ.
Then:
1. `12 is s ongly comple e wi h espec o B∼
12.
2. I `12 is he jux aposi ion o `1and `2, hen `12 is s ongly sound wi h
espec o B∼
12.
3. I `12 is consis en , hen he e is a cohe en non i ial jux aposed model
based on B∼
12.
4. B∼
12 is a class o jux aposed uni al s uc u es jus in case o e e y
i∈ {1,2}and nonemp y Γ⊆Sen (C12,P12) consis en wi h espec o
`12,∼Γ
iis uni al sui able o Ci,`12 and Γ.
3.3.5 Applying he esul s o Classical and In ui ion-
is ic Logics
A e ob aining his ple ho a o me alogical esul s, Schech e himsel makes
use o he me hod o jux aposi ion in o de o apply i o he cases o com-
bining classical and in ui ionis ic logics, since, as al eady poin ed ou , i is
85
Mixed Reasoning and Combining Logics
o hose cases ha he collapse heo ems ha e been p o ed in he li e a u e.
Thus, he s a s de eloping he jux aposi ion o classical and in ui ionis ic
logics as ollows.
Le P1=P2=P12 be a coun ably in ini e se o sen ence symbols and o
i= 1,2le Cibe he signa u e con aining hese se s o connec i es:
•C1
i={¬i}
•C2
i={∧i,∨i,→i,↔i}
So, C12 is he se con aining wo copies o each o he p oposi ional con-
nec i es. Le `i
1and `c
2be he in ui ionis ic and classical consequence ela-
ions o Sen (C1,P1) and Sen (C2,P2) espec i ely. We say ha `ic is he
in ui ionis -classical jux aposed consequence ela ion o Sen (C12,P12). We
can also ha e consequence ela ions o languages wi h wo copies o classical
connec i es o wo copies o in ui ionis ic connec i es. We call hese, `cc and
`ii, he bi-classical and bi-in ui ionis consequence ela ions espec i ely.
In he sec ion abou in e ac ion p inciples we ad anced some ough de -
ini ions o collapse and weak collapse. Le us, now, gi e some mo e p ecise
de ini ions needed in o de o s udy `cc,`ii and `ic. A consequence ela ion,
`12, o Sen (C12,P12)collapses jus in case o e e y δ, δ0∈Sen (C12,P12)
exac ly alike excep pe haps o some o all o hei subsc ip s, {δ} `12 δ0.
Le be a bijec ion om Sen (C1,P12) o Sen (C2,P12) ha maps each
sen ence α∈Sen (C1,P12) o he sen ence ha esul s om uni o mly sub-
s i u ing each connec i e in αwi h he co esponding connec i e om C2.
We say ha a consequence ela ion, `12, o Sen (C1,P12)weakly collapses
jus in case o e e y Γ⊆Sen (C1,P12) and α∈Sen (C1,P12), Γ`12 αjus
in case (Γ) `12 (α).
We conside he e he case o in ui ionis -classical consequence ela ion,
`ic. On one hand, o any non i ial Boolean algeb a, hB, ≤i, he e is a
co esponding uni al s uc u e, hB, {1},Φi, whe e Bis he se o seman ic
alues, {1} is he g ea es elemen o he Boolean algeb a gi en by he o de
≤, and o e e y a, b ∈B:
Φ(¬)(a) = −a
Φ(a, b)(∧) = aub
Φ(a, b)(∨) = a b
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Mixed Reasoning and Combining Logics
Φ(a, b)(→) = −a b
Φ(a, b)(↔) = (−a b)u(−b a)
Wi h −,u, as he complemen , in imum and sup emum ope a ions in
he Boolean algeb a. We call hese s uc u es “Boolean s uc u es” and he
classical consequence ela ion is s ongly de e mined, i.e., is sound and com-
ple e, wi h espec o he class o Boolean s uc u es. I is consis en , has
heo ems and i is le -ex ensional.
On he o he hand, o any non i ial Hey ing algeb a, hB, ≤i, he e is
a co esponding uni al s uc u e, hB, {1},Φi, whe e Bis he se o seman ic
alues, {1} is he g ea es elemen o he Hey ing algeb a and o e e y
a, b ∈B:
Φ(¬)(a) = a⇒0
Φ(a, b)(∧) = aub
Φ(a, b)(∨) = a b
Φ(a, b)(→) = a⇒b
Φ(a, b)(↔) = (a⇒b)u(b⇒a)
Wi h u, ,⇒as he in imum, sup emum and implica ion ope a ions in
he Hey ing algeb a and 0as i s leas elemen .We call hese s uc u es “Hey -
ing s uc u es” and he in ui ionis consequence ela ion is s ongly de e -
mined wi h espec o he class o Hey ing s uc u es. I is also consis en ,
has heo ems and i is le -ex ensional.
By P oposi ion 3.3.4 `ic is consis en and by P oposi ion 3.3.3 `ic is
a s ong conse a i e ex ension o `iand `c. An in e es ing poin ha
Schech e jus men ions wi hou ge ing in o he de ails is ha he jux a-
posed consequence ela ion `ic can be axioma ized “using a copy o any
na u al deduc ion-s yle axioma iza ion o in ui ionis logic and a copy o
any na u al deduc ion-s yle axioma iza ion o classical logic, each es ic ed
so ha he ules o one s ock o connec i es canno be applied wi hin any
subde i a ion used in he applica ion o a me a ule go e ning a connec i e
om he o he s ock” (Schech e ,2011, p. 595).
87
Mixed Reasoning and Combining Logics
AHey ing-Boolean s uc u e is a jux aposed uni al s uc u e hB1,B2i
such ha B1is a Hey ing s uc u e and B2is a Boolean s uc u e. By
P oposi ion (P oposi ion 6.34 o co olla y 6.33 in Schech e (2011)), `ic is
s ongly de e mined wi h espec o he class o Hey ing-Boolean s uc u es.
The i s non-collapse esul ha one can easily check is ha he in ui ionis -
classical jux aposed consequence ela ion does no collapse. Jus no ice ha
since `ic is a s ong conse a i e ex ension o bo h `cand `iwe will ha e
0ic p∨1¬1pbu `ic p∨2¬2p. Mo eo e , we can p o e ha no co esponding
connec i es o C12 a e in e de i able in `cc, so nei he in he weake ela ions
`ic and `ii.
P oposi ion 3.3.6 (P oposi ion 7.1 in Schech e (2011)).In `cc, no pai o
co esponding connec i es a e in e subs i u able. In pa icula :
•{¬1p}0cc ¬2p
•{p∨1q}0cc p∨2q
•{p→1q}0cc p→2q
•{p↔1q}0cc p↔2q
•{¬2(p∧1q)}0cc ¬2(p∧2q)
P oo . In o de o p o e his, we ha e o build a cohe en jux aposed coun-
e model ha in alida es each o he p e ious en ailmen s. We look, hus, o
a model based on bi-Boolean s uc u es. As i is usually done, we ep esen
he Boolean algeb as using Hasse diag ams. Conside he ollowing Boolean
algeb as B1and B2:
≤1
1B1
0B1
≤2
1B2
0B2
ab
88

Mixed Reasoning and Combining Logics
Le us choose he ollowing alua ions. V1(p1) = V1(p2) = V1(p3) = 0B1,
V2(p1) = V2(p2) = a,V2(p3) = band le Vi(p) = 1Bi o e e y o he p∈
P12. Since M1=hB1, V1iand M2=hB2, V2idesigna e he same sen ence
symbols, we know by P oposi ion 3.3.1 ha he e is a cohe en jux aposed
model M12. Since `cc is s ongly sound wi h espec o he class o bi-
Boolean s uc u es, we make use o he model, M12, in o de o in alida e
he en ailmen s abo e.
In M12,||¬1p1||1= 1B1and ||¬2p1||2=b, so {¬1p1}0cc ¬2p1.||p1∨2
p3||2= 1B2bu ||p1∨1p3||1= 0B1so {p1∨2p3}0cc p1∨1p3and by symme y
{p1∨1p3}0cc p1∨2p3.||p1→1p3||1= 1B1and ||p1→2p3||2=b, he e o e
{p1→1p3}0cc p1→2p3.||p1↔1p3||1= 1B1and ||p1↔2p3||2= 0B2, so
{p1↔1p3}0cc p1↔2p3. And he las one, which in ol es an embedded
connec i e. ||¬2(p1∧1p2)||2=−2||p1∧1p2||2. Since he 1- alue o p1∧1p2
is 0B1and i is a 2-a om, we choose in he cons uc ion o he cohe en
jux aposed model V2(p1∧1p2) = 0B2, he e o e ||¬2(p1∧1p2)||2= 1B2. Bu ,
||¬2(p1∧2p2)||2=−2(a) = b, so {¬2(p1∧1p2)}0cc ¬2(p1∧2p2).
As al eady said, since `ii and `ic a e weake ela ions han `cc he esul
applies o hem as well. No ice ha despi e ∧1and ∧2no being in e sub-
s i u able in gene al, hey a e in e subs i u able as main connec i es e en in
`ii. This is because {p∧1q} `ii p,{p∧1q} `ii qand {p, q} `ii p∧2q, so
{p∧1q} `ii p∧2q. Since `ic and `cc a e s onge han `ii, his also holds o
hem.
Ano he in e es ing esul is ha `cc is no le -ex ensional, he e o e, `ii
and `ic a e no le -ex ensional ei he .
P oposi ion 3.3.7.`cc is no le -ex ensional.
P oo . Le B1and B2be he Boolean s uc u es o he p e ious p oo .
Take V1(p1) = V1(p2)=1B1,V2(p1) = V2(p2)=1B2and Vi(p)=0Bi o
e e y o he p∈P12. Since M1=hB1, V1iand M2=hB2, V2idesigna e
he same sen ence symbols, we know by P oposi ion 3.3.1 ha he e is a
cohe en jux aposed model M12. We cons ue he cohe en model in such a
way ha we can ha e ||p2||2,||p1||2,||¬2¬1p1||2∈D2while ||¬2¬1p2||2/∈D2.
Since V1(p1) = V1(p2) = 1B1,||¬1p1||1=||¬1p2||1= 0B1. Bu bo h ¬1p1
and ¬1p2a e 2-a oms, so ake V2(¬1p1) = 0B2and V2(¬1p2) = a espec ing
cohe ence. Now we ha e ha ||¬2¬1p1||2= 1B2and ||¬2¬1p2||2=b, so
||p2||2,||p1||2,||¬2¬1p1||2∈D2and ||¬2¬1p2||2/∈D2as desi ed. The e o e,
p1, p2,¬2¬1p10cc ¬2¬1p2and, so, `cc is no le -ex ensional. 
89
Mixed Reasoning and Combining Logics
Le us now ocus on he case o weak collapse. I is clea om he de ini-
ion ha bo h `ii and `cc weakly collapse. No ice also ha i a consequence
ela ion `collapses, hen i weakly collapses. Thus, since we know ha he
ib ing o `iand `ccollapses, i does also weakly collapse. Howe e , he
jux aposi ion o `iand `cdoes no weakly collapse, because `ic is a s ong
conse a i e ex ension o bo h `iand `c. Tha is, we will ha e, o ins ance,
¬c¬cp`ic pbu ¬i¬ip0ic p.
As I said be o e, he e is a weal h o esul s ha Schech e p o ed in
his pape and he e we ha e jus seen some o he mos ele an ones, bu
Schech e ’s pape con ains o he in e es ing esul s. My nex s ep, hough,
is o p esen some new, u he applica ions o jux aposi ion.
3.3.6 Fu he applica ions o jux aposi ion
Since Schech e ’s seminal pape on jux aposi ion, no u he de elopmen s
ha e been made on i (I belie e ha i has no e en been applied in he
li e a u e). In his sec ion, we apply he me hod o jux aposi ion in o de
o ob ain u he esul s. Some o hese ha e al eady been explo ed in he
li e a u e (see Béziau and Coniglio (2011) and Coniglio (2007)), bu no
unde he me hod o jux aposi ion. We s a wi h a esul ha I ha e al eady
men ioned abo e.
P oposi ion 3.3.8.The jux aposi ion o he logic o conjunc ion, L∧, and
he logic o disjunc ion, L∨, is no dis ibu i e. Fu he mo e, L∧∨ is no he
logic o la ices since abso p ion does no hold.
P oo . Le P1=P2=P12 be a coun ably in ini e se o sen ence symbols
and le C1and C2be he signa u es con aining hese se s o connec i es:
•C2
1={∧}
•C2
2={∨}
Le , hen, `∧and `∨be he consequence ela ions o Sen (C1,P1) and
Sen (C2,P2) espec i ely. Thus, `∧∨ is he jux aposed consequence ela ion
o he se o sen ences Sen (C12,P12).
In o de o p o e his, le us make use again o he algeb as we employed
in he p oo o P oposi ion 3.3.6, since, as Boolean algeb as, hey a e bo h
join and mee -semila ices.
90
Mixed Reasoning and Combining Logics
≤1
1∧
0∧
≤2
1∨
0∨
ab
We ha e o build cohe en jux aposed models in which no dis ibu i i y
nei he abso p ion hold. Tha is:
•Dis ibu i i y o ∧o e ∨:p∧(q∨ )0∧∨ (p∧q)∨(p∧ )
•Dis ibu i i y o ∨o e ∧:p∨(q∧ )0∧∨ (p∨q)∧(p∨ )
•Abso p ion-1: p∨(p∧q)0∧∨ p
•Abso p ion-2: p∧(p∨q)0∧∨ p
Le us ake he ollowing alua ion V1(p) = 1∧,V1(q) = V1( ) = 0∧,V2(p) =
1∨,V2(q) = aand V2( ) = b. We know by P oposi ion 3.3.1 ha he e is a
cohe en jux aposed model M12. In his model, ||p∧(q∨ )||1= 1∧uV1(q∨ ).
Since ||q∨ ||2= 1∨,V1(q∨ ) = 1∧, he e o e, ||p∧(q∨ )||1= 1∧. Bu ,
in his e y same model, ||(p∧q)∨(p∧ )||2=||p∧q||2 ||p∧ ||2. Since
||p∧q||1=||p∧ ||1= 1∧u0∧= 0∧, we can ake V2((p∧q)) = V2((p∧ )) = 0∨
and we ge ha he p emise is designa ed in he model while he conclusion is
no . Then, he splicing o he logic o conjunc ion and he logic o disjunc ion
by jux aposi ion does no p o e dis ibu i i y o ∧o e ∨.
Le us now show ha abso p ion-1 does no hold ei he . Take V1(p) =
V1(q)=0∧and V2(p) = a. In his model, ||p∨(p∧q)||2=a V2(p∧q).
Since ||p∧q||1= 0∧, ake V2(p∧q) = b. Thus, we ge ||p∨(p∧q)||2= 1∨,
while he conclusion is no designa ed. So, he jux aposi ion o he logics
o a mee -semila ice and a join-semila ice does no esul in he logic o a
la ice.

The eade can easily check by playing a ound wi h he alua ions ha
he e a e jux aposed coun e models o dis ibu i i y o ∨o e ∧and abso p ion-
2 as well.
91
Mixed Reasoning and Combining Logics
The nex esul s ha e o do wi h eco e ing a logic om i s agmen s.
Mo e speci ically, we deal wi h he case o classical logic.
P oposi ion 3.3.9.The jux aposi ion o he logic o (classical) nega ion,
L¬, and he logic o (classical) condi ional, L→, does no eco e classical
logic.
P oo . We p o e his p oposi ion by showing ha o he jux aposed con-
sequence ela ion, `¬→, despi e Ex Con adic ione Quodlibe holding as a
ule, i.e. p, ¬p`¬→ q, we do no ge he P inciple o Pseudo-Sco us, `¬→
p→(¬p→q), which is alid in CL.
Le P1=P2=P12 be a coun ably in ini e se o sen ence symbols and le
C1and C2be he signa u es con aining hese se s o connec i es:
•C1
1={¬}
•C2
2={→}
Le , hen, `¬and `→be he consequence ela ions o Sen (C1,P1) and
Sen (C2,P2) espec i ely. Thus, `¬→ is he jux aposed consequence ela ion
o he se o sen ences Sen (C12,P12).
We make use, again, o he p e ious Boolean algeb as:
≤1
1¬
0¬
≤2
1→
0→
ab
In o de o p o e 0¬→ p→(¬p→q)we look o a cohe en jux aposed
coun e model based on he abo e pai o s uc u es. Thus, le us ake a
alua ion such ha V1(p) = 1¬,V1(q) = 0¬,V2(p) = 1→and V2(q) = 0→.
Since ||¬p||1= 0¬, espec ing cohe ence we ake V2(¬p) = ain o de o
ha e ||p→(¬p→q)||2=−1→ (−a 0→) = b, which is no designa ed,
he e o e, 0¬→ p→(¬p→q)as desi ed. Howe e , we do ha e p, ¬p`¬→ q
as announced. Jus no ice ha since jux aposi ion is a s ong conse a i e
ex ension and p, ¬p`¬qholds, we will also ha e ha in e ence in `¬→.
Mo eo e , he e is no jux aposed model sa is ying bo h pand ¬p, so he
a gumen is i ially alid. 
92
Towa ds a Solu ion o he P oblem o Mixed In e ences
di e ence is ha we dis inguish, now, he se s o sen ence symbols. So,
Pc6=Piand Pc∪Pi=Pic.
Le us s a wi h he easies case, namely, Mix, o see how i is done.
Fi s , we o malize he a gumen in he jux aposed language and, hen, we
apply he me hod o jux aposi ion as he alidi y c i e ion.
1Mix:We ca s a e unny
Ei he snow is whi e o we ca s a e no unny
Snow is whi e
Recall he assump ion ha he discou se abou humou is an e alua i e
discou se in which easoning is bes cap u ed by in ui ionis ic logic, while
discou se abou middle-sized objec s, such as snow, is a discou se in which
easoning is bes cap u ed by classical logic. Unde his assump ion, le us
o malize he a gumen by ansla ing i in o ou jux aposed language.
pi
qc∨x¬ipi
qc
(x=i, c)
The e is a i s di icul y igh away. I seems na u al o ansla e he
nega ion as he in ui ionis ic one, since i is being applied o an in ui ionis ic
p oposi ion1, i.e., ‘we ca s a e unny’. Howe e , which disjunc ion should
we use in o de o o malize he a gumen ? Since i is a mixed sen ence
wi h one disjunc om he classical domain and ano he disjunc om he
in ui ionis ic, i is no ob ious whe he he disjunc ion should be classical
o in ui ionis ic. Remembe , hough, ha Mix is among he a gumen s ha
a e in ui i ely alid and, as we will see, he choice o he disjunc ion is no
innocuous in his espec .
1This is no a s ic ule and he con ex usually plays a majo ole in de e mining
which he co ec in e p e a ion is, as i also happens wi h a single s ock o connec i es
when, o ins ance, he use o an ‘i ’ in a sen ence migh be ambiguous and u he con ex
migh be needed in o de o ansla e i as a condi ional o a bicondi ional. Thus, he e
migh be cases in which wha he na u al exp ession is ying o con ey is, say, a classical
nega ion applied o an in ui ionis ic p oposi ion. We will discuss mo e hese issues o
ansla ion la e on.
99

Towa ds a Solu ion o he P oblem o Mixed In e ences
Indeed, i we ansla e he disjunc ion as he classical disjunc ion, he
a gumen is in alid. An easy way o see his is no icing ha , since he dis-
junc ion is classical, ¬ipiis a c-a om (classical a om) and so, he logical o m
o he a gumen is his: pi, qc∨c `ic qc. ‘ ’ is no a o ally independen
a iable because i ||pi||x∈Dx, hen ||¬ipi||x=|| ||x6∈ Dx, bu his con-
s ain is no enough o make he a gumen alid. Le us o e a jux aposed
coun e model o show i .
P oposi ion 4.1.1.pi, qc∨c¬ipi`ic qcis in alid.2
P oo . We build a cohe en jux aposed coun e model ha in alida es he
a gumen . We look, hen, o a model based on a Hey ing-Boolean s uc u e.
Conside he ollowing algeb as:
1B
0B
abc
1H
0H
We wan o ind alua ions such ha ||pi||x∈Dx,||qc∨c¬ipi||x∈Dxand
||qc||x6∈ Dx3. Since ||pi||x∈Dx,Vi(pi)=1Hand Vc(pi)=1B, so ||¬ipi||i=
0H. Gi en cohe ence, Vc(¬ipi)6∈ Dc, so ake Vc(¬ipi) = b. Now we can
choose Vc(qc) = a. Wi h hese alua ions we ge ||pi||c= 1B,||qc∨c¬ipi||c=
a cb= 1Band ||qc||c=a. So, we ha e buil a cohe en jux aposed model
in which ||pi||x∈Dx,||qc∨c¬ipi||x∈Dxand ||qc||x6∈ Dxas desi ed. 
Bu his canno be he end o he s o y wi h Mix, o he wise jux aposi ion
would no be e en a pa ial solu ion o he p oblem o mixed in e ences
because i would no be able o explain he alidi y o he easies o he cases
ha Lynch and W enn conside . In ac , we ha e a way o explaining he
in ui i e alidi y o Mix, and his equi es ha we ansla e he disjunc ion
as he in ui ionis ic one. Tha way, he logical o m o he a gumen is ha o
2This could al eady be p oblema ic o jux aposi ion. In ac , I will y o co ec his
when imp o ing he me hod.
3O cou se, he alua ions ha we a e going o ake ha e o espec cohe ence, in o de
o comply wi h he su icien condi ion o he exis ence o cohe en non i ial jux aposed
models (P oposi ion 3.3.1).
100
Towa ds a Solu ion o he P oblem o Mixed In e ences
an in ui ionis ic disjunc i e syllogism and, so, he in ui i ely alid a gumen
will be alida ed by jux aposi ion.
P oposi ion 4.1.2.pi, qc∨i¬ipi`ic qcis alid.
P oo . We wan o show ha o e e y cohe en jux aposed model ha
designa es he p emises, he conclusion will also be designa ed. So, suppose
ha ||pi||x∈Dxand ||qc∨i¬ipi||x∈Dx. Fo ||pi||x∈Dx,Vi(pi)=1H, so
||¬ipi||i= 0H. Now, ||qc∨i¬ipi||i∈Di, so ||qc∨i¬ipi||i= 1H, bu ha ing
||¬ipi||i= 0H he disjunc ion is designa ed only i ||qc||i∈Di. Thus, i he
p emises a e designa ed he conclusion has o be designa ed oo. Then, he
a gumen is alida ed by jux aposi ion. 
Le us conside , now, he mixed in e ence dubbed Essen ial Mix:
2Essen ial Mix:
Ei he i ’s no he case ha snow isn’ whi e o
we ca s a en’ unny
We ca s a e unny
Snow is whi e
We o malize he a gumen as:
¬c¬cpc∨x¬iqi
qi
pc
He e, again, we ha e a simila si ua ion. I we ansla e he disjunc-
ion as he classical disjunc ion, i u ns ou ha he a gumen is in alid.
The p e ious s uc u es oge he wi h alua ions Vc(pc) = a,Vi(qi) = 1H,
Vc(¬ipi) = byield a cohe en jux aposed coun e model. Howe e , we can
explain he in ui i e alidi y o he a gumen i we ansla e he disjunc ion
as he in ui ionis ic one.
P oposi ion 4.1.3.¬c¬cpc∨i¬iqi,qi`ic pcis alid.
P oo . Le us p o e he alidi y o he a gumen , in his case using he
jux aposed na u al deduc ion calculus, in he way ha Schech e desc ibes
i .
101
Towa ds a Solu ion o he P oblem o Mixed In e ences
1¬c¬cpc∨i¬iqi
2qi
3¬c¬cpc
4¬c¬cpcIden i y, 3
5¬iqi
6⊥i¬iE, 2,5
7¬c¬cpc⊥iE, 6
8¬c¬cpc∨iE, 1–8
9pc¬c¬cE, 8
So, we ha e a de i a ion o he conclusion om he p emises showing ha
he a gumen is alid. 
Remembe ha W enn’s imp o emen on Lynch’s modes y c i e ion was
able o explain he in ui i e alidi y o hese a gumen s we ha e jus consid-
e ed oo. Howe e , W enn comes up wi h he a gumen ‘Disjunc i e Mix’,
which is allegedly he knockdown mixed in e ence agains localis p oposals.
He claims ha , i he p oposal is modes enough, which needs o be in o de
o in alida e a gumen s like ‘Nix’, hen i will no be able o accoun o he
alidi y o ‘Disjunc i e Mix’. We will show, now, ha his is no in ac he
case.
3Disjunc i e Mix:
Ei he i ’s no he case ha snow isn’ whi e
o we ca s a e unny
Ei he snow is whi e o we ca s a e unny
We o malize he a gumen as:
¬c¬cpc∨xqi
pc∨xqi
(x=i, c)
102
Towa ds a Solu ion o he P oblem o Mixed In e ences
No ice ha in his mixed in e ence he ansla ion ha would p ese e he
logical o m o a alid a gumen is he one wi h a classical disjunc ion. Unde
his ansla ion ||¬c¬cpc||c=||pc||cand, so, ||¬c¬cpc∨cqi||c=||pc∨cqi||c,
which makes he a gumen alid, con a y o wha W enn p edic s (gi en
ha we a e able o explain he in ui i e in alidi y o ‘Nix’, as we will see
below).
Ne e heless, le us see ha he a gumen would no be alid i we ans-
la ed i as:
¬c¬cpc∨iqi
pc∨iqi
P oposi ion 4.1.4.¬c¬cpc∨iqi`ic pc∨iqiis in alid.
P oo . We build a cohe en jux aposed coun e model ha in alida es he
a gumen . We look, hen, o a model based on a Hey ing-Boolean s uc u e.
Conside , o ins ance, he ollowing algeb as:
1B
0B
ab
1H
0H
cd
We wan alua ions such ha ||¬c¬cpc∨iqi||i∈Diand ||pc∨iqi||i6∈
Di. Take he ollowing alua ion, Vi(qi) = d,Vi(pc) = 0H,Vi(¬c¬cpc) =
c,Vc(pc)=0Band Vc(qi) = a. I is easy o check ha hose s uc u es
oge he wi h hese alua ions gi e a cohe en jux aposed coun e model o
he a gumen . 
Up o his poin we ha e shown ha jux aposi ion is no oo modes
wi h espec o accoun ing o he in ui i e alidi y o he mixed in e ences
ha W enn poses as a challenge o localism. Now, we ha e o see ha
jux aposi ion is no oo immodes . Tha is, we ha e o be able o explain
he in ui i e in alidi y o Nix.
103
Towa ds a Solu ion o he P oblem o Mixed In e ences
4Nix:
I i is no he case ha o ensi e jokes a e unny,
hen g ass is no g een
G ass is g een
O ensi e jokes a e unny
We o malize he a gumen as:
¬ipi→x¬cqc
qc
pi
(x=i, c)
The e is an impo an di e ence o no ice in his case. We canno simply
ind a ansla ion in which he a gumen is (in) alid, as we did in he p e ious
cases. Now, bo h ansla ions o he condi ional ha e o be in alida ed by
jux aposi ion, o he wise jux aposi ion would lea e open a way o alida ing
an a gumen ha seems o be in ui i ely in alid4. Le us see how o ind
coun e models o each case.
P oposi ion 4.1.5.¬ipi→c¬cqc,qc`ic piis in alid.
P oo . We build a cohe en jux aposed coun e model based on he ollow-
ing Hey ing-Boolean s uc u e:
4This asymme y be ween wha is equi ed o alida ing an a gumen (ha ing a ans-
la ion o which he a gumen is alid) and in alida ing an a gumen ( ha no ansla ion
makes i alid) is no a biza e ea u e o jux aposi ion. Take he ypical a gumen con-
cluding Soc a es’ mo ali y, which, ob iously, is in ui i ely alid. I we we e o ansla e
i o p oposi ional logic, he esul ing a gumen would no be alid, bu we can ansla e
i o i s -o de p edica e logic in o de o accoun o i s alidi y. On he con a y, i we
ha e an a gumen in na u al language which is in ui i ely in alid (say, ‘2 = 2, he e o e,
he e is a McDonald’s on he da k side o he moon’) we would expec ha any easonable
o maliza ion o he language o a legi ima e logic sys em would in alida e he a gumen .
So, he p oblem ha jux aposi ion migh ha e wi h his asymme y is no because o he
asymme y pe se, bu because he o he possible ansla ions also seem in ui i ely alid
and jux aposi ion does no accoun o i .
104

Towa ds a Solu ion o he P oblem o Mixed In e ences
1B
0B
a
1H
0H
Choose Vc(qc) = 1B,Vi(pi) = a. Then, ||¬cqc||c= 0Band ||¬ipi||i= 0H.
Gi en cohe ence, since ||¬ipi||i6∈ Di,Vc(¬ipi)6∈ Dc, so Vc(¬ipi) = 0B.
The e o e, ||¬ipi→c¬cqc||c= 1B(0B→c0B)and he a gumen is no
alid, because ¬ipi→c¬cqc∈Dx,qc∈Dxbu pi/∈Dx.
P oposi ion 4.1.6.¬ipi→i¬cqc,qc`ic piis in alid.
P oo . Conside he same Hey ing-Boolean s uc u e and, in o de o build
he model, ake Vc(qc)=1B,Vi(pi) = a. Then, ||¬cqc||c= 0B, so 6∈ Dc.
Gi en cohe ence, Vi(¬cqc)6∈ Di, so choose Vi(¬cqc) = a. Since, Vi(pi) =
a,||¬ipi||i= 0H. So, ||¬ipi→i¬cqc||i= 1H(0H⇒a)5. The e o e, o
he alua ion Vi(pi) = a,Vc(qc) = 1Band Vi(¬cqc) = a, he p emises a e
designa ed while he conclusion is no , making he a gumen no alid as
desi ed. 
This concludes he applica ion o he me hod o jux aposi ion o he
mixed in e ences ha W enn p esen s agains localis p oposals. Le me
summa ize he esul s:
•Mix:qc∨i¬ipi,pi`ic qcand qc∨c¬ipi,pi0ic qc
•Essen ial Mix:¬c¬cpc∨i¬iqi,qi`ic pcand ¬c¬cpc∨c¬iqi,qi0ic pc
•Disjunc i e Mix:¬c¬cpc∨cqi`ic pc∨cqiand ¬c¬cpc∨iqi0ic pc∨iqi
•Nix:¬ipi→x¬cqc,qc0ic pi(x=i, c)
5Recall ha a⇒bis he ela i e pseudo-complemen o a wi h espec o b, which is
he g ea es elemen xsuch ha a∧x≤b. Since ¬a=a⇒ ⊥ he alue o ¬ais he
g ea es xsuch ha a∧x=⊥.
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Towa ds a Solu ion o he P oblem o Mixed In e ences
Rema k 4.1.1.Le me b ie ly commen on hese esul s. Fi s , I eckon ha
he esul s make i clea ha W enn’s claim agains localism is, a leas ,
oo has y. The applica ion o he me hod o jux aposi ion has allowed us o
accoun o he (in) alidi y o he mixed in e ences espec ing ou in ui ions
abou hem. This should be mo e han enough o a oid he di ec conclusion
ha he p oblem o mixed in e ences ules localism ou . E en i i is no he
pe ec and de ini i e solu ion, jux aposi ion is a i s s epping s one owa ds
a mo e sa is ac o y explana ion o how we migh combine di e en logics and
sys ema ise wha ollows om wha in a mixed discou se, while adhe ing o
a localis philosophy o logic.
The e is a good eason o jux aposi ion o be in a good le el o (im)modes y:
he jux aposi ion o wo consequence ela ions is he minimal conse a i e
ex ension o each o hem. The e o e, he jux aposed consequence ela ion
will con ain e e y in e ence ha was al eady alid in each logic o i s own
s ock o connec i es and will no c ea e new in e ac ions o in e ences o
hose s ocks o connec i es. As we saw when p esen ing jux aposi ion, his is
wha gua an ees ha he combined consequence ela ion does no collapse.
Howe e , ecall ha his minimali y and conse a i eness was also he
eason o blocking he appea ance o b idge p inciples, such as he dis ibu-
i i y o conjunc ion o e disjunc ion when jux aposing he logics L∧and L∨.
Now, I belie e ha we a e in a si ua ion in which jux aposi ion, despi e being
a i s s ep owa ds a solu ion, alls sho o being a conclusi e answe , p e-
cisely due o he incapaci y o allowing in e es ing b idge p inciples. Which
b idge p inciples a e hose? Luckily enough we al eady ha e hem a hand.
Conside again he mixed in e ences ha we e supposed o be in ui i ely
alid acco ding o W enn. Take he case o Essen ial Mix, o ins ance, and
y o hink abou he e dic gi en by jux aposi ion (i.e., ¬c¬cpc∨i¬iqi,
qi`ic pcand ¬c¬cpc∨c¬iqi,qi0ic pc) om a localis poin o iew. Tha is,
om he poin o iew o a pe son (mos likely a philosophe ) ha claims
ha he logic o e alua i e discou se is in ui ionis ic logic and he logic o
middle-sized objec s is classical logic. Knowing ha he se o alid in e ences
o IL is included in he se o alid in e ences o CL, ha he seman ics o
IL (Hey ing algeb as, K ipkean possible wo ld seman ics,...) gene alizes ha
o CL (Boolean algeb as, u h-condi ional wo- alued seman ics, ...) and
ha he no ion o ‘cons uc ion’, which is a he cen e o IL seman ics,
se s a highe epis emological s anda d han ha o ‘ u h’, which is c ucial
o CL seman ics, how can we possibly jus i y ha ou me hod alida es
¬c¬cpc∨i¬iqi,qi`ic pcbu in alida es he a gumen when changing he
106
Towa ds a Solu ion o he P oblem o Mixed In e ences
in ui ionis ic disjunc ion by he classical one?
Look a i om his pe spec i e: i we had a cons uc ion o qiwe would
know ha he alue o ¬iqiis he bo om elemen in he Hey ing algeb a,
and so ha i we had a cons uc ion o ¬c¬cpc∨i¬iqii mus be because we
had a cons uc ion o ¬c¬cpc. Now, i one is a localis and i s in ui ionis ic
seman ics is elling you ha you ha e a cons uc ion o he p oposi ion qi
how can you no assign o ¬iqi he bo om alue o you classical seman ics?
And simila ly, i when ha ing he in ui ionis ic disjunc ion, he disjunc ion is
designa ed because we ha e a cons uc ion o he disjunc ¬c¬cpc, how can
i be ha changing he disjunc ion o he classical one allows you o ha e
¬c¬cpcno designa ed? I is an awkwa d si ua ion o say he leas .
The ac is ha ¬c¬cpc∨c¬iqi,qi`ic pcis a b idge p inciple, i.e. a new
in e ac ion p inciple be ween connec i es o IL and CL ha seems o be
in ui i ely alid on he ace o he examples ha we ha e been conside ing,
oge he wi h he philosophy o he logics in play in he mixed in e ence
and a localis s andpoin . And an iden ical easoning wo ks o he in ui i e
alidi y o qc∨c¬ipi,pi`ic qc.
The case o ¬c¬cpc∨iqi`ic pc∨iqiis no so clea , hough. Since, he
in ui ionis ic s anda ds o designa ion a e highe han hose o classical logic,
i is no so ob ious ha he classical ac ha a p oposi ion and i s double
nega ion ha e he same seman ic alue should be p ese ed once we swi ch a
classical connec i e in o an in ui ionis ic one. Mo eo e , gi en ha he se o
alid in e ences o IL is included in ha o CL, pe haps i is no so awkwa d
ha an in e ence ha was alid wi h a CL connec i e u ns ou o be in alid
when changed by i s co esponding in ui ionis ic one. Ne e heless, I eckon
ha he e a e legi ima e localis posi ions ha could make a case o he
alidi y o ¬c¬cpc∨iqi`ic pc∨iqi oo.
La e on, I will y o p esen a modi ied me hod o combina ion in o de
o be able o accoun o i as well. Fo he momen , le us y o make oom
o he mo e ob iously alid b idge p inciples o Mix and Essen ial Mix.
107
Towa ds a Solu ion o he P oblem o Mixed In e ences
4.2 Imp o ing on he me hod o jux aposi-
ion: coo dina ing logics o mixed in e -
ences
Le me e y in o mally sugges he mo i a ion behind his imp o emen wi h
an analogy. Imagine you a e ekking in an alpine e ain. You ca y a map
o he a ea and a compass in case you ge los . The e a e si ua ions in which,
only wi h he map, one could ind he way back home. Suppose you a e
heading owa ds a shel e o spend he nigh o e he e and suddenly you
s op seeing he miles ones ha ma k he ail. You look a ound ying o
look o some hing o o ien you sel . Fa in he dis ance you ecognise he
shape o a summi and look o i in he map. You shel e is a he oo o
he moun ain, so you know which di ec ion you mus ake. Now, he e a e
also si ua ions in which jus he compass will do, o ins ance, i you know
ha he shel e is o he sou h-wes o whe e you a e, you migh be able
o ind you way ou . Bu he e a e o he si ua ions in which, nei he he
compass no he map alone will be enough. Tha is, si ua ions in which you
need o use he map and he compass in combina ion because each o hem
can ha e mo e applica ions when used coo dina ed wi h he o he . When
used in coo dina ion, you can calcula e he exac cou se you mus ollow,
e en i you do no know much abou he geog aphical ea u es su ounding
you.
Following he analogy, jux aposi ion is like ha ing he map and he com-
pass in you backpack bu using hem as i hey whe e independen ins u-
men s. You ha e, say, in ui ionis ic and classical logic, you ha e e e y alid
in e ence o hei espec i e languages, bu you do no allow p ope ly new
in e ac ions. Howe e , he e seem o be si ua ions, in ou case, con ex s in
which we eason ac oss domains employing mixed in e ences, ha equi e
new in e ac ions. Tha is, he logics in combina ion can po en ially gene a e
mo e alid in e ences han only by hemsel es, jus like he map and he
compass a e use ul in mo e si ua ions when hey a e used in combina ion
han by hei own. In ac , he e y name o ‘jux aposi ion’ sugges s his
si ua ion o ha ing wo o mo e hings oge he bu no in e ac ing. Tha
is why, e en i i is a modi ica ion o jux aposi ion, I will call he imp o ed
me hod ‘coo dina ion’ (C) in o de o make clea ha he combina ion we
a e looking o allows o he eme gence o b idge p inciples.
The goal, hen, is o modi y he combina ion mechanism in such a way
108
Towa ds a Solu ion o he P oblem o Mixed In e ences
4P ese a ion heo ems o Coo dina ion
Now ha we know he combina ion mechanism, le us o e some me a-
heo e ical esul s. Some o hem a e di ec consequences o hose gi en
by Schech e and some o he ’s equi e mo e o less signi ican modi ica ions
o be p o ed. We begin by p o ing he exis ence o coo dina ed non i ial
models. Fi s , we de ine he seman ic no ion o a coo dina ion o wo models
jus as Schech e de ines a jux aposi ion o wo models.
Suppose Mi=hBi, Viiis a model o e Ciand Piand Mc=hBc, Vci
is a model o e Ccand Pc. A coo dina ion o he models Miand Mcis a
coo dina ed model hBi, V +
i,Bc, V +
cio e Ci,Ccand Pic such ha :
•I p∈Pi, V +
i(p) = Vi(p)and
•I p∈Pc, V +
c(p) = Vc(p)
No ice ha i MC
ic is a coo dina ion o Miand Mc, o any α∈Sen (Cx,
Px) ( o x=i, c), ||α||MC
ic
x=||α||Mx. The e o e, MC
ic αjus in case Mxα.
We now p o ide a necessa y and su icien condi ion o he exis ence o
coo dina ed models.
P oposi ion 4.2.1 (Exis ence o Coo dina ed Non i ial Models).Suppose
Ciand Cca e disjoin signa u es. Suppose Mi=hBi, Viiis a model o e Ci
and Piand Mc=hBc, Vciis a model o e Ccand Pc, sa is ying he condi ion
ha i he ca dinali y o he se o seman ic alues o Biis wo, |Bi|= 2,
hen |Bc|= 210. Then he e is a coo dina ed non i ial model, MC
ic, o e Cic
and Pic based on Bic, jus in case o e e y p∈Pi∩Pc,Mipjus in case
Mcpand i ||p||Mi=⊥i, hen ||p||Mc=⊥c.
P oo . Suppose he e is some p∈Pi∩Pcsuch ha , ei he Mipand
Mc2po Mi2pand Mcpo ||p||Mi=⊥ibu ||p||Mc6=⊥c, hen he e is
no coo dina ion o Miand Mc.
Now, suppose ha o e e y p∈Pi∩Pc,Mipjus in case Mcpand
i ||p||Mi=⊥i, hen ||p||Mc=⊥c. We show ha he e is a weak coo dina ion
o Miand Mc.
10This is because, i we allowed |Bc|>2when |Bi|= 2, i α∈Sen (Cc,Pc) and ||α||Mcis
non-designa ed bu no -bo om, when doing he coo dina ion, we would ha e o e alua e
his in ui ionis ic a om as ⊥i, since i is he only non-designa ed alue in he wo-elemen
Boolean algeb a. Bu , hen, we would no ge a coo dina ed model, since we would ha e
||α||MC
ic
i=⊥iand ||α||MC
ic
c=bc6=⊥c, whe e bcis a non-designa ed non-bo om alue.
115

Towa ds a Solu ion o he P oblem o Mixed In e ences
Le >xbe he op alue o Bx,⊥xbe he bo om alue o Bxand le bx
be an elemen o Bx− {>x}. We induc i ely de ine, [ ]x, he unc ion om
Sen (Cic,Pic) o Bxsuch ha :
•I p∈Px,[p]x=Vx(p);
•I p∈Pi−Pc,[p]c=>ci Vi(p) = >i,[p]c=⊥ci Vi(p) = ⊥iand
[p]c=bco he wise;
•I p∈Pc−Pi,[p]i=>ii Vc(p) = >c,[p]i=bii Vc(p) = ⊥cand
[p]i=bi(6=⊥i)o he wise;
•I c∈Cn
x,[cα1...αn]x= Φx(c)([α1]x...[αn]x);
•I c∈Cn
i,[cα1...αn]c=>ci Φi(c)([α1]i...[αn]i) = >i,[cα1...αn]c=⊥c
i Φi(c)([α1]i...[αn]i) = ⊥iand [cα1...αn]c=bco he wise;
•I c∈Cn
c,[cα1...αn]i=>ii Φc(c)([α1]c...[αn]c) = >c,[cα1...αn]i=bii
Φc(c)([α1]c...[αn]c) = ⊥cand [cα1...αn]i=bi(6=⊥i)o he wise;
I αis an x-a om, le V+
x(α) = [α]x. Le Mic =hBi, V +
i,Bc, V +
ci. We
show ha o e e y α∈Sen (Cic,Pic), ||α||Mic
i=>ii ||α||Mic
c=>cand
i ||α||Mic
i=⊥i, hen ||α||Mic
c=⊥c. Tha is, [α]i=>ii [α]c=>cand i
[α]i=⊥i hen [α]c=⊥c.
I p∈Pi∩Pc,[p]i=Vi(p). So, [p]i=>ii Vi(p) = >ii Vc(p) = >ci
[p]c=>c, and [p]i=⊥ii Vi(p) = ⊥i. Bu , i Vi(p) = ⊥i, hen Vc(p) = ⊥c
and, so, [p]c=⊥c.
I p∈Pi−Pc,[p]i=>ii Vi(p) = >ii [p]c=>c. Also, [p]i=⊥ii
Vi(p) = ⊥iand i Vi(p) = ⊥i, hen [p]c=⊥c.
I p∈Pc−Pi,[p]c=>ci Vc(p) = >ci [p]i=>i. Also, i [p]i=⊥i hen
Vc(p) = ⊥c= [p]c11.
I c∈Cn
iand α1...αn∈Sen (Cic,Pic), [cα1...αn]i=>ii Φi(c)([α1]i...[αn]i) =
>ii [cα1...αn]c=>c12. Also, [cα1...αn]i=⊥ii Φi(c)([α1]i...[αn]i) = ⊥i
and [cα1...αn]c=⊥ci Φi(c)([α1]i...[αn]i) = ⊥i.
11No ice ha he e is no o he op ion. I [p]i=⊥i hen i can only be ha Vc(p) = ⊥c.
The easoning is by con aposi ion. I i had any o he non-designa ed classical alue hen
pcould no ha e he in ui ionis ic bo om alue.
12Since Ciand Cca e disjoin , he only eason o ha ing [cα1...αn]c=>cis because
Φi(c)([α1]i...[αn]i) = >i. Again, easoning wi h he con aposi i e migh be help ul.
116
Towa ds a Solu ion o he P oblem o Mixed In e ences
I c∈Cn
cand α1...αn∈Sen (Cic,Pic), [cα1...αn]c=>ci Φc(c)([α1]c...[αn]c) =
>ci [cα1...αn]i=>i. I [cα1...αn]i=⊥i hen Φc(c)([α1]c...[αn]c) = [cα1...αn]c=
⊥c.
Since we a e doing he coo dina ion o in ui ionis ic and classical models,
i is ob ious ha his coo dina ed model will be non i ial (because he e
a e α1, α2∈Sen (Cic,Pic) such ha MC
ic α1and MC
ic 2α2). 
Now, we p oceed wi h he p oo o s ong soundness o he coo dina ed
consequence ela ion (axioma ized by he coo dina ed na u al deduc ion cal-
culus), `ic
C, wi h espec o he class o coo dina ed Hey ing-Boolean s uc-
u es, Bic. Be o e p o ing he main esul we need wo auxilia y lemmas
(simila o Lemmas 5.3 and 5.4 in Schech e (2011)).
Lemma 4.2.1.Suppose Bic is he class o coo dina ed s uc u es o e Ciand
Cc. Then, Bic
Cis a consequence ela ion o Sen (Cic,Pic).
P oo . We ha e o show ha Bic
Csa is ies Iden i y, Weakening, Cu and
Uni o m Subs i u ion. Fo he i s h ee he p oo is basically ha o
Schech e in Lemma 5.3. We ocus, hen, on Uni o m Subs i u ion, i.e.,
i ΓBic
Cα hen Γ[β/p]Bic
Cα[β/p], o ex end and cla i y wha Schech e
does.
We p o e i s con aposi i e. So, suppose Γ[β/p]2Bic
Cα[β/p]. This means
ha he e is a coo dina ed model, MC
ic, such ha
MC
ic Γ[β/p]and MC
ic 2α[β/p]. Wi h he help o his model, we build an-
o he coo dina ed model, M0C
ic =hBi, V 0
i,Bc, V 0
ci, by le ing V0
x(δ)=||δ[β/p]||MC
ic
x
whene e δis an x-a om. We show ha ||δ||M0C
ic
x=||δ[β/p]||MC
ic
x o e e y δ∈
Sen (Cic,Pic), by induc ion on he complexi y o he o mulas:
•Base case: we ha e de ined V0
x(δ) in such a way ha i δis an x-a om
he equali y holds.
•Induc i e Hypo hesis (IH): Assume ha he equali y holds o all o -
mulas less complex han α. We show ha he equali y holds o any
possible α.
Assume ha α=¬xγ. By IH, we know ha ||γ||M0C
ic
x=
||γ[β/p]||MC
ic
x. So, clea ly ||¬xγ||M0C
ic
x=||¬xγ[β/p]||MC
ic
x.
Assume now ha α=γ∨xω. By IH, we know ha ||γ||M0C
ic
x=
||γ[β/p]||MC
ic
xand ||ω||M0C
ic
x=||ω[β/p]||MC
ic
x. Again, i is clea ha he
117
Towa ds a Solu ion o he P oblem o Mixed In e ences
same ope a ion, namely, ∨x, o e he same seman ic alues will yield
he same seman ic alue. So, ||γ∨xω||M0C
ic
x=||(γ∨xω)[β/p]||MC
ic
x. One
can easily check ha he same holds o α=γ∧xωand α=γ→xω.
The e o e, M0C
ic is a coo dina ed model such ha M0C
ic Γand M0C
ic 2
α.
Lemma 4.2.2.Suppose Bic is he coo dina ion o Biand Bc. I ΓBiαo
ΓBcα, hen ΓBic
Cα.
P oo . Simila o he p oo o Schech e o Lemma 5.4. The idea is ha
i ΓBxα, wi h Γ∪α⊆Sen (Cx,Px), hen we ake a coo dina ed model
MC
ic =hBi, Vi,Bc, Vcisuch ha MC
ic Γand Mx|Px=hBx, Vx|Pxiis he
es ic ion o Mx o Px, making ||β||Mx|Px=||β||MC
ic
x o e e y β∈Sen (Cx,
Px). Wi h his and knowing ha Mx|Pxis based on Bx, we ge ha MC
ic α.
So, ΓBic
Cα.
Now we ha e he ing edien s o p o e he main esul .
Theo em 4.2.1 (S ong Soundness).The coo dina ed na u al deduc ion cal-
culus is s ongly sound wi h espec o he class o Hey ing-Boolean s uc-
u es. Tha is,
Γ`ic
Cα⇒ΓBic
Cα.
P oo . By he p e ious lemmas, we know ha Bic
Cis a consequence e-
la ion and ha i ΓBiαo ΓBcα, hen ΓBic
Cα. We also know ha
he jux aposed na u al deduc ion o in ui ionis ic and classical logics is he
minimal conse a i e ex ension o hem and ha IL and CL a e s ongly
sound w. . . he classes o Hey ing and Boolean algeb as espec i ely. F om
he e, we ge ha , o he jux aposed na u al deduc ion de i a ion, o any Γ
and α, i Γ`ic α hen ΓBic α. The e o e, we know ha Γ`ic α⇒Γ`ic
Cα
(because we ha e all he p e ious ules and some o hem ha e ewe e-
s ic ions), we also know ha Γ`ic α⇒ΓBic α( om Schech e ’s p oo
o Soundness) and, inally, also ha ΓBic α⇒ΓBic
Cα(because e e y
coo dina ed model is a jux aposed model bu no ice e sa). Thus, now we
jus need o conside he ules ha we ha e changed, i.e., elaxed, o gi ing
he coo dina ed na u al deduc ion calculus o in ui ionis ic and classical log-
ics, and check ha hey a e s ongly sound wi h espec o he coo dina ed
seman ics.
118
Towa ds a Solu ion o he P oblem o Mixed In e ences
The p oo is, as usual, by induc ion on he leng h o he de i a ion. Le
us s a wi h he base case, namely, p oo s o size k= 1:
•Base case (k= 1): i Γ`ic
Cαand he leng h o he de i a ion is k= 1,
hen α∈Γ. Since α∈Γ, o e e y coo dina ed model, MC
ic, i MC
ic Γ,
hen MC
ic α. The e o e, ΓBic
Cα.
•Induc i e Hypo hesis (IH): assume ha i Γ`ic
Cαand he leng h o he
de i a ion is ≤k, hen ΓBic
Cα.
We ha e o show ha his also holds o he new ules wi h de i a ions o
leng h k+ 1. Le me s a wi h ¬cI:
(¬cI)γ1
.
.
.
γj
Π1
α
Π2
k⊥i
k+ 1 ¬cα
Recall ha , wi hin Π2only IL ules a e allowed o epe i ions o φsuch
ha (Γ`ic
Cφ)∈Π113. The de i a ion ends wi h an applica ion o ¬cI in line
k+ 1, bu p e iously, we ge o ⊥i om undischa ged assump ions Γ∪ {α},
in ≤klines. The e o e, Γ∪ {α} `ic
C⊥iand, by IH, Γ∪ {α}Bic
C⊥i.14
Now, we wan o show ha o e e y model MC
ic designa ing e e y o mula
in Γ,|| Vx(Γ)||MC
ic
x=>x, he alue o αhas o be bo om, ||α||MC
ic
i=⊥i,
and, also, ha wi hin Π1we could ha e used ules bo h om CL and IL
13O cou se, his includes any o he γ∈Γand, in ac , in mos o he cases in which I
will apply he ules, wha is going o be epea ed wi hin Π2is one o he p emisses in Γ.
14No ice ha wi h Γ∪ {α} `ic
C⊥i⇒Γ∪ {α}Bic
C⊥iwe a e in he same scena io as in
a s anda d in ui ionis ic nega ion in oduc ion ule, since in Π2we ha e only applied IL
ules. So, he easoning o showing ha ||α||i=⊥ishould be he same.
119
Towa ds a Solu ion o he P oblem o Mixed In e ences
na u al deduc ions. So, no ice ha o wha e e ule we applied in Π1, i
|| Vx(Γ)||MC
ic
x=>xand ΓBic
Cφ(because (Γ`ic
Cφ)∈Π1in ≤klines), hen
||α||MC
ic
x=>x15. So, i some o mula φwas used oge he wi h α o ob aining
⊥i, ha o mula akes he alue >iin he Hey ing algeb a o he models
ha make || Vi(Γ)||MC
ic
i=>i.
Since om assuming αonly IL ules we e used in Π2and hese a e sound
wi h espec o he class o Hey ing algeb as, whe e o any ∆and ω,∆HA ω
i || V(∆)||HA ≤ ||ω||HA, hen, Γ∪ {α}Bic
C⊥ii || Vi(Γ)∧iα||MC
ic
i≤ ||⊥i||MC
ic
i
o e e y MC
ic. So, ake any coo dina ed model Mjsuch ha || Vi(Γ)||Mj
i=
>i. Gi en ha o e e y MC
ic,|| Vi(Γ) ∧iα||MC
ic
i=⊥i, o Mjin pa icula ,
|| Vi(Γ) ∧iα||Mj
i=⊥i. Tha is, ||α||Mj
i=⊥iand, by coo dina ion, ||α||Mj
c=
⊥c. The e o e, o e e y MC
ic, i || Vi(Γ)||MC
ic
i=>i, hen ||¬cα||MC
ic
c=>c.
Hence, ΓBic
C¬cαas desi ed.
The nex ule we ha e o conside is →cI. I no in ui ionis ic ule is applied
wi hin he subde i a ion, we know ha he ule is sound because he na u al
deduc ion calculus o CL is s ongly sound wi h espec o he class o
Boolean algeb as. When in ui ionis ic ules a e applied wi hin →cI, hen he
ule is as ollows and i is sound wi h espec o he coo dina ed seman ics:
(→cI)γ1
.
.
.
γj
Π1
α
Π2
⊥i
k δ
k+ 1 α→cδ
15This shows ha wha e e we used by epe i ion a e assuming α(any φsuch ha
(Γ`ic
Cφ)∈Π1) has alue >xin he coo dina ed models ha make || Vx(Γ)||MC
ic
x=>x.
120

Towa ds a Solu ion o he P oblem o Mixed In e ences
Jus as wi h ¬cI, no ice ha wi hin Π2only IL ules a e allowed o
epe i ions o φsuch ha (Γ`ic
Cφ)∈Π1. In his case, he de i a ion ends
wi h an applica ion o →cI in line k+1, bu p e iously, we ge o ⊥iand δ om
undischa ged assump ions Γ∪ {α}, in ≤klines. The e o e, Γ∪ {α} `ic
C⊥i
and, by IH, Γ∪ {α}Bic
C⊥i.
Now, no ice ha wha we need o do is exac ly wha we did in he p e ious
case, which is o p o e ha o e e y model MC
ic designa ing e e y o mula
in Γ,|| Vx(Γ)||MC
ic
x=>x, he alue o αhas o be bo om, ||α||MC
ic
i=⊥i. So,
by he same a gumen as be o e, which elied on he ac ha Γ∪ {α} `ic
C
⊥i⇒Γ∪ {α}Bic
C⊥i, we know ha o e e y MC
ic, i || Vi(Γ)||MC
ic
i=>i,
hen ||α||MC
ic
i=⊥iand, by coo dina ion, ||α||MC
ic
c=⊥c. Gi en he u h
condi ions o he classical condi ional, i in e e y coo dina ed model in which
Γis designa ed, he alue o αis ⊥c, hen o e e y coo dina ed model, i
MC
ic Γ hen MC
ic α→cδ. Hence, ΓBic
Cα→cδas desi ed.
We conclude by showing he soundness o he ule o ∨cE when IL ules
a e applied wi hin he subde i a ions:
121
Towa ds a Solu ion o he P oblem o Mixed In e ences
(∨cE)γ1
.
.
.
γj
Π1
α∨cβ
α
Π2
⊥i
δ
β
Π3
δ
k+ 1 δ
Like in he p e ious cases, wi hin Π2only IL ules a e allowed o epe i-
ions o φsuch ha (Γ`ic
Cφ)∈Π1. The de i a ion ends wi h an applica ion
o ∨cE in line k+ 1, bu p e iously, we ge o ⊥iand δ om undischa ged
assump ions Γ∪ {α}, in ≤k, and we also ge δ om undischa ged assump-
ions Γ∪ {β}, in ≤k. The e o e, Γ∪ {α} `ic
WC ⊥iand Γ∪ {β} `ic
WC δ
and, by IH, Γ∪ {α}Bic
C⊥iand Γ∪ {β}Bic
Cδ. And, again, no ice ha
he same a gumen ha we used abo e can be applied he e o conclude ha
o e e y MC
ic, i || Vi(Γ)||MC
ic
i=>i, hen ||α||MC
ic
i=⊥iand, by coo dina ion,
||α||MC
ic
c=⊥c.
Now, we also ha e ha Γ∪{β}Bic
Cδ, om which we know ha e e y co-
o dina ed model ha sa is ies Γ∪{β}also sa is ies δ. Thus, we can conclude
ha e e y model sa is ying Γ∪ {α∨cβ}is a model sa is ying Γ∪ {β}and,
he e o e, also δ16. Hence, o e e y coo dina ed model, i MC
ic Γ∪{α∨cβ}
16No ice ha he e he seman ic clause o coo dina ion is c ucial, as in he p e ious ules.
This is wha gua an ees ha , since we concluded ha he in ui ionis ic seman ic alue o
122
Towa ds a Solu ion o he P oblem o Mixed In e ences
hen MC
ic δ, so, Γ∪ {α∨cβ}Bic
Cδ, as desi ed.
This concludes he s ong soundness p oo o weak coo dina ion.

4.2.2 Applying coo dina ion o mixed in e ences: a
be e eply o W enn
One o he mo i a ions o de eloping coo dina ion was o allow o he eme -
gence o b idge p inciples when combining in ui ionis ic and classical logics,
especially, hose cases o b idge p inciples belonging o Mix and Essen ial
Mix. I we a e capable o doing his while assuming a localis s and owa ds
logic, hen we a e in a good posi ion o mee ing W enn’s challenge agains
localism and, in gene al, o gi ing a localis accoun o mixed in e ences. Le
us show ha we can, in ac , accoun o he alidi y o Mix and Essen ial
Mix when hey a e o malized as b idge p inciples.
1Mix:We ca s a e unny
Ei he snow is whi e o we ca s a e no unny
Snow is whi e
Le us o malize he a gumen as a b idge p inciple by ansla ing i in o
ou coo dina ed language and ecall he assump ion ha he discou se abou
humou is an e alua i e discou se in which easoning is bes cap u ed by
in ui ionis ic logic, while discou se abou middle-sized objec s, such as snow,
is a discou se in which easoning is bes cap u ed by classical logic.
pi
qc∨c¬ipi
qc
I showed in P oposi ion 4.1.1 ha his a gumen was in alida ed by jux-
aposi ion. Now I will show ha coo dina ion makes he a gumen alid, as
desi ed.
P oposi ion 4.2.2.pi, qc∨c¬ipiic
Cqcis alid.
αhas o be ⊥i, i s classical seman ic alue, unde he scope o ∨c, has o be ⊥c.
123
Towa ds a Solu ion o he P oblem o Mixed In e ences
P oo . We wan o check ha o e e y coo dina ed model based on Bic, i
he p emises a e designa ed, hen he conclusion is designa ed oo. Suppose
hen, ha ||pi||x∈Dxand ||qc∨c¬ipi||x∈Dx.||pi||x∈Dxjus in case
||pi||i=>i. I ||pi||i=>i, hen ||¬ipi||i=⊥iand Vc(¬ipi) = ⊥c(by he
second condi ion o coo dina ion, i.e. i ||α||i=⊥i hen ||α||c=⊥c). Thus,
in o de o qc∨c¬ipi o be designa ed, ||qc||c=>c. The e o e, ||qc||c∈Dc
and, so, he a gumen is alida ed by coo dina ion. 
2Essen ial Mix:
Ei he i ’s no he case ha snow isn’ whi e o
we ca s a en’ unny
We ca s a e unny
Snow is whi e
We o malize he a gumen as:
¬c¬cpc∨c¬iqi
qi
pc
This a gumen oo was in alida ed by jux aposi ion. Le us see ha we
can now accoun o i s alidi y applying coo dina ion.
P oposi ion 4.2.3.¬c¬cpc∨c¬iqi,qiic
Cpcis alid.
P oo . We p o e i , his ime, using ou coo dina ed na u al deduc ion,
since we know ha i is s ongly sound wi h espec o he class o Hey ing-
Boolean s uc u es.
124
Towa ds a Solu ion o he P oblem o Mixed In e ences
1c¬ip
2c p
3⊥i¬iE, 1,2
4¬cp¬cI, 2–3
5¬ip→c¬cp!!→cI, 1–4

The p oblem wi h his de i a ion is ha i does no espec he es ic-
ions ha we ha e es ablished in o de o apply IL ules wi hin CL sub-
de i a ions. On one hand, he ules allow he applica ion o IL ules wi hin
CL subde i a ions bu , in ha case, e e y ule applied wi hin he CL sub-
de i a ion mus be in ui ionis ic. Howe e , in his de i a ion bo h ¬iE and
¬cI a e applied wi hin →cI. On he o he hand, i IL ules a e applied wi hin
→cI, he only way o closing is wi h ⊥iand δin he ou e mos subde i a ion
(in his case he one ha ing ¬ipas an assump ion). Bu we do no ge ha
in his subde i a ion and, he e o e, he ule →cI is badly applied.
No ice, hough, ha once pis o ced o ha e a seman ic alue ha educes
he possible ansla ions be ween s uc u es (i.e. ||p||i=>ii ||p||c=>c
o i ||p||i=⊥i hen ||p||c=⊥c) and, he e o e, he lexibili y o ge ing
coun e models, he a gumen migh be alid. Fo ins ance, we can pu pin
he p emises o ge a alid a gumen :
•pic
C¬ip→c¬cp
P oo . Suppose ||p||i=>i. Then, ||¬ip||i=⊥i, so ||¬ip||c=⊥c. This
is enough o see ha he conclusion will be designa ed oo, i.e. ||¬ip→c
¬cp||c=>c.
And wi h he na u al deduc ion,
131

Towa ds a Solu ion o he P oblem o Mixed In e ences
1p
2c¬ip
3⊥i¬iE, 1,2
4¬cp⊥iE, 3
5¬ip→c¬cp→cI, 1–4

•¬ip, q, ¬i , (p∨i )∨c(q→is)ic
Cs
P oo . Suppose ha ||¬ip||x=||q||x=||¬i ||x=||(p∨i )∨c(q→is)||x=
>x. Since, ||¬ip||i=||¬i ||i=>i, hen ||p||i=|| ||i=⊥iand, so, ||p∨i ||i=
⊥i. By coo dina ion, ||p∨i ||c=⊥cwhene e ||p∨i ||i=⊥i. The e o e,
||(p∨i )∨c(q→is)||x=>xi ||q→is||x=>xi ||s||x=>x(because we
ha e supposed ||q||x=>x).
And by he coo dina ed na u al deduc ion,
132
Towa ds a Solu ion o he P oblem o Mixed In e ences
1¬ip
2q
3¬i
4 (p∨i )∨c(q→is)
5¬ip∧i¬i ∧iI, 1,3
6¬i(p∨i )De Mo gan, 5
7c p ∨i
8⊥i¬iE, 6,7
9q→is⊥iE, 8
10 c q →is
11 q→isIden i y, 10
12 q→is∨cE, 4,7–11
13 s→iE, 2,12

I will end up wi h a couple mo e cases o make su e ha di e en ypes o
in e ences a e a ailable o he eade in o de o enhance he comp ehension
o he me hod.
•p, q ic
C(p→i¬iq)→c¬cq
133
Towa ds a Solu ion o he P oblem o Mixed In e ences
P oo .
1p
2q
3c p →i¬iq
4¬iq→iE, 1,3
5⊥i¬iE, 2,4
6¬cq⊥iE, 5
7 (p→i¬iq)→c¬cq→cI, 3–6

Le me conclude wi h an a gumen which I belie e migh be qui e sugges-
i e, since i is he ‘mi o image’ o a e sion o Mix. Conside he ollowing
a gumen :
Snow is whi e
Ei he we ca s a e unny o snow isn’ whi e
We ca s a e unny
So, we o malize he a gumen like his:
pc
qi∨x¬cpc
qi
(x=i, c)
In his case, he disjunc i e syllogism is occu ing wi h he classical pa
o he logic while in he o iginal Mix, we had an in ui ionis ic disjunc i e
syllogism. In his second e sion, i we ansla e he a gumen wi h a classical
disjunc ion, he a gumen is clea ly alid (al eady in jux aposi ion and a
o io i in coo dina ion), since i is an ins ance o a classically alid a gumen .
Howe e , i we ansla e he a gumen wi h an in ui ionis ic disjunc ion, we
can gi e a coo dina ed coun e model o i . So,
•pc, qi∨i¬cpc2ic
Cqi
134
Towa ds a Solu ion o he P oblem o Mixed In e ences
P oo . Take he pai o s uc u es ha we ha e been using in his sec ion
and alua ions Vc(pc) = >c,Vi(qi) = c.||¬cpc||c=⊥c, so by coo dina ion,
||¬cpc||i/∈Di, so ake Vi(¬cpc) = d. Then, ||qi∨i¬cpc||i=>i, so ||pc||x∈Dx,
||qi∨i¬cpc||x∈Dx, while ||qi||x/∈Dx.
And, om he na u al deduc ion side, i is qui e easy o see ha we canno
make he de i a ion because ha equi es he applica ion o a classical ule
wi hin an in ui ionis ic subde i a ion.
Howe e , he impo an ques ion is whe he ou combined logic sys em
o classical and in ui ionis ic logic should make his in e ence alid o no .
So, is he a gumen in ui i ely alid? O a e he e philosophical and localis
easons ha could be p o ided in a ou o he alidi y o he a gumen ? O ,
ano he way o pu ing i , should ou combined sys em aim o his kind o
b idge p inciples oo? Is coo dina ion s ill oo modes ?
To my mind, i is no ob ious ha i is a b idge p inciple o which he e
is good enough jus i ica ion. In ac , some o he easons ha we ga e o
he clause ‘i ||α||i=⊥i hen ||α||c=⊥c’ and so, o le ing he eme gence
o b idge p inciples like qc∨c¬ipi,piic
Cqc, would speak agains accep ing
his new b idge p inciple. The in ui ionis ic epis emic s anda d is highe
han he classical, in ui ionis ic logic is weake han (i.e. i is included in)
classical logic, ha ing a p oo o he absu di y o a p oposi ion seems o
imply he alsi y o ha p oposi ion, e c. So, he e migh be localis s who
will ind hese easons appealing enough as o s ick wi h coo dina ion and
a oid going beyond i by le ing mo e suspicious b idge p inciples eme ge.
Ye , he e migh be some o he localis s who ind he b idge p inciple
appealing o o he easons. To s a wi h, i is ue ha he epis emic
s anda d is highe o in ui ionism han o classical logic, bu ha migh
ha e less o do wi h he op and bo om alues o he algeb as and mo e
wi h he s uc u al ea u es o he in e media e ones. Tha is, wi h he ac
ha no o e e y seman ic alue, x, in a Hey ing algeb a, A,¬¬x=xand
¬x∨x=>. In ac , despi e he di e en epis emic s anda ds and seman ic
concep ions o wha i akes o a p oposi ion o ge he op alue, he localis
de ending coo dina ion is al eady accep ing ha ||α||i=>ii ||α||c=>c,
so why no accep also ha ||α||i=⊥ii ||α||c=⊥c? This is enough o
b idge p inciples like pc, qi∨i¬cpcic
Cqi o eme ge, indeed.
In he ollowing sec ion I will explo e his new s eng hening o jux a-
posi ion ha I call ‘s ong coo dina ion’. I hope i is clea enough ha I
am no using ‘s ong’ o he lack o an adjec i e as a alue judgemen .Fo
135
Towa ds a Solu ion o he P oblem o Mixed In e ences
he momen , I am explo ing he di e en possibili ies and analysing how
o make hem echnically iable while assessing hei possible philosophical
consequences. I eckon i is a complex and delica e ma e how o weigh he
echnical and philosophical i ues and sho comings o each o hem, bu
hope ully he analysis will shed some ligh on his new issue be o e us.
4.2.4 S ong Coo dina ion (SC)
Le me in oduce a u he s eng hening o he jux aposi ion, which is also
a s eng hening o coo dina ion. F om he syn ac ic poin o iew, s ong
coo dina ion is a consequence ela ion and, like coo dina ion, i is a pa icula
case o jux aposed consequence ela ion. Tha is, he s ongly coo dina ed
consequence ela ion, `ic
SC, is a jux aposed consequence ela ion ha ex ends
`i,`c,`ic and `ic
C. Since we canno appeal o minimali y, as i is done
wi h he jux aposed consequence ela ion, in o de o disc imina e `ic
SC om
o he jux aposed consequence ela ions, we will cha ac e ize i by means o
i s seman ic and syn ac ic p ope ies.
1Seman ics o S ong Coo dina ion
Wi h espec o he seman ics he app oach is almos he same. The
s ong coo dina ion o he s uc u es Bi, o e Ci, and Bc, o e Cc, is he
jux aposi ion o he s uc u es, hBi,Bci, and he s ong coo dina ion o he
classes o s uc u es Biand Bcis he Ca esian p oduc Bi×Bc, which is he
jux aposi ion o he classes o s uc u es.
As ongly coo dina ed model, MSC
ic =hBi, Vi,Bc, Vcio e Cic and Pic,
based on he s ongly coo dina ed s uc u e hBi,Bci(o , mo e gene ally,
based on he class o s ongly coo dina ed s uc u es Bic) is a cohe en jux-
aposed model sa is ying he ollowing p ope y:
•S ong Coo dina ion: A model is s ongly coo dina ed when o
e e y α∈Sen (Cic,Pic),
1. ||α||i=>ii ||α||c=>c
2. ||α||i=⊥ii ||α||c=⊥c
Again, pa o he mo i a ion o he s eng hening o he second clause
has come om he ac ha b idge p inciples like pc, qi∨i¬cpc`qiwe e
136

Towa ds a Solu ion o he P oblem o Mixed In e ences
in alida ed by coo dina ion. We ha e seen some easons o his a gumen o
be in alid and o he clause o coo dina ion, bu I ha e also poin ed ou some
easons ha migh jus i y s ong coo dina ion. In any case, a e p esen ing
he me hod and some o i s esul s, we will come back o i s adequacy as a
localis solu ion o mixed in e ences and o i s jus i ica ion.
2A Na u al Deduc ion calculus o S ong Coo dina ion
As one migh expec , jus as wi h coo dina ion he na u al deduc ion
calculus e lec ed he asymme y o he seman ic clauses, now he s ongly
coo dina ed na u al deduc ion calculus has o be modi ied in such a way ha
we ge he symme y o he seman ic clauses o s ong coo dina ion in he
calculus. The e o e, wha we need o do is jus o elax he in ui ionis ic ules
opening subde i a ions in he same way ha we elaxed he classical ones.
Tha is, on op o he coo dina ed na u al deduc ion calculus, we allow he
applica ion o classical ules wi hin ¬iI, →iI and ∨iE wi h he same ca ea s
ha had hei analogous classical ules.
Le us conside each in ui ionis ic ule opening subde i a ions and speci y,
jus in case, how o apply classical ules wi hin hem. Conside , i s , he
ule o in ui ionis ic nega ion in oduc ion:
(¬iI)α
Π
⊥i
¬iα
I no applica ion o a classical ule occu ed wi hin ¬iI, he ule is jus
he s anda d in ui ionis ic ule. I he e is an applica ion o a classical ule
wi hin ¬iI, hen he subde i a ion has o be closed like his:
(¬iI)α
Π
⊥c
¬iα
137
Towa ds a Solu ion o he P oblem o Mixed In e ences
Analogously, he e we equi e ha he only ules ha can be applied in he
de i a ion Πa e classical ules, oge he wi h epe i ions o o mulas de i ed
om he p emisses o he a gumen .
Fo →iI and ∨iE, le me jump di ec ly o he elaxed e sions o he
s anda d ules. The de ails o applying he ules mi o exac ly hose o he
coo dina ed na u al deduc ion calculus.
(→iI)α
Π
⊥c
β
α→iβ
(∨iE)α∨iβ
α
Π1
⊥c
δ
β
Π2
δ
δ
3P ese a ion heo ems o S ong Coo dina ion
Jus as we did in he case o coo dina ion, le me p esen now some
me a- heo e ical esul s conce ning s ong coo dina ion. We s a , as we did
be o e, p o ing he exis ence o s ongly coo dina ed non i ial models. This
ime, we skip he de ini ion o a s ong coo dina ion o he models Miand
Mc, since i emains he same as be o e.
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Towa ds a Solu ion o he P oblem o Mixed In e ences
P oposi ion 4.2.4 (Exis ence o S ongly Coo dina ed Non i ial Models).
Suppose Ciand Cca e disjoin signa u es. Suppose Mi=hBi, Viiis a model
o e Ciand Piand Mc=hBc, Vciis a model o e Ccand Pc, sa is ying he
condi ion ha |Bi|= 2 jus in case |Bc|= 220. Then he e is a s ongly
coo dina ed non i ial model, MSC
ic , o e Cic and Pic based on Bic, jus in
case o e e y p∈Pi∩Pc,Mipjus in case Mcpand ||p||Mi=⊥ii
||p||Mc=⊥c.
P oo . Suppose he e is some p∈Pi∩Pcsuch ha , ei he Mipand
Mc2po Mi2pand Mcpo ||p||Mi=⊥iand ||p||Mc6=⊥co ||p||Mc=⊥c
and ||p||Mi6=⊥i, hen he e is no s ong coo dina ion o Miand Mc.
Now, suppose ha o e e y p∈Pi∩Pc,Mipjus in case Mcp
and ||p||Mi=⊥ijus in case ||p||Mc=⊥c. We show ha he e is a s ong
coo dina ion o Miand Mc.
Le >xbe he op alue o Bx,⊥xbe he bo om alue o Bxand le bx
be an elemen o Bx− {>x,⊥x}. We induc i ely de ine, [ ]x, he unc ion
om Sen (Cic,Pic) o Bxsuch ha :
•I p∈Px,[p]x=Vx(p);
•I p∈Pi−Pc,[p]c=>ci Vi(p) = >i,[p]c=⊥ci Vi(p) = ⊥iand
[p]c=bco he wise;
•I p∈Pc−Pi,[p]i=>ii Vc(p) = >c,[p]i=⊥ii Vc(p) = ⊥cand
[p]i=bio he wise;
•I c∈Cn
x,[cα1...αn]x= Φx(c)([α1]x...[αn]x);
•I c∈Cn
i,[cα1...αn]c=>ci Φi(c)([α1]i...[αn]i) = >i,[cα1...αn]c=⊥c
i Φi(c)([α1]i...[αn]i) = ⊥iand [cα1...αn]c=bco he wise;
•I c∈Cn
c,[cα1...αn]i=>ii Φc(c)([α1]c...[αn]c) = >c,[cα1...αn]i=⊥i
i Φc(c)([α1]c...[αn]c) = ⊥cand [cα1...αn]i=bio he wise;
I αis an x-a om, le V+
x(α) = [α]x. Le Mic =hBi, V +
i,Bc, V +
ci. We
show ha o e e y α∈Sen (Cic,Pic), ||α||Mic
i=>ii ||α||Mic
c=>cand
||α||Mic
i=⊥ii ||α||Mic
c=⊥c. Tha is, [α]i=>ii [α]c=>cand [α]i=⊥i
i [α]c=⊥c.
20The eason is analogous o ha gi en o he exis ence o coo dina ed non i ial mod-
els.
139
Towa ds a Solu ion o he P oblem o Mixed In e ences
I p∈Pi∩Pc,[p]i=Vi(p). So, [p]i=>ii Vi(p) = >ii Vc(p) = >ci
[p]c=>c, and [p]i=⊥ii Vi(p) = ⊥ii Vc(p) = ⊥ci [p]c=⊥c.
I p∈Pi−Pc,[p]i=>ii Vi(p) = >ii [p]c=>c. Also, [p]i=⊥ii
Vi(p) = ⊥ii [p]c=⊥c.
I p∈Pc−Pi,[p]c=>ci Vc(p) = >ci [p]i=>i. Also, [p]i=⊥ii
Vc(p) = ⊥c= [p]c.
I c∈Cn
iand α1...αn∈Sen (Cic,Pic), [cα1...αn]i=>ii Φi(c)([α1]i...[αn]i)
=>ii [cα1...αn]c=>c. Also, [cα1...αn]i=⊥ii Φi(c)([α1]i...[αn]i) = ⊥ii
[cα1...αn]c=⊥c.
I c∈Cn
cand α1...αn∈Sen (Cic,Pic), [cα1...αn]c=>ci Φc(c)([α1]c...[αn]c)
=>ci [cα1...αn]i=>i. Also, [cα1...αn]c=⊥ci Φc(c)([α1]c...[αn]c) = ⊥c
i [cα1...αn]i=⊥i.
Since we a e doing he s ong coo dina ion o in ui ionis ic and classical
models, i is ob ious ha his s ongly coo dina ed model will be non i ial
(because he e a e α1, α2∈Sen (Cic,Pic) such ha MSC
ic α1and MSC
ic 2
α2). 
In o de o p o e he s ong soundness o he s ongly coo dina ed na u al
deduc ion calculus, we ely on he same lemmas ha we did o coo dina ion,
app op ia ely adap ed o s ong coo dina ion.
Lemma 4.2.3.Suppose Bic is he class o s ongly coo dina ed s uc u es
o e Ciand Cc. Then, Bic
SC is a consequence ela ion o Sen (Cic,Pic).
Lemma 4.2.4.Suppose Bic is he s ong coo dina ion o Biand Bc. I ΓBiα
o ΓBcα, hen ΓBic
SC α.
The p oo s o hese lemmas a e almos iden ical o he ones gi en o
coo dina ion. Since we ha e jus elaxed h ee mo e ules compa ed o he
na u al deduc ion calculus ha we had o coo dina ion, i is enough o p o e
ha , o he in e ences ha hose new ules allow in he calculus, we ha e
designa ion p ese a ion in he seman ics.
Theo em 4.2.2 (S ong Soundness).The s ongly coo dina ed na u al de-
duc ion calculus is s ongly sound wi h espec o he class o Hey ing-
Boolean s uc u es. Tha is,
Γ`ic
SC α⇒ΓBic
SC α.
P oo . Immedia e om he p oo o S ong Soundness o coo dina ion. I
is enough o change he subindexes ‘c’ and ‘i’.
140