JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS3deductionmeta-rulesintheirfullgen - PDF document

JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS3deductionmeta-rulesintheirfullgenerality.8Thenaturaldeductionrulesen-codemorelogicalstrengththanonemighthaveexpected.Perhapssurprisingly,thecollapseresultsrelyonthisextrastrength.Thus,thecollapseresultsturnouttobeveryfragile.Indeed,theissueofwhenexactlylogicscollapseturnsouttoberatherdelicate.Thepurposeofthispaperistwo-fold.First,Idevelopanewframeworkforcombininglogicalsystems,called\juxtaposition".Iprovegeneralmetalogicalresultsconcerningthecombinationoflogicsbyjuxtaposition.Second,Iexaminetheparticularcaseofcombiningclassicalandintuitionistlogics.Ishowhowthegeneralresultscanbeappliedtoshedlightonthephenomenonofcollapse.Idemonstratethatthecollapseresultsaremuchmorelimitedthanonemighthaveexpected.Thepaperwillproceedasfollows.Inthenextsection,Iintroducethegen-erallogicalapparatusthatwillbeemployed.Inthispaper,Ifocusonthecaseofpropositionallogics.Theapproachtosemanticsemployedhereisbroadlyalgebraic.Iconsidertwosemanticframeworks.Therstinvolvessetsoflogi-calmatrices,algebraswithanarbitrarysetofdesignatedvalues.Thesecond,moreinteresting,frameworkinvolvessetsofunitalmatrices,algebraswithasingledesignatedvalue.9Insectionthree,Ipresentthemainconstructionsforcombining(\juxtaposing")logicalsystems.Juxtaposingconsequencerelationsisstraightforward{thejuxtapositionoftwoconsequencerelationsistheleastconsequencerelationthatextendstheoriginalconsequencerelations.(Inthispaper,consequencerelationsarerequiredtoobeytheusualstructuralrulesandtobesubstitutioninvariant.)Thejuxtapositionoftwoalgebraicstructuresisonlyslightlymorecomplicated.Ajuxtaposedmodelisanorderedpairofmod-els,eachofwhichisbasedontherespectivealgebraicstructure.Therearetwomodicationstothisbasicideathatareneededtogetthesemanticstowork.First,eachofthetwomodelsmustprovidesemanticvaluesforsentencesoftheentirelanguage.Therefore,eachmodeltreatssentenceswithmainconnectivesgovernedbytheotherlogicasthoughtheywereadditionalsentencesymbols.Second,thetwomodelsmustagreeonwhichsentencesgetassigneddesignatedvalues.Inthisway,juxtaposedmodelsmustbe\coherent".Insectionfour,Icomparejuxtapositiontotwoothermethodsofcombininglogics{algebraicbringandmodulatedbring.Sectionveisdevotedtopre-sentingbasicmetalogicalresultsconcerningjuxtaposition.Inparticular,Ishowthatunderreasonableconditions,juxtapositionpreservesstrongsoundness.Ishowthatunderreasonableconditions,juxtapositionpreservesconsistency.Ialsoshowthatunderreasonableconditions,thejuxtapositionoftwoconsequencerelationsisastrongconservativeextensionoftheoriginalrelations.Insectionsix,Iturntostrongcompleteness.Inthissection,Ipresentdirectproofsof 8Ameta-ruleisarulethatgovernsrelationsamongentailmentsbutdonotthemselvesstateentailments.ConditionalIntroduction,ReasoningbyCases,andClassicalReductioaremeta-rules.ConjunctionIntroductionandElimination,DisjunctionElimination,ModusPonens,andDoubleNegationEliminationarenotmeta-rulesinthissense.9Thereareothernaturalapproachestoalgebraicsemantics.Forexample,wecouldmakeuseofclassesoflogicalmatriceswherethedesignatedvaluescanbecharacterizedbyasetofequations.SeeBlok&Pigozzi(1989). 4JOSHUAB.SCHECHTERstrongcompletenessthatapplyinawiderangeofcases.Indeed,Iprovidenec-essaryandsucientconditionsforthecasewherethetwostocksofconnectivesaredisjoint.Finally,insectionseven,Iturntothephilosophicallyimportantcaseofcombiningclassicalandintuitionistlogics.Applyingthegeneralmet-alogicalresults,Ishowthatalogicwithtwostocksofclassicalconnectivesisconsistent,conservative,andstronglysoundandstronglycompletewithrespecttoaparticularclassofjuxtaposedstructures{the\bi-Boolean"structures.Ishowthatalogicwithtwostocksofintuitionistconnectivesisconsistent,con-servative,andstronglysoundandstronglycompletewithrespecttotheclassof\bi-Heyting"structures.10Alogicwithonestockofintuitionistconnectivesandonestockofclassicalconnectivesisconsistent,conservative,andstronglysoundandstronglycompletewithrespecttotheclassof\Heyting-Boolean"structures.Iprovethatnoneoftheselogicscollapse.Ialsoinvestigatethequestionofwhichrules(andmeta-rules)leadtocollapsewhenaddedtotheselogics.x2.BasicNotions.2.1.Syntax.Alanguageforpropositionallogiccanbespeciedbyasigna-tureandasetofsentencesymbols.AsignatureC=fCngn2Nisanindexedfamilyofsetsoverthenaturalnumbers.11Foreachn2N,Cnisthe(possiblyempty)setofconnectivesofarityn.Asetofsentencesymbols,P,isanon-emptyset.Forconvenience,weonlyworkwithinnitesetsofsentencesymbols.Toavoidambiguity,weassumethattheelementsofeachCnandParenotthem-selvessequences.WealsoassumethatCmandCnaredisjointifm6=nandthateachCnisdisjointwithP.SupposeCandC0aretwosignatures.WesaythatCandC0aredisjointjustincaseforeachn2N,CnandC0naredisjoint.Otherwise,wesaythatCandC0overlap.WesaythatC0isasub-signatureofCjustincaseforeachn2N,C0nCn.GivenasignatureCandasetofsentencesymbolsP,thesetofsentencesgeneratedbyCandP,Sent(C;P),isinductivelydenedtobetheleastsetsuchthat:If2Pthen2Sent(C;P);Ifc2Cnand1;:::;n2Sent(C;P)thenc1:::n2Sent(C;P).Wewrite,, ,and(sometimeswithsuperscripts)tostandforsentences.12Wewrite�and
tostandforsetsofsentences.Wewritep,q,andrtostandforsentencesymbols.Wewrite[=p]tostandfortheresultofuniformlysubstitutingeachoccur-renceofpinwith.Wewrite�[=p]tostandforthesetf [=p]j 2�g.LetbeanyfunctionfromasubsetofPtosomeset.Wewritetostandfortheresultofuniformlysubstitutingeachoccurrenceofanypinthedomainofinwith(p).Wewrite�tostandforthesetf j 2�g. 10Thisisadierentusageof\bi-Heyting"thantheonefamiliarfromSkolem.11Inthispaper,superscriptsareusedasindicesandnotasexponents.12Followingstandardpractice,wedon'tconformtostrictconventionsgoverninguseandmentionwhenthereislittledangerofconfusion. JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS5Foreaseofcomprehension,whendisplayingsentencesinthelanguageofclas-sicalpropositionallogic,weuseinxratherthanprexnotation.2.2.ConsequenceRelations.Aconsequencerelation,`,forasetofsen-tencesSent(C;P)isarelationholdingbetweensubsetsofSent(C;P)andele-mentsofSent(C;P)suchthatthefollowingconditionsobtainforevery,,�,
,andp:Identity.fg`;Weakening.If�`then�[
`;Cut.If�`and
[fg`then�[
`;UniformSubstitution.If�`then�[=p]`[=p].13Inthispaper,wedonotrequirethatconsequencerelationsbecompact.Thatis,itneednotbethecasethatif�`thenthereisanite
�suchthat
`.SupposeC�isasub-signatureofCandP�isasubsetofP.Suppose`�isaconsequencerelationforSent(C�;P�)and`isaconsequencerelationforSent(C;P).Wesaythat`extends`�justincaseforevery�Sent(C�;P�)and2Sent(C�;P�),if�`�then�`.Wesaythat`isastrongconservativeextensionof`�justincaseforevery�Sent(C�;P�)and2Sent(C�;P�),�`�justincase�`.Let`beaconsequencerelationforSent(C;P).Wesaythatisdeduciblefrom�justincase�`.Wesaythatisatheoremof`justincase;`.Asusual,wewrite`for;`.Wesaythat�Sent(C;P)isconsistentwithrespectto`justincasethereisan2Sent(C;P)suchthat�0.14Wesaythat`isconsistentjustincasethereisan2Sent(C;P)suchthat0.Wesaythat`isnon-trivialjustincasethereisanon-empty�Sent(C;P)andan2Sent(C;P)suchthat�0.Wesaythat`hasnomerefollowersjustincase`whenever�`foreverynon-empty�Sent(C;P).15Wesaythat`hastheoremsjustincasethereisatleastonetheoremof`.Thesenotionsarerelatedasfollows:Anyconsequencerelationthathastheoremshasnomerefollowers.Anyconsistentconsequencerelationthathasnomerefollowersisnon-trivial.Anynon-trivialconsequencerelationisconsistent.2.3.Semantics.Theapproachtosemanticsweemployisbroadlyalgebraic.AstructureoverasignatureCisanorderedtripleB=hB;D;isuchthatBisthecarriersetofthestructure,Disanon-emptypropersubsetofB,andforeveryc2Cn,(c)isafunctionfromthen-thCartesianpowerofBtoB.BisthesetofsemanticvaluesofB.DisthesetofdesignatedvaluesofB.isthedenotationfunctionofB.SinceDisanon-emptypropersubsetofB,Bmusthaveatleasttwoelements.IfDhasonlyasingleelement,wesaythatBisaunitalstructureandwewrite1tostandforthesingledesignatedvalue. 13Aconsequencerelationisthusrequiredtobestructuralinthesenseof Los&Suszko(1958).14Consistencyisdenedhereasnon-explosion.Inaparaconsistentlogic,aconsistentsetmaycontainbothasentenceanditsnegation.15Iborrowtheterm\nomerefollowers"fromHumberstone(2011),pages459{460. 6JOSHUAB.SCHECHTERGivenastructureB=hB;D;iandasetofsentencesymbolsP,avaluationforBandPisafunctionfromPtoB.AmodeloverasignatureCandasetofsentencesymbolsPisanorderedpairM=hB;VisuchthatBisastructureoverC,andVisavaluationforBandP.WesaythatthemodelMisbasedonthestructureB.GivenaclassofstructuresBoverC,wesaythatMisbasedonBjustincaseMisbasedonsomeelementofB.IfMisbasedonaunitalstructure,wecallMaunitalmodel.SupposeM=hB;ViisamodeloverCandPbasedonB=hB;D;i.Forany2Sent(C;P),thevalueofinM,kkM,isrecursivelydenedasfollows:kkM=V()if2P;kc1:::nkM=(c)(k1kM;:::;knkM)ifc2Cnand1;:::;n2Sent(C;P).GivenamodelM,wewriteMtomeanthathasadesignatedvalueinM.Forshort,wesaythatMdesignatesthesentence.ThisobtainsjustincasekkM2D.WewriteM�tomeanthatMdesignateseachofthesentencesin�.Thisobtainsjustincaseforevery 2�,M .IfamodelMoverCandPeitherdesignateseveryelementofSent(C;P)ordesignatesnoelementofSent(C;P),wesaythatMistrivial.Otherwise,wesaythatMisnon-trivial.ForanystructureBoverCandanysetofsentencesymbolsP,thereisanon-trivialmodeloverCandPbasedonB.GivenaclassofstructuresBoverC,wewriteBtomeanthatisvalidinB.ThisobtainsjustincaseforeverymodelMoverCandPbasedonB,M.Wewrite�Btomeanthat�entailsinB.Thisobtainsjustincase,foreverymodelMoverCandPbasedonB,ifM�thenM.162.4.SoundnessandCompleteness.Givenaconsequencerelation`forthesetofsentencesSent(C;P)andaclassofstructuresBoverC,wesaythat`isstronglysoundwithrespecttoBjustincaseforevery�Sent(C;P)and2Sent(C;P),if�`then�B.Wesaythat`isstronglycompletewithrespecttoBjustincaseforevery�Sent(C;P)and2Sent(C;P),if�Bthen�`.Wesaythat`isstronglydeterminedwithrespecttoBjustincaseforevery�Sent(C;P)and2Sent(C;P),�Bjustincase�`.Wesaythat`isstronglysound,stronglycomplete,orstronglydetermined(simpliciter)justincase`isstronglysound,stronglycomplete,orstronglydetermined(respectively)withrespecttosomenon-emptyclassofstructures.Wesaythat`isstronglyunitalsound,stronglyunitalcomplete,orstronglyunitaldetermined(simpliciter)justincase`isstronglysound,stronglycomplete,orstronglydetermined(respectively)withrespecttosomenon-emptyclassofunitalstructures. 16Strictlyspeaking,thesedenitionsrequirePtobexedbycontext.However,whetherisvalidinBdoesnotdependonPsolongasPcontainsallofthesentencesymbolsthatoccurin.Similarly,whether�entailsinBdoesnotdependonPsolongasPcontainsallofthesentencesymbolsthatoccurin�and. 12JOSHUAB.SCHECHTERbothBooleanandHeytingalgebras.ButthesemustallbeBooleanalgebras.Thus,theresultinglogicendsupbehavingpurelyclassicallyforbothstocksofconnectives.Indeed,wecanshowthatcorrespondingconnectivesbecomeintersubstitutable.Thelogiccollapses.Thereisasecondwaytoseetheproblem.Supposewehavealanguagewithtwoofeachoftheusuallogicalconnectives.Suppose`iiistheconsequencerelationforthislanguagethatobeysalloftheintuitionisttheoremsandentailmentsforeachstockofconnectives.Aswewillseebelow,anyconsequencerelationthatextends`iiandobeysEntailmentCongruencecollapses.ButtheproofofthestrongdeterminationresultforalgebraicbringcruciallydependsonthecombinedlogicobeyingEntailmentCongruence.Sothereisnowaytoavoidcollapse.Therearemethodsofcombininglogicsdesignedtoavoidcollapse.31Perhapsthemostwellworkedoutismodulatedbring.32Themethodofcombiningconsequencerelationsandthemethodofcombiningclassesofstructuresaresignicantlymorecomplicatedformodulatedbringthanforalgebraicbringorforjuxtaposition.Butthebasicideaisstraightforward.AmodulatedstructureisaquadruplehB;;f1g;isuchthathB;f1g;iisastructureandisapre-orderwithnitemeetsandtopelement1.EachmodulatedstructureB12inthemodulatedbringB12oftwoclassesofmodulatedstructuresB1andB2correspondstoapair(orpairs)ofmodulatedstructureshB1;B2i,whereB12B1andB22B2.ThesemanticvaluesofB12aretheunionofthesemanticvaluesoftheoriginaltwostructures.Togettheconstructiontowork,however,arestrictionhastobeimposedonwhichpairsofmodulatedstructuresyieldamodulatedstructureinB12.Inparticular,theremustbetranslationsbetweenthesemanticvaluesofB1andthesemanticvaluesofB2.Thesetranslationsareestablishedbysomethingcalledabridge,whichisprovidedasaninputtothemodulatedbringprocedure.TheprooftheoryformodulatedbringreliesonavariantofHilbertcalculicalledmodulatedHilbertcalculi.ThestrongdeterminationresultforalgebraicbringreliesuponthefactthatanymodulatedHilbertcalculusthathasthe-oremsandobeysEntailmentCongruenceisstronglydeterminedwithrespecttosomeclassofmodulatedstructures.33Thestrongdeterminationresultformodulatedbringisasfollows:Themodulatedbring(byanadequatebridge)oftwomodulatedHilbertcalculithateachhavetheoremsandobeyEntailmentCongruenceisstronglydeterminedwithrespecttoaclassofmodulatedstruc-tures.Inparticular,itisstronglydeterminedwithrespecttothemodulatedbring(bythebridge)oftheclassofallmodulatedstructuresinwhich`1issoundandtheclassofmodulatedstructuresinwhich`2issound.34Moreover,asucientconditionfortherebeinganadequatebridgeisthatthetwoconse-quencerelationsarefordisjointlanguages.35Sothereisastrongdetermination 31See,forinstance,crypto-bringasdenedinCaleiro&Ramos(2007).32SeeSernadasetal.(2002).AlsoseeCarniellietal.(2008),chapter8.33SeeTheorem5.6inSernadasetal.(2002)andTheorem8.5.10inCarniellietal.(2008)foracloselyrelatedresult.34SeeTheorem5.12inSernadasetal.(2002)andTheorem8.5.16inCarniellietal.(2008).35SeeExample5.13inSernadasetal.(2002)andExample8.5.17inCarniellietal.(2008). JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS13resultforanypairofmodulatedHilbertcalculifordisjointlanguageswhichhavetheoremsandobeyEntailmentCongruence.Asbefore,thisresultisweakerthanthestrongdeterminationresultforjux-taposition.However,itcanbeusedtocombinetheclassicalandintuitionistconsequencerelations.Moreover,itcanbeshownthattheresultingmodulatedHilbertcalculusdoesnotcollapse.Somodulatedbringdoesprovideawaytoavoidthecollapseofclassicalandintuitionistlogics.Thismaysoundstrangegivenourresultthatanyconsequencerelationthatextends`iiandobeysEntailmentCongruencecollapses.Modulatedbringisde-signedtopreserveEntailmentCongruence.SothemodulatedbringofclassicalandintuitionistlogicobeysEntailmentCongruence.Whydoesn'titcollapse?Theansweristhattheresultofmodulatedbringclassicalandintuitionistlogic(overdisjointlanguagesbyanadequatebridge)isnotaconsequencerela-tioninthesensedenedabove.Theresultingrelationobeystheusualstructuralrules,butitisnotsubstitutioninvariant.AmodulatedHilbertcalculuscomeswithasetof\safesubstitutions"andisonlyguaranteedtobesubstitutionin-variantwithrespecttothosesubstitutions.Fortheparticularcaseofcombiningclassicalandintuitionistlogic,therelevantsetofsafesubstitutionsonlypermitustouniformlysubstitutesentenceswithaintuitionistmainconnective(oranintuitionistsentencesymbol)intotheintuitionistaxiomsandrules.Thus,whilemodulatedbringenablesustocombineclassicalandintuitionistlogicwithoutcollapse,itdoesnotyieldasubstitutioninvariantconsequencerelation.Aswewillseebelow,juxtapositionenablesustocombineclassicalandintuitionistlogicinawaythatpreservessubstitutioninvariance.ThecostisthattheresultingconsequencerelationdoesnotobeyEntailmentCongruence.Butthatseemstobeasmallercosttobear.x5.PreservationTheorems.Inthissection,weprovegeneralresultsaboutthemetalogicalpropertiesofjuxtaposition.Inparticular,weshowthatifthetwosignaturesaredisjoint,juxtapositionpreservesstrongsoundnessandstrongunitalsoundness.Weshowthatunderreasonableconditions,juxtapositionpre-servesconsistency.Wealsoshowthatunderreasonableconditions,thejuxtapo-sitionoftwoconsequencerelationsisastrongconservativeextensionofeachofthem.Inwhatfollows,inthissectionandthenext,weassumethatC1andC2aretwosignaturesandC12istheirjuxtaposition.Unlessotherwisespecied,C1andC2mayoverlap.WeassumethatP1andP2aresetsofsentencesymbolsandP12istheirjuxtaposition.Weassumethat`1isaconsequencerelationforSent(C1;P1),`2isaconsequencerelationforSent(C2;P2),and`12isaconsequencerelationforSent(C12;P12).Unlessotherwisespecied,`12neednotbethejuxtapositionof`1and`2,orevenajuxtaposedconsequencerelationover`1and`2.WealsoassumethatCisasignature,C�isasub-signatureofC,Pisasetofsentencesymbols,and`isaconsequencerelationforSent(C;P).5.1.TheExistenceofCoherentNon-TrivialModels.Werstprovideasimplesucientconditionforwhenthereisacoherentnon-trivialjuxtaposedmodelbasedonajuxtaposedstructure.Toprovethisresult,weintroducean-othersemanticnotion,thatofajuxtapositionoftwomodels. 14JOSHUAB.SCHECHTERSupposeM1=hB1;V1iisamodeloverC1andP1andM2=hB2;V2iisamodeloverC2andP2.AjuxtapositionofthemodelsM1andM2isajuxtaposedmodel,hB1;V+1;B2;V+2i,overC1,C2,andP12suchthat:Ifp2P1,V+1(p)=V1(p);andIfp2P2,V+2(p)=V2(p).Noticethatajuxtapositionoftwounitalmodelsisajuxtaposedunitalmodel.Ajuxtapositionoftwomodelsneednotbecoherent.Giventwomodels,thereneednotbeacoherentjuxtapositionofthem.Ifthereisacoherentjuxtaposition,itneednotbeunique.SupposeM12isajuxtapositionofM1andM2.Itisroutinetoshowthatforany2Sent(Ci;Pi),kkM12i=kkMi.Therefore,ifM12iscoherentthenforany2Sent(Ci;Pi),M12justincaseMi.Thefollowinglemmaprovidesasimplesucientconditiononwhentwomodelshaveacoherentjuxtaposition.Lemma5.1.SupposeC1andC2aredisjointsignatures.SupposeM1=hB1;V1iisamodeloverC1andP1andM2=hB2;V2iisamodeloverC2andP2.ThenthereisacoherentjuxtapositionofM1andM2justincaseforeveryp2P1\P2,M1pjustincaseM2p.Proof.Supposethereissomep2P1\P2suchthatM1pandM22p.V1(p)2D1andV2(p)62D2.SothereisnocoherentjuxtapositionofM1andM2.Similarly,ifM12pandM2p,thereisnocoherentjuxtapositionofM1andM2.Nowsupposeforeveryp2P1\P2,M1pjustincaseM2p.WeshowthatthereisacoherentjuxtapositionofM1andM2.LetdibeanelementofDiandletaibeanelementofBi�Di.Let[]ibethefunctionfromSent(C12;P12)toBiinductivelydenedasfollows:Ifp2Pi,[p]i=Vi(p);Ifp2P1�P2,[p]2=d2ifV1(p)2D1and[p]2=a2otherwise;Ifp2P2�P1,[p]1=d1ifV2(p)2D2and[p]1=a1otherwise;Ifc2Cni,[c1:::n]i=i(c)([1]i:::[n]i);Ifc2Cn1,[c1:::n]2=d2if1(c)([1]1:::[n]1)2D1and[c1:::n]2=a2otherwise;Ifc2Cn2,[c1:::n]1=d1if2(c)([1]2:::[n]2)2D2and[c1:::n]1=a1otherwise.Ifisani-atom,letV+i()=[]i.LetM12=hB1;V+1;B2;V+2i.Clearly,M12isajuxtapositionofM1andM2.Weshowthatforeach2Sent(C12;P12),kkM1212D1justincasekkM1222D2.NoticethatkkM12i=[]i.Soweneedonlytoshowthat[]12D1justincase[]22D2.Ifp2P1\P2,[p]12D1justincaseV1(p)2D1justincaseM1pjustincaseM2pjustincaseV2(p)2D2justincase[p]22D2.Ifp2P1�P2,[p]12D1justincaseV1(p)2D1justincase[p]2=d2justincase[p]22D2.Ifp2P2�P1,[p]22D2justincaseV2(p)2D2justincase[p]1=d1justincase[P]12D1.Ifc2Cn1and1:::n2Sent(C12;P12),[c1:::n]12D1justincase1(c)([1]1:::[n]1)2D1justincase[c1:::n]2=d2(sinceC1andC2aredisjoint)justincase[c1:::n]22D2.Ifc2Cn2and1:::n2 16JOSHUAB.SCHECHTERcoherentjuxtaposedmodelbasedonB12thatdesignateseachelementof�[
alsodesignates.So�[
B12.Cut:Suppose�B12and
[fgB12.SupposeM12isacoherentjuxtaposedmodelbasedonB12thatdesignateseachelementof�[
.SinceeverycoherentjuxtaposedmodelbasedonB12thatdesignateseachelementof�alsodesignates,M12designates.SinceeverycoherentjuxtaposedmodelbasedonB12thatdesignateseachelementof
[fgalsodesignates,M12designates.So�[
B12.UniformSubstitution:Suppose�[=p]2B12[=p].Since�[=p]2B12[=p],thereisacoherentjuxtaposedmodelM12=hB1;V1;B2;V2ibasedonB12thatdesignateseachelementof�[=p]butdoesnotdesignate[=p].LetV0i()=k[=p]kM12iifisani-atom.LetM012=hB1;V01;B2;V02i.Clearly,M012isajux-taposedmodelbasedonB12.Byasimpleinduction,forany2Sent(C12;P12),kkM012i=k[=p]kM12i.SinceM12iscoherent,soisM012.M012designateseachelementof�butdoesnotdesignate.Therefore,�2B12.Lemma5.4.SupposeB12isthejuxtapositionofB1andB2.If�B1or�B2,then�B12.Proof.Suppose�Biforsomei2f1;2g,�Sent(Ci;Pi),and2Sent(Ci;Pi).SupposeM12=hB1;V1;B2;V2iisacoherentjuxtaposedmodelbasedonB12suchthatM12�.LetVijPibetherestrictionofVitoPi.LetMijPi=hBi;VijPii.MijPiisamodeloverCiandPi.Forevery2Sent(Ci;Pi),kkMijPi=kkM12i.SoMijPi�.SinceMijPiisbasedonBi,MijPi.SoM12.Therefore,�B12.Theorem5.5(PreservationofStrongSoundness).Suppose`12isthejux-tapositionof`1and`2andB12isthejuxtapositionofB1andB2.If`1isstronglysoundwithrespecttoB1and`2isstronglysoundwithrespecttoB2,then`12isstronglysoundwithrespecttoB12.Proof.Bythedenitionof`12,itsucestoshowthatB12isaconsequencerelationforSent(C12;P12)suchthatif�`1or�`2then�B12.ByLemma5.3,B12isaconsequencerelationforSent(C12;P12).Bystrongsound-ness,if�`1then�B1andif�`2then�B2.ByLemma5.4,if�B1or�B2then�B12.Therefore,if�`1or�`2then�B12.Thisresultisfullygeneral{inparticular,itdoesnotrequirethatC1andC2bedisjoint.However,toshowthatjuxtapositionpreservesstrongsoundnessorstrongunitalsoundness(simpliciter),wemustshowthatthereisacoherentnon-trivialjuxtaposedmodelbasedonB12.IfC1andC2aredisjoint,wecanapplyProposition5.2.Indeed,ifC1andC2aredisjoint,wecanshowthatthejuxtapositionoftwoconsequencerelationsisstronglysoundjustincasebothofthemare.Toprovethisstrongerresult,werstneedtoproveasimplelemma.Lemma5.6.SupposeC1andC2aredisjointsignatures.SupposeB12isthejuxtapositionofB1andB2,eachofwhichisanon-emptyclassofstructures.Foranyi2f1;2g,�Sent(Ci;Pi),and2Sent(Ci;Pi),if�B12,then�Bi. JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS17Proof.Suppose�Sent(C1;P1)and2Sent(C1;P1).Suppose�2B1.SothereisamodelM1basedonB1suchthatM1�andM12.SinceB2isnon-empty,therearemodelsbasedonit.ByLemma5.1,thereisacoherentjuxtaposedmodelM12basedonB12thatisthejuxtapositionofM1withsomeM2basedonB2.(WemustpickM2sothatitdesignatesthesameelementsofP1\P2asM1.)SoM12�andM122.Therefore,�2B12.Thecasewhere�Sent(C2;P2)and2Sent(C2;P2)isanalogous.Corollary5.7.SupposeC1andC2aredisjointsignatures.Suppose`12isthejuxtapositionof`1and`2.Then:1.`12ishC1;C2i-stronglysoundjustincase`1and`2areeachstronglysound;2.`12ishC1;C2i-stronglyunitalsoundjustincase`1and`2areeachstronglyunitalsound.Proof.Suppose`1isstronglysoundwithrespecttoB1and`2isstronglysoundwithrespecttoB2,whereB1andB2arenon-emptyclassesofstructures.LetB12bethejuxtapositionofB1andB2.ByTheorem5.5,`12isstronglysoundwithrespecttoB12.SinceB1andB2arenon-empty,soisB12.ByProposition5.2,thereisacoherentnon-trivialjuxtaposedmodelbasedonB12.Therefore,`12ishC1;C2i-stronglysound.Nowsuppose`12isstronglysoundwithrespecttoB12,anon-emptyclassofstructures.Suppose�Sent(Ci;Pi)and2Sent(Ci;Pi).Suppose�2Bi.SinceB12isnon-empty,soareB1andB2.ByLemma5.6,�2B12.Bystrongsoundness,�012.So�0i.So`iisstronglysoundwithrespecttoBi,anon-emptyclassofstructures.Therefore,`1and`2areeachstronglysound.B12isaclassofjuxtaposedunitalstructuresjustincaseB1andB2areeachclassesofunitalstructures.Therefore,`12ishC1;C2i-stronglyunitalsoundjustincase`1and`2areeachstronglyunitalsound.5.3.ConservativenessandConsistency.Ournextgeneralresultscon-cernconservativenessandthepreservationofconsistency.Werstprovideasucientconditiononwhenthejuxtapositionoftwocon-sequencerelationsisastrongconservativeextensionoftheoriginalrelations.Proposition5.8.SupposeC1andC2aredisjointsignatures.Suppose`12isthejuxtapositionof`1and`2.If`1and`2areeachstronglydetermined,then`12isastrongconservativeextensionofeachof`1and`2.Proof.Suppose�Sent(Ci;Pi)and2Sent(Ci;Pi).Suppose�`i.Bythedenitionof`12,�`12.Nowsuppose�`12.Suppose`1isstronglydeterminedwithrespecttoB1and`2isstronglydeterminedwithrespecttoB2,whereB1andB2arenon-emptyclassesofstructures.LetB12bethejuxtapositionofB1andB2.ByTheorem5.5,`12isstronglysoundwithrespecttoB12.So�B12.ByLemma5.6,�Bi.Therefore,bystrongcompleteness,�`i.Conservativenessiscloselytiedtoconsistency.If�isconsistentwithrespecttosomeconsequencerelation,�isconsistentwithrespecttoanystrongconservativeextensionofit.Thus,wehavethefollowing: 18JOSHUAB.SCHECHTERProposition5.9.SupposeC1andC2aredisjointsignatures.Suppose`12isthejuxtapositionof`1and`2.Suppose`1and`2areeachstronglydeter-mined.If(fori=1or2)�Sent(Ci;Pi)isconsistentwithrespectto`i,then�isconsistentwithrespectto`12.CombiningtheseresultswithTheorem2.1,thefollowingresultsareimmediate:Theorem5.10(PreservationofConsistency).SupposeC1andC2aredis-jointsignatures.Supposeeachof`1and`2isconsistentandhasnomerefollowers.Suppose`12isthejuxtapositionof`1and`2.Then`12isconsistent.If�Sent(Ci;Pi)isconsistentwithrespectto`i,�isconsistentwithrespectto`12.Theorem5.11(StrongConservativeness).SupposeC1andC2aredisjointsignatures.Supposeeachof`1and`2isconsistentandhasnomerefollowers.Suppose`12isthejuxtapositionof`1and`2.Then`12isastrongconservativeextensionofeachof`1and`2.36x6.StrongCompletenessandStrongDetermination.Asusual,itissomewhatmorediculttoprovecompletenessresults.Inthissection,wepresentdirectproofsofstrongcompletenessandstrongunitalcompletenessthatapplyinawiderangeofcases.37ThegeneralstrategyofproofreliesonamodicationofthefamiliarLindenbaum-Tarskiconstructions.38WerstproveaveryabstractcompletenessresultthatrequirestheretobesuitableequivalencerelationswithwhichtobuildourLindenbaum-Tarskimodels.Wetheninvestigatewhensuchrelationsexist.6.1.TheLindenbaum-TarskiConstruction.LetbeanequivalencerelationonSent(C;P).WesaythatisacongruenceoverC�justincase:Foreveryc�2C�n,1;:::;n;2Sent(C;P),andk2f1;:::;ng,ifkthenc�1:::k:::nc�1::::::n.Wesaythatiscompatiblewith`and�Sent(C;P)justincase:Forevery;2Sent(C;P),ifthen�`justincase�`.Wesaythatisstronglycompatiblewith`and�Sent(C;P)justincaseiscompatiblewith`and�and:Forevery;2Sent(C;P),ifboth�`and�`then.WesaythatissuitableforC�,`,and�justincaseisacongruenceoverC�compatiblewith`and�.WesaythatisunitalsuitableforC�,`,and�justincaseisacongruenceoverC�stronglycompatiblewith`and�.We 36SeeCruz-Filipeetal.(2007),Proposition2.17,foraslightlystrongerresult.Inparticular,thejuxtapositionoftwonon-trivialconsequencerelationsoverdisjointsignaturesisastrongconservativeextensionofeachofthem.Theirproofofthisresultispurelyproof-theoretic,relyingonaxed-pointargument.37SeeZanardoetal.(2001)andSernadasetal.(2002)forcompletenessresultsconcerningalgebraicbringandmodulatedbring,respectively.38SeeRasiowa(1974)andRasiowa&Sikorski(1970)fortheapplicationoftheLindenbaum-Tarskimethodtologicsthatcontainconditionals.SeeBlok&Pigozzi(1989)fortheapplicationofthismethodinamoregeneralsetting. 22JOSHUAB.SCHECHTERNoticethatinthecasethat`12ishC1;C2i-stronglysoundwithrespecttoB12,wealsogettheconverseofthisresult:Supposethereisacoherentnon-trivialjuxtaposedmodelM12basedonB12.SinceM12isnon-trivial,forsome2Sent(C12;P12),M122.Bystrongsoundness,012.So,`12isconsistent.Summarizingourresults,wehavethefollowing:Theorem6.7.Suppose`12hasnomerefollowers.Supposeforeveryi2f1;2gandnon-empty�Sent(C12;P12)consistentwithrespectto`12,�iisanequivalencerelationonSent(C12;P12)suitableforCi,`12,and�.Then:1.`12isstronglycompletewithrespecttoB12;2.If`12isthejuxtapositionof`1and`2,then`12isstronglysoundwithrespecttoB12;3.If`12isconsistent,thenthereisacoherentnon-trivialmodelbasedonB12;4.B12isaclassofjuxtaposedunitalstructuresjustincaseforeveryi2f1;2gandnon-empty�Sent(C12;P12)consistentwithrespectto`12,�iisunitalsuitableforCi,`12,and�.6.2.StrongDetermination.Theorem6.7raisesthequestionofwhenthereisanequivalencerelationonSent(C12;P12)suitableorunitalsuitableforCi,`12,and�.Thecaseofsuitabilityisstraightforward.Thefollowingresultiseasytoprove:Lemma6.8.Suppose�Sent(C;P).ThentheidentityrelationonSent(C;P)isanequivalencerelationonSent(C;P)suitableforC�,`,and�.LetB=12betheLindenbaum-TarskiclassofjuxtaposedstructuresforC1,C2,and`12,usingtheidentityrelationonSent(C12;P12)forthesuitableequivalencerelations.CombiningLemma6.8withTheorem6.7,wearriveatthefollowingresult:Proposition6.9.Suppose`12hasnomerefollowers.Then:1.`12isstronglycompletewithrespecttoB=12;2.If`12isthejuxtapositionof`1and`2,then`12isstronglysoundwithrespecttoB=12;3.If`12isconsistent,thenthereisacoherentnon-trivialmodelbasedonB=12.TheidentityrelationonSent(C;P)isthemostne-grainedequivalencerela-tionsuitableforC�,`,and�.Wecanalsocharacterizethemostcoarse-grainedsuchrelation.WesaythatpstrictlyC�-occursinjustincasepdoesnotoccurwithinthescopeofanyconnectivenotfromC�.Suppose�Sent(C;P).WedeneabinaryrelationonSent(C;P)asfollows:h;i2 �C�justincaseforevery2Sent(C;P)andpthatstrictlyC�-occursin;�`[=p]justincase�`[=p]:Thisisamodicationofthedenitionofthewell-knownLeibnizcongruence.40Wewrite(mod �C�)tostandfortheclaimthath;i2 �C�. 40TheLeibnizcongruencemaybedenedasfollows:(mod �)justincaseforevery2Sent(C;P)andpoccurringin,�`[=p]justincase�`[=p].SeeBlok&Pigozzi(1989)fordiscussionofthiscongruence. JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS25Indeningthisrelation,itturnsouttobehelpfultoworkwithanexpansionofourlanguage.LetPbeacountablyinnitesetofsentencesymbolsdisjointwithP.(Inwhatfollows,wealwayschoosePtobedisjointwithwhateversetsofsentencesymbolsweareworkingwith.)Let`betheleastconsequencerelationforSent(C;P[P)thatextends`.Lemma6.17.`existsandisastrongconservativeextensionof`.Proof.LetbeanyfunctionfromPtoP.Let�`justincase�`.Itisroutinetoshowthat`isaconsequencerelationforSent(C;P[P)thatextends`.ItisalsoroutinetoshowthattheintersectionofallconsequencerelationsforSent(C;P[P)thatextend`isitselfaconsequencerelationforSent(C;P[P).Therefore,`exists.Suppose�`.Since`extends`,�`.Nowsuppose�0.AgainletbeanyfunctionfromPtoP.Itiseasytoseethat�0.So�0.Therefore,`isastrongconservativeextensionof`.Suppose�Sent(C;P).WedeneabinaryrelationonSent(C;P)asfollows:h;i2�C�justincaseforevery2Sent(C�;P[P)andpoccurringin;�[f[=p]g`[=p]and�[f[=p]g`[=p]:Wewrite(mod�C�)tostandfortheclaimthath;i2�C�.Inthisdenition,isrestrictedtosentenceswithconnectivesfromC�.Thisiswhatenablesustoimproveourstrongunitaldeterminationresult.However,thisisalsowhatmotivatestheuseofanextrasetofsentencesymbols.BymakinguseoftheelementsofP,wecanapplyUniformSubstitutiontoshowthat�C�isacongruenceoverC�evenwhen�containsoccurrencesofeveryelementinP.42Lemma6.18.Suppose�Sent(C;P).Then�C�isanequivalencerelationonSent(C;P)suitableforC�,`,and�.Proof.Suppose�Sent(C;P).Clearly,�C�isanequivalencerelationonSent(C;P).CongruenceoverC�:Supposec�2C�n;1;:::;n;2Sent(C;P);andk2f1;:::ng.Supposek(mod�C�).Suppose2Sent(C�;P[P)andpoccursin.Letp1;:::;pnbedistinctelementsofPthatdonotoccurin.�[f[c�p1:::pk:::pn=p][k=pk]g`[c�p1:::pk:::pn=p][=pk].Thatis,�[f[c�p1:::k:::pn=p]g`[c�p1::::::pn=p].ByUniformSubstitution,�[f[c�1:::k:::n=p]g`[c�1::::::n=p].Byanalogousreasoning,�[ 42Thereisanalternativeapproachthatwecouldinsteadhaveadopted.Theideaistomakeuseofthefollowingequivalencerelation:h;i20�C�justincaseforevery2Sent(C�;P)andpoccurringin,�[f[=p]g`[=p]and�[f[=p]g`[=p].UsingUniformSubstitution,wecanshowthatif�lacksoccurrencesofinnitelymanyelementsofPthen0�C�isacongruenceoverC�.Ifweretoadoptthisapproach,wewouldhavetomodifyourdenitionoftheLindenbaum-Tarskiclassofjuxtaposedstructuressothatitcontainsajuxtaposedstructureforeverynon-empty�Sent(C12;P12)consistentwithrespectto`12thatlacksoccurrencesofinnitelymanyelementsofP12.Ineect,wewouldbeworkingwithrestrictionsofthelanguageratherthananexpansionofit. 26JOSHUAB.SCHECHTERf[c�1::::::n=p]g`[c�1:::k:::n=p].Hence,c�1:::k:::nc�1::::::n(mod�C�).Compatibilitywith`and�:Suppose(mod�C�).Letp2P.So�[fp[=p]g`p[=p].Thatis,�[fg`.ByLemma6.17,�[fg`.Similarly,�[fg`.ByCut,�`justincase�`.Beforeweprovideconditionsforwhen�C�isunitalsuitableforC�,`,and�,werstprovetwosimpleresultsconcerningleft-extensionalityoveraset:Lemma6.19.Suppose
Sent(C;P).Suppose`isleft-extensionalover
.Thenanyconsequencerelationthatextends`isleft-extensionalover
.Proof.SupposeCisasub-signatureofC+andPisasubsetofP+.Suppose`+isaconsequencerelationforSent(C+;P+)thatextends`.Suppose;2Sent(C+;P+);2
;andpoccursin.LetqandrbedistinctelementsofPthatdonotoccurin.Since`isleft-extensionalover
,fq;r;[q=p]g`[r=p].Since`+extends`,fq;r;[q=p]g`+[r=p].ByUniformSubstitution,f;;[=p]g`+[=p].Therefore,`+isleft-extensionalover
.Lemma6.20.SupposeP�isaninnitesubsetofP.Then`isleft-exten-sionaloverSent(C�;P)justincase`isleft-extensionaloverSent(C�;P�).Proof.Suppose`isleft-extensionaloverSent(C�;P).SinceSent(C�;P�)isasubsetofSent(C�;P),`isleft-extensionaloverSent(C�;P�).Nowsuppose`isleft-extensionaloverSent(C�;P�).Supposethat2Sent(C�;P);;2Sent(C;P);andpoccursin.LetPcontainthoseel-ementsofP�P�thatoccurin.LetPcontainthoseelementsofP�thatdonotoccurin,,or.LetbeaninjectivefunctionfromPtoP.SuchafunctionexistssincePisniteandPisinnite.Byleft-extensionalityoverSent(C�;P�),f;;[=p]g`[=p].ByUniformSubstitution,f;;[=p]g`[=p].Therefore,`isleft-extensionaloverSent(C�;P).Makinguseoftheseresults,wecanprovethefollowingresult:Lemma6.21.`isleft-extensionaloverSent(C�;P)justincaseforeverynon-empty�Sent(C;P)consistentwithrespectto`,�C�isunitalsuitableforC�,`,and�.Proof.Suppose`isleft-extensionaloverSent(C�;P).Supposethat�Sent(C;P).ByLemma6.18,forevery�Sent(C;P),�C�issuitableforC�,`,and�.Suppose�`and�`.Suppose2Sent(C�;P[P)andpoccursin.ByLemma6.19,`isleft-extensionaloverSent(C�;P).ByLemma6.20,`isleft-extensionaloverSent(C�;P[P).Sof;;[=p]g`[=p].Since`extends`,�`and�`.ByCut,�[f[=p]g`[=p].Similarly,�[f[=p]g`[=p].So(mod�C�).So�C�isunitalsuitableforC�,`,and�.Nowsupposeforeverynon-empty�Sent(C;P)consistentwithrespectto`,�C�isunitalsuitableforC�,`,and�.Suppose;2Sent(C;P);2Sent(C�;P);andpoccursin.Iff;gisinconsistentwithrespectto`,thenf;;[=p]g`[=p].Suppose,then,thatf;gisconsistentwithrespectto`.Sof;gC�isunitalsuitableforC�,`,andf;g.ByIdentity 28JOSHUAB.SCHECHTERM12basedonit.SinceM12isnon-trivial,forsome;2Sent(C12;P12),M12andM122.Sofg2B12.Bysoundness,fg012.So`12isnon-trivial.Itimmediatelyfollowsthat`12ishC1;C2i-stronglysoundonlyif`12iscon-sistent.Proposition6.25.`12ishC1;C2i-stronglycompletewithrespecttoaclassofjuxtaposedunitalstructuresonlyif`12isleft-extensionaloverSent(C1;P1)andoverSent(C2;P2).Proof.Suppose`12isstronglycompletewithrespecttoB12,aclassofjuxta-posedunitalstructuresoverC1andC2.Leti=1or2.Suppose2Sent(Ci;Pi)and;2Sent(C12;P12).SupposeM12isacoherentjuxtaposedunitalmodeloverC1,C2,andP12basedonB12suchthatM12,M12,andM12[=p].InM12,kki=kki=k[=p]ki=1i.Byinductiononthecom-plexityof,k[=p]ki=k[=p]ki.Sok[=p]ki=1i.SoM12[=p].Sof;;[=p]gB12[=p].Bystrongcompleteness,f;;[=p]g`12[=p].Therefore,`12isleft-extensionaloverSent(Ci;Pi).Proposition6.26.SupposeC1andC2aredisjoint.`12ishC1;C2i-stronglydeterminedwithrespecttoaclassofjuxtaposedstructuresonlyif`12hasnomerefollowers.Proof.Suppose`12isstronglydeterminedwithrespecttoB12,aclassofjuxtaposedstructuresoverC1andC2.Suppose012.LetpbeanelementofP12thatdoesnotoccurin.Weshowthatfpg012.Bycompleteness,2B12.SothereisacoherentjuxtaposedmodelM12=hB1;V1;B2;V2ioverC1,C2,andP12basedonB12suchthatM122.LeteachBi=hBi;Di;ii.LetdibeanelementofDiandletaibeanelementofBi�Di.Let[]ibethefunctionfromSent(C12;P12)toBiinductivelydenedasfollows:Ifhasnooccurrenceofp,[]i=kkM12i;[p]i=di;Ifc2Cniandpoccursinatleastoneof1;:::;n,[c1:::n]i=i(c)([1]i:::[n]i);Ifc2Cn1andpoccursinatleastoneof1;:::;n,[c1:::n]2=d2if1(c)([1]1:::[n]1)2D1and[c1:::n]2=a2otherwise;Ifc2Cn2andpoccursinatleastoneof1;:::;n,[c1:::n]1=d1if2(c)([1]2:::[n]2)2D2and[c1:::n]1=a1otherwise.SinceC1andC2aredisjoint,[]iiswell-dened.Ifisani-atom,letV0i()=[]i.LetM012=hB1;V01;B2;V02i.Clearly,M012isajuxtaposedmodelbasedonB12.ItisstraightforwardtoshowthatinM012,kki=[]i.ItisalsostraightforwardtoshowthatM012iscoherent.M012pandM0122.Sofpg2B12.Bystrongsoundness,fpg012.CombiningCorollaries6.13and6.23withPropositions6.24,6.25,and6.26,wearriveatthefollowingniceresult: JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS29Corollary6.27.SupposeC1andC2aredisjoint.Suppose`12isthejux-tapositionof`1and`2.Then:1.`12ishC1;C2i-stronglydeterminedjustincase`12isconsistentandhasnomerefollowers;2.`12ishC1;C2i-stronglyunitaldeterminedjustincase`12isconsistent,hasnomerefollowers,andisleft-extensionaloverSent(C1;P1)andoverSent(C2;P2).6.6.JuxtaposedConsequenceRelations.Ourmaininterestinthispaperconcernsthecombinationoflogics.Anaturalquestiontoaskis:Whatpropertiesof`1and`2suceforajuxtaposedconsequencerelationover`1and`2tobehC1;C2i-stronglydeterminedorhC1;C2i-stronglyunitaldetermined?Theresultsaboveenableustoprovideafairlycomprehensiveanswertothisquestion.Werstdeneanewnotion.Suppose2Sent(C;P).Wesaythatanoccurrenceofinisamaximalnon-C�-occurrencejustincasethemainconnectiveofisnotfromC�andtheoccurrenceisnotproperlywithinthescopeofanyconnectivenotfromC�.Lemma6.28.Suppose`isC�-left-extensional.Thenanyconsequencere-lationthatextends`isC�-left-extensional.Proof.SupposeCisasub-signatureofC+andPisasubsetofP+.Suppose`+isaconsequencerelationforSent(C+;P+)thatextends`.Suppose;;2Sent(C+;P+)andpstrictlyC�-occursin.Withoutlossofgenerality,wecanassumethat2Sent(C+;P).(WecanuseUniformSubstitutiontocoverthecasewhereitdoesn't.)Let�betheresultofreplacingthemaximalnon-C�-occurrencesinwithdistinctelementsofPthatdonotoccurin;,or.LetqandrbedistinctelementsofPthatdonotoccurin�.So�2Sent(C;P)andpstrictlyC�-occursin.Since`isC�-left-extensional,fq;r;�[q=p]g`�[r=p].Since`+extends`,fq;r;�[q=p]g`+�[r=p].SincepstrictlyC�-occursin,byUniformSubstitutionfq;r;[q=p]g`+[r=p].AgainusingUniformSubstitution,f;;[=p]g`+[=p].Therefore,`+isC�-left-extensional.Proposition6.29.Suppose`12isajuxtaposedconsequencerelationover`1and`2.Supposeatleastoneof`1or`2hastheorems.Then:1.`12isstronglycompletewithrespecttoeachofB=12,B 12,andB12;2.If`12isthejuxtapositionof`1and`2,then`12isstronglysoundwithrespecttoeachofB=12,B 12,andB12;3.If`12isconsistent,thentherearecoherentnon-trivialmodelsbasedoneachofB=12,B 12,andB12;4.If`1and`2areeachleft-extensional,thenB 12andB12areclassesofjuxtaposedunitalstructures.Proof.Ifatleastoneof`1and`2hastheorems,`12hastheorems,andso`12hasnomerefollowers.Claims1{3thenfollowfromPropositions6.9,6.12,and6.22.If`iisleft-extensional,thenbyLemma6.28,`12isCi-left-extensional(andthus,isleft-extensionaloverSent(Ci;Pi)).Claim4thenfollowsfromPropositions6.15and6.22.Thefollowingcorollaryisimmediate: 30JOSHUAB.SCHECHTERCorollary6.30.Suppose`12isthejuxtapositionof`1and`2.Suppose`12isconsistent.Supposeatleastoneof`1and`2hastheorems.Then:1.`12ishC1;C2i-stronglydetermined;2.If`1and`2areeachleft-extensional,then`12ishC1;C2i-stronglyunitaldetermined.InthecasewhereC1andC2aredisjoint,wehavethefollowingniceresults:Proposition6.31.SupposeC1andC2aredisjoint.Suppose`12isthejuxtapositionof`1and`2.Supposeeachof`1and`2isconsistentandhasnomerefollowers.Supposeatleastoneof`1and`2hastheorems.Then:1.`12isstronglydeterminedwithrespecttoeachofB=12,B 12,andB12;2.Thereisacoherentnon-trivialmodelbasedoneachofB=12,B 12,andB12;3.B 12andB12areclassesofjuxtaposedunitalstructuresjustincase`1and`2areeachleft-extensional.Proof.ByTheorem5.10,`12isconsistent.Ifatleastoneof`1and`2hastheorems,`12hastheorems,andso`12hasnomerefollowers.Claims1{3thenfollowfromPropositions6.9,6.12,and6.22If`iisleft-extensional,thenbyLemma6.28,`12isCi-left-extensional(andthus,isleft-extensionaloverSent(Ci;Pi)).ByTheorem5.11,if`12isleft-extensionaloverSent(Ci;Pi),then`iisleft-extensional.Claim4thenfollowsfromPropositions6.15and6.22.Corollary6.32.SupposeC1andC2aredisjoint.Suppose`12isthejux-tapositionof`1and`2.Supposeatleastoneof`1and`2hastheorems.Then:1.`12ishC1;C2i-stronglydeterminedjustincaseeachof`1and`2iscon-sistentandhasnomerefollowers;2.`12ishC1;C2i-stronglyunitaldeterminedjustincaseeachof`1and`2isconsistent,hasnomerefollowers,andisleft-extensional.Proof.Theright-to-leftdirectionofeachclaimfollowsfromProposition6.31.Theleft-to-rightdirectionfollowsfromPropositions6.24,6.25,and6.26com-binedwithTheorem5.11.UsingTheorems2.1and2.2,thefollowingresultisimmediate:Corollary6.33.SupposeC1andC2aredisjoint.Suppose`12isthejux-tapositionof`1and`2.Supposeatleastoneof`1and`2hastheorems.Then:1.`12ishC1;C2i-stronglydeterminedjustincaseeachof`1and`2isstronglydetermined;2.`12ishC1;C2i-stronglyunitaldeterminedjustincaseeachof`1and`2isstronglyunitaldetermined.Thereisanadditionalresultworthstating.Wewillmakeuseofthisresultinourdiscussionofclassicalandintuitionistlogicbelow.WesaythataclassofstructuresBisfullfortheconsequencerelation`justincase`isstronglysoundwithrespecttoBandBcontainsatleastonerepresentativefromeveryisomorphismclassofstructuresfor`thathasasetofsemanticvalueswithcardinalityatmostthatofthelanguageof`.44WesaythatBisunitalfullfor 44ThisisaweakeningofDenition4.4inZanardoetal.(2001)andDenition3.3.11inCarniellietal.(2008). JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS33Proof.Suppose�Sent(C;P).ByRe exivity,Symmetry,Transitivity,Weakening,andCut,�isanequivalencerelationonSent(C;P).CongruenceoverC�:Supposec�2C�n;1;:::;n;2Sent(C;P);andk2f1;:::ng.Supposek(mod�).So�`(k;).Sincesat-isesCongruenceoverC�,(k;)`(c�1:::k:::n;c�1::::::n).ByCut,�`(c�1:::k:::n;c�1::::::n).Hence,c�1:::k:::nc�1::::::n(mod�).Compatibilitywith�and`:Suppose(mod�).So�`(;).Suppose�`.ByModusPonens,fg[(;)`.ByCut,�`.Similarly,if�`,then�`.So�`justincase�`.Lemma6.38.SupposeisanequivalencesetoverC�for`.ThenisaregularequivalencesetoverC�for`justincaseforeverynon-empty�Sent(C;P)consistentwith`,�isunitalsuitableforC�,`,and�.Proof.SupposeisaregularequivalencesetoverC�for`.Suppose�Sent(C;P).Suppose�`and�`.ByRegularity,f;g`(;).ByCut,�`(;).So(mod�).Therefore,�isunitalsuitableforC�,`,and�.Nowsupposeforeverynon-empty�Sent(C;P)consistentwith`,�isunitalsuitableforC�,`,and�.Suppose;2Sent(C;P).Iff;gisinconsistentwithrespectto`,f;g`(;).Suppose,then,thatf;gisconsistentwithrespectto`.Sof;gisunitalsuitableforC�,`,andf;g.ByIdentityandWeakening,f;g`andf;g`.Bystrongcompatibilitywith`andf;g,(modf;g).So,again,f;g`(;).Therefore,isaregularequivalencesetoverC�for`.Suppose1isanequivalencesetoverC1for`12and2isanequivalencesetoverC2for`12.LetBh1;2i12betheLindenbaum-TarskiclassofjuxtaposedstructuresforC1,C2,and`12builtusingthe�irelations.Anyconsequencerelationthatisequivalentialoversomesignaturehastheorems.CombiningthisfactwithLemmas6.37and6.38andTheorem6.7,wearriveatthefollowingresult:Proposition6.39.Suppose1isanequivalencesetoverC1for`12and2isanequivalencesetoverC2for`12.Then:1.`12isstronglycompletewithrespecttoBh1;2i12;2.If`12isthejuxtapositionof`1and`2,then`12isstronglysoundwithrespecttoBh1;2i12;3.If`12isconsistent,thenthereisacoherentnon-trivialmodelbasedonBh1;2i12;4.Bh1;2i12isaclassofjuxtaposedunitalstructuresjustincase1isaregularequivalencesetoverC1for`12and2isaregularequivalencesetoverC2for`12.Thereisaparticularlywell-behavedkindofequivalencesetoverC�.WesaythatisaninternalequivalencesetoverC�for`justincaseisanequivalencesetoverC�for`andSent(C�;fq;rg).Wesaythatisanregularinternalequivalencesetif,inaddition,satisestheRegularitycondition.Wesaythat JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS35Indeed,itisstraightforwardtoshowthatif`isinternallyequivalentialandregularlyequivalential,theneveryinternalequivalencerelationisregular.)Suppose`isinternallyequivalentialoverC�.Let�C�standfortheuniqueequivalencerelationdenedonSent(C;P)generatedbyanyoftheinternalequiv-alencesetsoverC�for`.Wecanshowthat�C�= �C�=�C�isthemostcoarse-grainedequivalencerelationonSent(C;P)suitableforC�,`,and�.Lemma6.43.1.IfisanequivalencesetoverC�for`,thenisanequivalencesetoverC�foranyconsequencerelationthatextends`;2.IfisaregularequivalencesetoverC�for`,thenisaregularequiva-lencesetoverC�foranyconsequencerelationthatextends`.Proof.SupposeCisasub-signatureofC+andPisasubsetofP+.Suppose`+isaconsequencerelationforSent(C+;P+)thatextends`.Supposep,q,andraredistinctelementsofP.Suppose;; 2Sent(C+;P+).Re exivity:`(p;p).So`+(p;p).ByUniformSubstitution,`+(;).Symmetry:(p;q)`(q;p).So(p;q)`+(q;p).ByUniformSubstitution,(;)`+(;).Transitivity:(p;q)[(q;r)`(p;r).So(p;q)[(q;r)`+(p;r).ByUniformSubstitution,(;)[(; )`+(; ).ModusPonens:fpg[(p;q)`q.Sofpg[(p;q)`+q.ByUniformSubsti-tution,fg[(;)`+.CongruenceoverC�:Letc�2C�n.Letp1;:::pn;qbedistinctelementsofP.Let1;:::;n;2Sent(C+;P+)andk2f1;:::;ng.(pk;q)`(c�p1:::pk:::pn,c�p1:::q:::pn).So(pk;q)`+(c�p1:::pk:::pn;c�p1:::q:::pn).ByUniformSubstitution,(k;)`+(c�1:::k:::n;c�1::::::n).Regularity:ByRegularity,fp;qg`(p;q).Sofp;qg`+(p;q).ByUniformSubstitution,f;g`(;).Proposition6.44.Suppose`isinternallyequivalentialoverC�.Suppose�Sent(C;P).Then�C�= �C�=�C�isthemostcoarse-grainedequiva-lencerelationoverSent(C;P)suitableforC�,`,and�.Proof.SupposeisaninternalequivalencesetoverC�for`.Werstshowthat(mod�C�)justincase(mod �C�).Suppose(mod�C�).ByProposition6.11,(mod �C�).Nowsuppose(mod �C�).Suppose2.LetpbeanelementofPthatdoesnotoccurin.SinceisaninternalequivalencesetoverC�for`,pstrictlyC�-occursin(;p).So�`(;p)[=p]justincase�`(;p)[=p].Thatis,�`(;)justincase�`(;).ByRe exivityandWeakening,�`(;).So�`(;).Therefore,(mod�C�).Wenextshowthat(mod�C�)justincase(modC�).Suppose(mod�C�).So�`(;).Suppose2Sent(C�;P[P)andpoccursin.ByLemma6.43,isanequivalencesetoverC�for`.ByProposition6.35,f[=p]g[(;)`[=p].Since`extends`,�`(;).ByCut,�[f[=p]g`[=p].Moreover,bySymmetry,�`(;).Sobyanalogousreasoning,�[f[=p]g`[=p].Therefore,(mod�C�). 38JOSHUAB.SCHECHTERTheclassicalconsequencerelationisstronglydeterminedwithrespecttotheclassofBooleanstructures.47Itisconsistent,hastheorems,andisleft-extensional.ByTheorem5.10,`ccisconsistent.ByTheorem5.11,`ccisstronglyconserva-tiveover`c1and`c2.ItcanbeaxiomatizedusingtwocopiesofanyHilbert-styleaxiomatizationforclassicallogic.(Itcanalsobeaxiomatizedusingtwocopiesofanynaturaldeduction-styleaxiomatizationforclassicallogicrestrictedsothattherulesforonestockofconnectivescannotbeappliedwithinanysub-derivationusedintheapplicationofameta-rulegoverningaconnectivefromtheotherstock.)ByCorollary6.32,`ccishC1;C2i-stronglyunitaldetermined.Wecanextractadditionalinformationabout`ccusingProposition6.34.TheclassofallBooleanstructuresisunitalfullfortheclassicalconsequencerelation.WesaythatajuxtaposedunitalstructurehB1;B2iisabi-BooleanstructurejustincasebothB1andB2areBooleanstructures.ByProposition6.34,`ccisstronglydeterminedwithrespecttotheclassofallbi-Booleanstructures.Thecaseofthebi-intuitionistconsequencerelation,`iiisanalogous.Givenanynon-trivialHeytingalgebra,hB;i,thereisacorrespondingunitalstructurehB;f1g;iwhereBisthesamesetofsemanticvalues,1isthegreatestelementoftheHeytingalgebra,andforeverya;b2B:(:)(a)=a)0;(^)(a;b)=aub;(_)(a;b)=atb;(!)(a;b)=a)b;($)(a;b)=(a)b)u(b)a):Here,u,t,and),aretheinmum,supremum,andimplicationrelationsontheHeytingalgebra,and0isitsleastelement.Letuscallsuchstructures\Heytingstructures".GivenaHeytingstructure,thepartialorderofthecorrespondingHeytingalgebracanberecovered:abjustincase(!)(a;b)=1.TheintuitionistconsequencerelationisstronglydeterminedwithrespecttotheclassofHeytingstructures.48Itisconsistent,hastheorems,andisleft-extensional.ByTheorem5.10,`iiisconsistent.ByTheorem5.11,`iiisstronglyconservativeover`i1and`i2.ItcanbeaxiomatizedusingtwocopiesofanyHilbert-styleaxiomatizationforintuitionistlogic.(Itcanalsobeaxiomatizedusingtwocopiesofanynaturaldeduction-styleaxiomatizationforintuitionistlogicrestrictedsothattherulesforonestockofconnectivescannotbeappliedwithinanysub-derivationusedintheapplicationofameta-rulegoverningaconnectivefromtheotherstock.)ByCorollary6.32,`iiishC1;C2i-stronglyunitaldetermined. 47TheclassicalconsequencerelationisalsostronglydeterminedwithrespecttotheclassofstructuresdenedasaboveexceptthatthesetofdesignatedvaluesisallowedtobeanyproperlterontheBooleanalgebra.48TheintuitionistconsequencerelationisalsostronglydeterminedwithrespecttotheclassofstructuresdenedasaboveexceptthatthesetofdesignatedvaluesisallowedtobeanyproperlterontheHeytingalgebra. JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS39TheclassofallHeytingstructuresisunitalfullfortheintuitionistconsequencerelation.WesaythatajuxtaposedunitalstructurehB1;B2iisabi-Heytingstruc-turejustincasebothB1andB2areHeytingstructures.ByProposition6.34,`iiisstronglydeterminedwithrespecttotheclassofallbi-Heytingstructures.Finally,considertheintuitionist-classicalconsequencerelation,`ic.ByThe-orem5.10,`icisconsistent.ByTheorem5.11,`icisstronglyconservativeover`i1and`c2.ItcanbeaxiomatizedbycombiningaHilbert-styleaxiomatizationforclassicallogicandaHilbert-styleaxiomatizationforintuitionistlogic.(Itcanalsobeaxiomatizedusingacopyofanynaturaldeduction-styleaxiomatizationforintuitionistlogicandacopyofanynaturaldeduction-styleaxiomatizationforclassicallogic,eachrestrictedsothattherulesforonestockofconnectivescannotbeappliedwithinanysub-derivationusedintheapplicationofameta-rulegoverningaconnectivefromtheotherstock.)ByCorollary6.32,`icishC1;C2i-stronglyunitaldetermined.WesaythatajuxtaposedunitalstructurehB1;B2iisaHeyting-Booleanstruc-turejustincaseB1isaHeytingstructureandB2isaBooleanstructure.AgainusingProposition6.34,`icisstronglydeterminedwithrespecttotheclassofallHeyting-Booleanstructures.7.2.Non-CollapseResults.Theseresultsalreadyhaveimplicationsforthecollapseofclassicalandintuitionistlogics.Considerthecaseof`ic.Wehaveshownthatthisconsequencerelationisstronglyconservativeover`i1and`c2.Thissucestoshowthat`icdoesnotcollapse.49Sincep_:pisatheoremofclassicallogicbutnotatheoremofintuitionistlogic,0icp_1:1pand`icp_2:2p.Similarly,Peirce'slaw,((p!q)!p)!p,canbeusedtoshowthat!1and!2arenotintersubstitutable.DoubleNegationEliminationcanbeusedtoshowthat:1and:2arenotintersubstitutable,either.Theinterestoftheseresultsshouldnotbeunderemphasized.Juxtapositionisanaturalwaytocombineclassicalandintuitionistlogic.`ichasalloftheentailmentsofintuitionistlogic(forthe1-connectives)andalloftheentailmentsofclassicallogic(forthe2-connectives).Indeed,itisastrongconservativeextensionofbothintuitionistandclassicallogic.Itobeystheusualstructuralrules,andissubstitutioninvariant.Yet,theclassicalandintuitionistconnectivesarenotintersubstitutable.Thereisnocollapse.Noticethatthenon-collapseresultalsoappliesto`ii.Thisisastrictlyweakerrelationthan`ic.Thus,ifcorrespondingconnectivesarenotintersubstitutablein`ic,theyarenotintersubstitutablein`ii,either.Whataboutbi-classicallogic?Doesthislogicavoidcollapse?Theanswerisyes,butprovingthistakesabitmorework.Inparticular,toshowthat`ccdoesnotcollapse,wemakeuseofLemma5.1tobuildacoherentbi-Booleanmodel.ABooleanmodelcanbespeciedbyspecifyingaBooleanalgebraandavalu-ation.BooleanalgebrascanbevisuallyrepresentedusingHassediagrams.Suchadiagramcontainsnodes{eachrepresentingadistinctelementofthecarriersetofthealgebra{andlinesegmentsconnectingpairsofnodes.Giventwoelementsofthecarrierset,aandb,abjustincasethereisapathfromato 49Thiscanalsobeprovedusingthetechniqueofcryptobring.SeeCaleiro&Ramos(2007). 42JOSHUAB.SCHECHTER�itomakeB�12intoaHeytingstructure.Hence,�1and�2mapcorrespondingconnectivestotheverysamefunctions.Itfollowsthatanytwosentencesthatareexactlyalikeexceptfortheirsubscriptswillhavethesamei-valuesinM�12.Let;02Sent(C12;P12)besentencesthatareexactlyalikeexceptforsomeoralloftheirsubscripts.Iffgisinconsistentwithrespectto`12,fg`120.Suppose,then,thatfgisconsistentwithrespectto`12.ByLemma6.2,Mfg12.Bytheabovereasoning,and0havethesamei-valuesinMfg12.SoMfg120.AgainusingLemma6.2,fg`120.Therefore,`12collapses.Thus,ifweenrich`ii(orastrongerlogic)withtherulethatsaysthat!1and!2areintersubstitutableasmainconnectives,theresultinglogiccollapses.Wecanleveragethisresulttogetfurthercollapseresults.Inintuitionistlogic,!isequivalentto$(^).Soitfollowsthatifweenrich`ii(orastrongerlogic)with(i)therulethatsaysthat$1and$2areintersubstitutableasmainconnectivesaswellas(ii)therulethatsaysthat^1and^2areintersub-stitutableingeneral,theresultinglogiccollapses.Inintuitionistlogic,!isalsoequivalentto$(_).Soifweenrich`ii(orastrongerlogic)with(i)therulethatsaysthat$1and$2areintersubstitutableasmainconnectivesaswellas(ii)therulethatsaysthat_1and_2areintersubstitutableingeneral,theresultinglogiccollapses.Wecanproveadditionalcollapseresultsforenrichmentsofbi-classicallogic.Forexample:Proposition7.4.Suppose`12isaconsequencerelationforSent(C12;P12)thatextends`cc.Supposeforany;2Sent(C12;P12),f_1g`12_2andf_2g`12_1.Then`12collapses.Proof.f!1g`cc:2_1(!1).f:2_1;:2_1(!1)g`cc:2_1.`cc:2_2andf:2_2g`12:2_1.ByCut,f!1g`12:2_1.f:2_1g`12:2_2.f:2_2g`cc!2.ByCut,f!1g`12!2.Similarly,f!2g`12!1.ByProposition7.3,`12collapses.Thus,ifweenrich`cc(orastrongerlogic)withtherulethatsaysthat_1and_2areintersubstitutableasmainconnectives,theresultinglogiccollapses.Inclassicallogic,!isequivalentto:(^:).Soifweenrich`cc(orastrongerlogic)with(i)therulethatsaysthat:1and:2areintersubstitutableingeneralaswellas(ii)therulethatsaysthat^1and^2areintersubstitutableingeneral,theresultinglogiccollapses.Wecanalsoprovenon-collapseresultsforenrichedlogics.Forexample:Proposition7.5.Suppose`12istheleastconsequencerelationthatextends`ccsuchthatforany;2Sent(C12;P12),(i)f:1g`12:2andf:2g`12:1;and(ii)f$1g`12$2andf$2g`12$1.Thenin`12,nopairofcorrespondingconnectivesareintersubstitutable.Inparticular:fp_1qg012p_2q;fp!1qg012p!2q;f:2(p$1q)g012:2(p$2q);f:2(p^1q)g012:2(p^2q);fp_1:1qg012p_1:2q. JUXTAPOSITION:ANEWWAYTOCOMBINELOGICS45Itfollowsfromthisresultthatifweenrich`icwithleft-extensionality,theresultingconsequencerelationdoesnotweaklycollapse.Takentogether,ourcollapseandnon-collapseresultsaresomewhatsurprising.Theyshowthattheissueofwhenalogiccollapses(orweaklycollapses)isverydelicate.Afullcatalogueofsuchresultsmustawaitanotheroccasion.7.5.Meta-Rules.Thereisanaltopicworthdiscussing{namely,thestatusofthefamiliarclassicalandintuitionistmeta-rules.Hereisalistofstandardnaturaldeductionmeta-rules:ConditionalIntroduction.If�[fg`then�`!;BiconditionalIntroduction.If�[fg`and�[fg`then�`$;ReasoningbyCases.If�[fg`and�[fg`then�[f_g`;IntuitionistReductio.If�[fg`and�[fg`:then�`:;ClassicalReductio.If�[f:g`and�[f:g`:then�`.Givenourresults,itiseasytoshowthatifweenrich`ii(orastrongerlogic)withConditionalIntroductionforbothof!1and!2,theresultinglogiccol-lapses:Proposition7.7.Let`12beaconsequencerelationforSent(C12;P12)thatextends`iiandobeysConditionalIntroductionforbothconditionals.Then`12collapses.50Proof.f;!1g`ii.ByConditionalIntroductionfor!2,f!1g`12!2.Similarly,f!2g`12!1.ByProposition7.3,`12collapses.Thisresulthasanimportantconsequence.Weknowthat`cc,`ii,and`icdonotcollapse.SononeoftheseconsequencerelationsobeyConditionalIn-troductionforbothconditionals.(Indeed,theydonotnotobeyConditionalIntroductionforeitherconditional.)Althoughjuxtapositionpreservesentail-ments,itdoesnotpreservethevalidityofmeta-rules.51Whatthisshowsisthatthemeta-rulesareinanimportantsensestrongerthanthecorrespondingentailmentsthattheylicense.Thestandardmeta-rulesarecloselyrelatedtooneanother:Proposition7.8.Suppose`isaconsequencerelationthatextendsthein-tuitionistconsequencerelation(inaperhapslargerlanguage).Suppose`obeysConditionalIntroduction.Then:1.`obeysBiconditionalIntroduction,ReasoningbyCases,andIntuitionistReductio;2.If`extendstheclassicalconsequencerelation,then`obeysClassicalRe-ductio.Proof.BiconditionalIntroduction:Suppose�[fg`and�[fg`.ByConditionalIntroduction,�`!and�`!.f!;!g`$.ByCut,�`$. 50SeeHarris(1982)foradierentproofofthisresult.51SeeConiglio(2007)foravariantofbringdesignedtopreservemeta-rules. 46JOSHUAB.SCHECHTERReasoningbyCases:Suppose�[fg`and�[fg`.ByConditionalIntroduction,�`!and�`!.f!;!;_g`.ByCut,�[f_g`.IntuitionistReductio:Suppose�[fg`and�[fg`:.ByConditionalIntroduction,�`!and�`!:.f!;!:g`:.ByCut,�`:.ClassicalReductio:Suppose�[f:g`and�[f:g`:.ByConditionalIntroduction,�`:!and�`:!:.Inclassicallogic,f:!;:!:g`.ByCut,�`:.Moreimportantlyforourpurposeshere,wealsohavethefollowingrelations:Proposition7.9.Suppose`isaconsequencerelationthatextendsthein-tuitionistconsequencerelation(inaperhapslargerlanguage).Then:1.If`obeysBiconditionalIntroductionorClassicalReductio,then`obeysConditionalIntroduction;2.If`extendstheclassicalconsequencerelationand`obeysReasoningbyCasesorIntuitionistReductio,then`obeysConditionalIntroduction.Proof.BiconditionalIntroduction:Suppose�[fg`.f;g`^.ByCut,�[fg`^.�[f^g`.ByBiconditionalIntroduction,�`$(^).f$(^)g`!.ByCut,�`!.ClassicalReductio:Werstshowthatif`obeysClassicalReductio,then`extendsclassicallogic.f::;:g`:.f::;:g`::.ByClassicalReductio,f::g`.Thus,`extendsclassicallogic.Wenextshowthat`obeysConditionalIntroduction.Suppose�[fg`.Inclassicallogic,�[f:(!)g`.ByCut,�[f:(!)g`.Inclassicallogic,�[f:(!)g`:.ByClassicalReductio,�`!.ReasoningbyCases:Suppose�[fg`.fg`!.ByCut,�[fg`!.Inclassicallogic,�[f:g`!.ByReasoningbyCases,�[f_:g`!.Inclassicallogic,`_:.ByCut,�`!.IntuitionistReductio:Suppose�[fg`.Inclassicallogic,�[f:(!)g`.ByCut,�[f:(!)g`.Inclassicallogic,�[f:(!)g`:.ByIntuitionistReductio,�`::(!).Inclassicallogic,f::(!)g`!.ByCut,�`!.Itfollowsthatifweenrich`ii(orastrongerlogic)withtwocopiesofBicondi-tionalIntroductionorwithtwocopiesofClassicalReductio,theresultinglogiccollapses.Ifweenrich`cc(orastrongerlogic)withtwocopiesofReasoningbyCasesortwocopiesofIntuitionistReductio,theresultinglogiccollapses.52Thereisanothermeta-ruleworthdiscussing.RecallthedenitionofEntail-mentCongruence:EntailmentCongruence.If�[fg`and�[fg`thenforany2Sent(C;P)andpoccurringin,�[f[=p]g`[=p].Thismeta-ruleisnotanintroductionoreliminationrule.Itdoesnotgovernthebehaviorofanyparticularconnective.Roughlyspeaking,itgovernshow 52SeeMcGee(2000)foraproofoftheclaimthatif`12isaconsequencerelationforSent(C12;P12)thatobeystheusualnaturaldeductionrulesforintuitionistlogicforeachstockofconnectives,then`12collapses. 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