Table Of ContentS
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One is often said to be reasoning well when one is reasoning a
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logically. Many attempts to say what logical reasoning is ir
o
have been proposed, but one commonly proposed system is a
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classical first-order logic. This Element will examine the basics
d
of classical first-order logic and discuss some surrounding K
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philosophical issues. The first half of the Element develops a S philosophy and logic
e
language for the system, as well as a proof theory and model l
theory. The authors provide theorems about the system they
developed, such as unique readability and the Lindenbaum
lemma. They also discuss the meta-theory for the system, and
provide several results there, including sketching a proof of the
soundness and completeness theorems. The second half of the
C Classical
Element compares classical first-order logic to other systems: la
s
classical higher-order logic, intuitionistic logic, and several sic
a
paraconsistent logics which reject the law of ex falso quodlibet. l F
ir First-order
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about the Series Series editors
This Cambridge Elements series provides Bradley
an extensive overview of the many and Armour-Garb
varied connections between philosophy SUNY Albany Stewart Shapiro and
and logic. Distinguished authors provide Frederick Kroon
an up-to-date summary of the results of The University of Teresa Kouri Kissel
current research in their fields and give Auckland
their own take on what they believe are
the most significant debates influencing
research, drawing original conclusions.
Cover image: J ESPER KLAUSEN /
SCIENCE PHOTO LIBRARY/ Getty Images ISSN 2516-418X (online)
https://doi.org/10.1017/9781108982009 Published online by Cambridge University Press ISSN 2516-4171 (print)
https://doi.org/10.1017/9781108982009 Published online by Cambridge University Press
Elements in Philosophy and Logic
editedby
BradleyArmour-Garb
SUNYAlbany
FrederickKroon
TheUniversityofAuckland
CLASSICAL FIRST-ORDER LOGIC
Stewart Shapiro
Ohio State University
Teresa Kouri Kissel
Old Dominion University
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DOI:10.1017/9781108982009
©StewartShapiroandTeresaKouriKissel2022
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https://doi.org/10.1017/9781108982009 Published online by Cambridge University Press
Classical First-Order Logic
ElementsinPhilosophyandLogic
DOI:10.1017/9781108982009
Firstpublishedonline:April2022
StewartShapiro
OhioStateUniversity
TeresaKouriKissel
OldDominionUniversity
Authorforcorrespondence:TeresaKouriKissel,[email protected]
Abstract:Oneisoftensaidtobereasoningwellwhenoneisreasoning
logically.Manyattemptstosaywhatlogicalreasoningishavebeen
proposed,butonecommonlyproposedsystemisclassicalfirst-orderlogic.
ThisElementwillexaminethebasicsofclassicalfirst-orderlogicand
discusssomesurroundingphilosophicalissues.Thefirsthalfofthe
Elementdevelopsalanguageforthesystem,aswellasaprooftheoryand
modeltheory.Theauthorsprovidetheoremsaboutthesystemthey
developed,suchasuniquereadabilityandtheLindenbaumlemma.They
alsodiscussthemeta-theoryforthesystem,andprovideseveralresults
there,includingsketchingaproofofthesoundnessandcompleteness
theorems.ThesecondhalfoftheElementcomparesclassicalfirst-order
logictoothersystems:classicalhigher-orderlogic,intuitionisticlogic,and
severalparaconsistentlogicswhichrejectthelawofexfalsoquodlibet.
Keywords:classicallogic,first-orderlogic,modeltheory,prooftheory,
meta-theoryhigher-orderlogic,intuitionisticlogic,paraconsistentlogic
©StewartShapiroandTeresaKouriKissel2022
ISBNs:9781108987004(PB),9781108982009(OC)
ISSNs:2516-418X(online),2516-4171(print)
https://doi.org/10.1017/9781108982009 Published online by Cambridge University Press
Contents
1 Introduction 1
ClassicalFirst-Order Logic 1
2 Formal System 1
3 Language 2
4 Deduction 10
5 Model-Theoretic Semantics 16
6 Meta-theory 20
Alternativesto Classical First-Order Logic 23
7 Classical Higher-Order Logic 23
8 Intuitionism 42
9 Paraconsistency: Demurring fromex falso quodlibet 53
10 Conclusion 64
References 66
https://doi.org/10.1017/9781108982009 Published online by Cambridge University Press
ClassicalFirst-OrderLogic 1
1 Introduction
Logic surely has something to do with reasoning. We say that someone has
reasoned poorly about something if they have not reasoned logically, or that
an argument is bad because it is not logically valid. There has been much
speculation over just what types of logical systems are appropriate for guid-
ingdeductivereasoning.Somehavesuggestedthatclassicalfirst-orderlogicis
theidealforguidingreasoning(forexample,seeQuine(1986),Resnik(1996)
or Rumfitt (2015)).1 Classical first-order logic has occasionally been dubbed
“theonetruelogic.”
Though there is much debate about the underlying issues, it is safe to say
that classical first-order logic has been prevalent in mathematics and philos-
ophy over the past century or so. There are good reasons for this. Classical
first-orderlogichasruleswhicharemoreorlessintuitive,anditissurprisingly
simplegivenitsstrength.Plus,itisbothsoundandcomplete.2 Asiscommon
inphilosophy,however,thereisnoconsensusontheprimarystatusofclassical
first-orderlogic.AsindicatedinSection6,therearecertainexpressivelimita-
tionstofirst-orderlogicand,asindicatedinthelatersections(7-9),thereare
serious,well-developedrivalstoclassicallogic.
This Element will examine classical first-order logic and then provide
a more thorough understanding of it by exploring some alternative logics.
In the first half, we provide the details of classical first-order logic. Then,
we consider three alternatives to the system we develop: classical higher-
order logic (Section 7), intuitionistic logic (Section 8), and para-consistent
logics(Section9).3
CLASSICAL FIRST-ORDER LOGIC
2 Formal System
For our purposes, a logic includes a formal language and a deductive sys-
tem and/or a model-theoretic semantics.4 A formal language is like a natural
1 Whendiscussingcriticalthinkingandargumentanalysis,classicalfirst-orderlogic,isusually,
butnotalways,thestartingpoint.See,forexample,Stebbing(1939)andWoods(2021).
2 Wesaywhatthismeansbelow.
3 Thereareanumberofalternativestoclassicalfirst-orderlogicthatwewillnothavespace
to cover here. In particular, we omit modal logics (see Marcus (1995), for example) and
substructurallogics(seeRestall(2000),amongstothers).
4 Sections2-6arebasedonShapiroandKouriKissel(2020).Morecompleteproofsandproof
sketchescanbefoundthere,orinmosttextbooksforformallogic.Weincludehereonlyafew
ofthecompleteproofs:thosethatarerequiredtoproveuniquereadability,asanexampleof
howproofsoftengo,andaproofoftheorem4.5(TheruleofCut)Notalllogicshavebotha
deductivesystemandmodeltheory,butwewilldevelopbothforclassicalfirst-orderlogichere.
https://doi.org/10.1017/9781108982009 Published online by Cambridge University Press
2 PhilosophyandLogic
language, such as English, Japanese, or Swahili, but with explicit and sim-
ple rules of composition. It has terms and components which come together
in a grammatical way. The deductive system is meant to capture reason-
ing presented in the language. The model theory gives meanings to the
terms in the language, and tells us which propositions are true in various
interpretations.
Section3developsaformallanguagewitharigoroussyntaxandgrammar.
The formal language is a recursively defined collection of strings on a fixed
alphabet.Itemsintheformallanguagedonotmeananythingontheirown–they
needthedeductivesystemandmodeltheorytogivethemmeaning.Someitems
correspondtoitemsinnaturallanguage,andthedeductivesystemandmodel
theoryaredesignedtopreservethiscorrespondence,atleastapproximately.An
argument is a non-empty collection of sentences in the formal language, one
ofwhichisdesignatedtobetheconclusion.Theothersentences(ifany)inan
argumentareitspremises.
Section4setsupadeductivesystemforthelanguage,inthespiritofnatural
deduction.Anargumentisderivableifthereisadeductionfromsomeorallof
itspremisestoitsconclusion.Section5providesamodel-theoreticsemantics.
An argument is valid if there is no interpretation (in the semantics) in which
itspremisesarealltrueanditsconclusionfalse.Thisreflectsthelong-standing
viewthatavalidargumentistruth-preserving.
InSection6,weexploretherelationshipbetweenthedeductivesystemand
themodel-theoreticsemantics.Wewillbeinterestedmostlyintherelationship
betweenderivabilityandvalidity.Wesketchtwoimportanttheorems.Thefirst,
soundness,willshowthatnodeductioncanstartwithatruepremiseandend
withafalseconclusion.So,ifanargumentisderivable,thenitisvalid.Prov-
ingthisshowsthatdeductionspreservetruth.Thesecond,completeness,isthe
converseofsoundness.Itshowsthatifanargumentisvalid,thenitisderivable.
Thisdemonstratesthatthereare“enoughdeductions”:allvalidargumentsare
derivable.Wealsobrieflyindicateotherfeaturesofthelogic,someofwhich
arecorollariestosoundnessandcompleteness.
3 Language
Here we develop the basics of the formal language we will use for classical
logic.Asnoted,aformallanguageisarecursivelydefinedsetofstringsona
fixed alphabet. Some parts of the formal language correspond to some parts
of natural language. This correspondence is not really a part of the formal
language,butnotingthesecorrespondencescanhelpmotivatethesystem.
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ClassicalFirst-OrderLogic 3
3.1 BuildingBlocks
Westartwithanaloguesofsingularterms,sometimesjustcalledterms.These
arelinguisticitemswhosefunctionistodenoteapersonorobject.Weassume
astockofindividualconstants.Thesearelower-caseletters,nearthebeginning
oftheRomanalphabet,withorwithoutnumericalsubscripts:
a,a ,b ,c,d ,etc.
1 23 22
Constantsareananalogueofpropernames.Weallowourselvesaninfinity
ofindividualconstants.Eachconstantisasinglecharacter,andsoindividual
constantsdonothaveaninternalsyntax.Thus,wehaveaninfinitealphabet.5
There are two roles that constants play. One is like that of proper names,
where a constant denotes a specific object or person (in each interpretation).
The other is to denote specific, but unspecified (or arbitrary), objects and
persons.6
Wealsoassumeastockofindividualvariables.Thesearelower-caseletters,
neartheendofthealphabet,withorwithoutnumericalsubscripts:
w,x,y ,z,z ,etc.
12 4
Variables are used to express generality. In ordinary language, some uses of
pronounsplaythelatterrole,asin“Whenadogisangry,itgrowls.”7
Wenextintroducefunctionsymbols.Theseallowcomplextermscorrespond-
ingto:“5+3”and“theMarvelcharacterplayedbyChadwickBoseman,”or
termscontainingvariables,like“thesisterofx”and“x(cid:157)y.”Functionsymbols
arelower-caseletters,nearthemiddleofthealphabet:
f,g,h,etc.
Eachfunctionhasanarity,thatis,thenumberofargumentsittakes.
Wenowgivearecursivedefinitionofthetermsofthelanguage:
1. Allindividualconstantsandallvariablesareterms.
2. Ift ,...t aretermsandfisann-placefunctionsymbol,thenft ...t isa
1 n 1 n
term.
3. That’sallfolks:everytermisconstructedinaccordancewith(1)and(2).
Atermisclosedifitcontainsnovariables.
For each natural number n, we introduce a stock of n(cid:0)place predicate let-
ters. These are upper-case letters at the beginning or middle of the alphabet.
5 Thiscouldbeavoidedbytakingaconstantliked22,forexample,toconsistofthreecharacters,
alowercase“d”followedbyapairofsubscript“2”s.
6 Someauthorsuse(free)variablesforthisrole;othersusedifferentsymbolsforthis,sometimes
called“individualparameters.”
7 Anotheruseofpronounsistodenoteaspecificobjectorperson,assuppliedbythecontext.The
formallanguagehasnoanalogueofthis.
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4 PhilosophyandLogic
A superscript indicates the number of places, and there may or may not be
a subscript. For example, A3,B3,D3,etc, are three-place predicate letters. We
2
oftenomitthesuperscriptwhennoconfusionwillresult.Wealsoaddaspecial
two-placepredicatesymbol“=”foridentity.
Zero-place predicate letters are sometimes called “sentence letters.” They
correspond to freestanding sentences whose internal structure does not mat-
ter.One-placepredicateletters,called“monadicpredicateletters,”correspond
to linguistic items denoting properties, adjectives, or common nouns like
“woman,” “large,” or “prime number.” Two-place predicate letters, called
“binarypredicateletters,”correspondtolinguisticitemsdenotingbinaryrela-
tions, like “is a parent of” or “is shinier than.” Three-place predicate letters
correspondtothree-placerelations,like“liesonastraightlinebetween.”And
soon.
Thenon-logicalterminologyofthelanguageconsistsofitsindividualcon-
stantsandpredicateletters.Thesymbol“=”,foridentity,isnotanon-logical
symbol.Intakingidentitytobelogical,weprovideexplicittreatmentforitin
thedeductivesystemandinthemodel-theoreticsemantics.Mostauthorsdothe
same,butthereissomecontroversyovertheissue(seeQuine(1986)Chapter
5).IfKisasetofconstantsandpredicateletters,thenwegivethefundamen-
tals of a language L1K= built on this set of non-logical terminology. It may
becalledthefirst-orderlanguagewithidentityonK.Asimilarlanguagethat
lacksthesymbolforidentity(orwhichtakesidentitytobenon-logical)maybe
calledL1K,thefirst-orderlanguagewithoutidentityonK.
3.2 AtomicFormulas
IfVisann-placepredicateletterinK,andt ,...,t aretermsofK,thenVt ...t
1 n 1 n
isanatomicformulaofL1K=.Examplesofatomicformulasinclude:
P4xaab,C1x,C1a,D0,A3aba.
The last one is an analogue of a statement that a certain relation „A” holds
betweentheobjects„a,b,a”.Ift andt areterms,thent =t isalsoanatomic
1 2 1 2
formulaofL1K=.Itcorrespondstoanassertionthatt isidenticaltot .
1 2
Ifanatomicformulahasnovariables,thenitiscalledanatomicsentence.If
itdoeshavevariables,itiscalledopen.Intheabovelistofexamples,thefirst
andsecondareopen;therestareatomicsentences.
3.3 Compound Formulas
Thefinalitemsofthelexiconare:
:,^,_,!,8,9,„,”
WegivearecursivedefinitionofaformulaofL1K=:
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