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An axiomatic theory of truth is a deductive theory of truth as aprimitive undefined predicate. Because of the liar and otherparadoxes, the axioms and rules have to be chosen carefully in orderto avoid inconsistency. Many axiom systems for the truth predicatehave been discussed in the literature and their respective propertiesbeen analysed. Several philosophers, including many deflationists, have endorsed axiomatic theories of truth in theiraccounts of truth. The logical properties of the formal theories arerelevant to various philosophical questions, such as questions aboutthe ontological status of properties, Gödel’s theorems,truth-theoretic deflationism, eliminability of semantic notions andthe theory of meaning.

  • 1. Motivations
  • 2. The base theory
  • 3. Typed theories of truth
  • 4. Type-free truth
  • 5. Non-classical approaches to self-reference

1. Motivations

There have been many attempts to define truth in terms of correspondence, coherence or other notions. However, it is far from clear that truth is a definable notion. Informal settings satisfying certain natural conditions, Tarski’stheorem on the undefinability of the truth predicate shows that adefinition of a truth predicate requires resources that go beyondthose of the formal language for which truth is going to be defined.In these cases definitional approaches to truth have to fail. By contrast,the axiomatic approach does not presuppose that truth can bedefined. Instead, a formal language is expanded by a new primitivepredicate for truth or satisfaction, and axioms for that predicate are then laiddown. This approach by itself does not preclude the possibility that the truth predicateis definable, although in many cases it can be shown that the truthpredicate is not definable.

In semantic theories of truth (e.g., Tarski 1935, Kripke 1975), incontrast, a truth predicate is defined for a language, the so-calledobject language. This definition is carried out in a metalanguage ormetatheory, which is typically taken to include set theory or at leastanother strong theory or expressively rich interpretedlanguage. Tarski’s theorem on the undefinability of the truthpredicate shows that, given certain general assumptions, the resourcesof the metalanguage or metatheory must go beyond the resources of theobject-language. So semantic approaches usually necessitate the use ofa metalanguage that is more powerful than the object-language forwhich it provides a semantics.

As with other formal deductive systems, axiomatic theories of truth can bepresented within very weak logical frameworks. These frameworks require veryfew resources, and in particular, avoid the need for a strong metalanguage andmetatheory.

Formal work on axiomatic theories of truth has helped to shed some light onsemantic theories of truth. For instance, it has yielded information on what isrequired of a metalanguage that is sufficient for defining a truth predicate.Semantic theories of truth, in turn, provide one with the theoretical toolsneeded for investigating models of axiomatic theories of truth and withmotivations for certain axiomatic theories. Thus axiomatic and semanticapproaches to truth are intertwined.

This entry outlines the most popular axiomatic theories of truth andmentions some of the formal results that have been obtained concerning them. Wegive only hints as to their philosophical applications.

1.1 Truth, properties and sets

Theories of truth and predication are closely related to theories of properties and property attribution. To say that an open formula (phi(x))is true of an individual (a) seems equivalent (in some sense)to the claim that (a) has the property ofbeing such that (phi) (this property is signified by the open formula).For example, one might say that ‘(x) is a poor philosopher’ is trueof Tom instead of saying that Tom has the property of being a poor philosopher.Quantification over definable properties can then be mimicked in a languagewith a truth predicate by quantifying over formulas. Instead of saying, forinstance, that (a) and (b) have exactly the same properties, onesays that exactly the same formulas are true of (a) and (b). Thereduction of properties to truth works also to some extent for sets ofindividuals.

There are also reductions in the other direction: Tarski (1935) hasshown that certain second-order existence assumptions (e.g.,comprehension axioms) may be utilized to define truth (see the entryon Tarski’s definition oftruth). The mathematical analysis of axiomatic theories of truthand second-order systems has exhibited many equivalences between thesesecond-order existence assumptions and truth-theoreticassumptions.

These results show exactly what is required for defining a truth predicatethat satisfies certain axioms, thereby sharpening Tarski’s insights intodefinability of truth. In particular, proof-theoretic equivalences described inSection 3.3 below make explicit to what extent ametalanguage (or rather metatheory) has to be richer than the object languagein order to be able to define a truth predicate.

The equivalence between second-order theories and truth theories also hasbearing on traditional metaphysical topics. The reductions of second-ordertheories (i.e., theories of properties or sets) to axiomatic theories of truthmay be conceived as forms of reductive nominalism, for they replace existenceassumptions for sets or properties (e.g., comprehension axioms) byontologically innocuous assumptions, in the present case by assumptions on thebehaviour of the truth predicate.

1.2 Truth and reflection

According to Gödel’s incompleteness theorems, the statement that Peano Arithmetic (PA)is consistent, in its guise as a number-theoretic statement (given thetechnique of Gödel numbering), cannot be derived in PAitself. But PA can be strengthened by adding this consistencystatement or by stronger axioms. In particular, axioms partiallyexpressing the soundness of PA can be added. These are known asreflection principles. An example of a reflection principle for PAwould be the set ofsentences (Bew_{PA}(ulcorner phi urcorner)rightarrow phi) where (phi) is a formula of the language ofarithmetic, (ulcorner phi urcorner) a name for (phi)and (Bew_{PA}(x)) is the standard provabilitypredicate for PA (‘(Bew)’ was introduced byGödel and is short for the German word‘beweisbar’, that is,‘provable’).

The process of adding reflection principles can be iterated: onecan add, for example, a reflection principle R for PA to PA; thisresults in a new theory PA+R. Then one adds the reflection principlefor the system PA+R to the theory PA+R. This process can be continuedinto the transfinite (see Feferman 1962 and Franzén 2004).

The reflection principles express—at leastpartially—the soundness of the system. The most natural and fullexpression of the soundness of a system involves the truth predicateand is known as the Global Reflection Principle (see Kreisel andLévy 1968). The Global Reflection Principle for a formal systemS states that all sentences provable in S are true:

[forall x(Bew_S (x) rightarrow Tx)]

(Bew_S (x)) expresses here provability ofsentences in the system S (we omit discussion here of the problems ofdefining (Bew_S (x))). The truth predicatehas to satisfy certain principles; otherwise the global reflectionprinciple would be vacuous. Thus not only the global reflectionprinciple has to be added, but also axioms for truth. If a naturaltheory of truth like T(PA) below is added, however, it is no longernecessary to postulate the global reflection principle explicitly, astheories like T(PA) prove already the global reflection principle forPA. One may therefore view truth theories as reflection principles asthey prove soundness statements and add the resources to express thesestatements.

Thus instead of iterating reflection principles that are formulatedentirely in the language of arithmetic, one can add by iteration newtruth predicates and correspondingly new axioms for the new truthpredicates. Thereby one might hope to make explicit all theassumptions that are implicit in the acceptance of a theory likePA. The resulting theory is called the reflective closure of theinitial theory. Feferman (1991) has proposed the use of a single truthpredicate and a single theory (KF), rather than a hierarchy ofpredicates and theories, in order to explicate the reflective closureof PA and other theories. (KF is discussed furtherin Section 4.4 below.)

The relation of truth theories and (iterated) reflection principlesalso became prominent in the discussion of truth-theoreticdeflationism (see Tennant 2002 and the follow-up discussion).

1.3 Truth-theoretic deflationism

Many proponents of deflationist theories of truth have chosen to treat truth as a primitive notionand to axiomatize it, often using some version ofthe (T)-sentences as axioms. (T)-sentences areequivalences of the form(Tulcorner phi urcorner leftrightarrow phi), where (T) is the truth predicate, (phi) is a sentenceand (ulcorner phi urcorner) is a name for thesentence (phi). (More refined axioms have also been discussed bydeflationists.) At first glance at least, the axiomatic approach seemsmuch less ‘deflationary’ than those more traditionaltheories which rely on a definition of truth in terms ofcorrespondence or the like. If truth can be explicitly defined, it canbe eliminated, whereas an axiomatized notion of truth may and oftendoes come with commitments that go beyond that of the basetheory.

If truth does not have any explanatory force, as some deflationistsclaim, the axioms for truth should not allow us to prove any newtheorems that do not involve the truth predicate. Accordingly, Horsten (1995), Shapiro (1998) and Ketland (1999) have suggested thata deflationary axiomatization of truth should be atleast conservative. The new axioms for truth are conservativeif they do not imply any additional sentences (free of occurrences ofthe truth-predicate) that aren’t already provable without thetruth axioms. Thus a non-conservative theory of truth adds newnon-semantic content to a theory and has genuine explanatory power,contrary to many deflationist views. Certain natural theories oftruth, however, fail to be conservative (see Section3.3 below, Field 1999 and Shapiro 2002 for furtherdiscussion).

According to many deflationists, truth serves merely the purpose ofexpressing infinite conjunctions. It is plain that not allinfinite conjunctions can be expressed because there are uncountablymany (non-equivalent) infinite conjunctions over a countablelanguage. Since the language with an added truth predicate has onlycountably many formulas, not every infinite conjunction can beexpressed by a different finite formula. The formal work on axiomatictheories of truth has helped to specify exactly which infiniteconjunctions can be expressed with a truth predicate. Feferman (1991)provides a proof-theoretic analysis of a fairly strong system. (Again,this will be explained in the discussion about KFin Section 4.4 below.)

2. The base theory

2.1 The choice of the base theory

In most axiomatic theories, truth is conceived as a predicate ofobjects. There is an extensive philosophical discussion on thecategory of objects to which truth applies: propositions conceived asobjects that are independent of any language, types and tokens ofsentences and utterances, thoughts, and many other objects have beenproposed. Since the structure of sentences considered as types isrelatively clear, sentence types have often been used as the objectsthat can be true. In many cases there is no need to make very specificmetaphysical commitments, because only certain modest assumptions onthe structure of these objects are required, independently fromwhether they are finally taken to be syntactic objects, propositionsor still something else. The theory that describes the properties ofthe objects to which truth can be attributed is called the basetheory. The formulation of the base theory does not involve thetruth predicate or any specific truth-theoretic assumptions. The basetheory could describe the structure of sentences, propositions and thelike, so that notions like the negation of such an object can then beused in the formulation of the truth-theoretic axioms.

In many axiomatic truth theories, truth is taken as a predicateapplying to the Gödel numbers of sentences. Peano arithmetic hasproved to be a versatile theory of objects to which truth is applied,mainly because adding truth-theoretic axioms to Peano arithmeticyields interesting systems and because Peano arithmetic is equivalentto many straightforward theories of syntax and even theories ofpropositions. However, other base theories have been considered aswell, including formal syntax theories and set theories.

Of course, we can also investigate theories which result by addingthe truth-theoretic axioms to much stronger theories like settheory. Usually there is no chance of proving the consistency of settheory plus further truth-theoretic axioms because the consistency ofset theory itself cannot be established without assumptionstranscending set theory. In many cases not even relative consistencyproofs are feasible. However, if adding certain truth-theoretic axiomsto PA yields a consistent theory, it seems at least plausible thatadding analogous axioms to set theory will not lead to aninconsistency. Therefore, the hope is that research on theories oftruth over PA will give an some indication of what will happen when weextend stronger theories with axioms for the truth predicate. However, Fujimoto (2012) has shown that some axiomatic truth theories over set theory differ from their counterparts over Peano arithmetic in some aspects.

Truth

2.2 Notational conventions

For the sake of definiteness we assume that the language ofarithmetic has exactly (neg , wedge) and (vee) as connectives and(forall) and (exists) as quantifiers. It has as individual constantsonly the symbol 0 for zero; its only function symbol is the unarysuccessor symbol (S); addition and multiplication are expressedby predicate symbols. Therefore the only closed terms of the languageof arithmetic are the numerals(0, S)(0), (S(S)(0)), (S(S(S)(0))),….

The language of arithmetic does not contain the unary predicatesymbol (T), solet (mathcal{L}_T) be thelanguage of arithmetic augmented by the new unary predicatesymbol (T) for truth. If (phi) is a sentenceof (mathcal{L}_T,ulcorner phi urcorner) is a name for (phi) in thelanguage (mathcal{L}_T);formally speaking, it is the numeral ofthe Gödel number of(phi). In general, Greek letters like (phi) and (psi) are variables ofthe metalanguage, that is, the language used for talking abouttheories of truth and the language in which this entry is written(i.e., English enriched by some symbols). (phi) and (psi) range overformulas of the formallanguage (mathcal{L}_T).

In what follows, we use small, upper case italic letterslike ({scriptsize A}, {scriptsize B},ldots) asvariables in (mathcal{L}_T)ranging over sentences (or their Gödel numbers, to beprecise). Thus(forall{scriptsize A}(ldots{scriptsize A}ldots))stands for(forall x(Sent_T (x)rightarrow ldots xldots)),where (Sent_T (x)) expresses in thelanguage of arithmetic that (x) is a sentence of the languageof arithmetic extended by the predicate symbol (T). Thesyntactical operations of forming a conjunction of two sentences andsimilar operations can be expressed in the language ofarithmetic. Since the language of arithmetic does not contain anyfunction symbol apart from the symbol for successor, these operationsmust be expressed by sutiable predicate expressions. Thus one can sayin the language (mathcal{L}_T)that a negation of a sentenceof (mathcal{L}_T) is true if andonly if the sentence itself is not true. We would write this as

[forall{scriptsize A}(T[neg{scriptsize A}] leftrightarrow neg T{scriptsize A}).]

The square brackets indicate that the operation of forming the negation of({scriptsize A}) is expressed in the language of arithmetic. Since thelanguage of arithmetic does not contain a function symbol representing thefunction that sends sentences to their negations, appropriate paraphrasesinvolving predicates must be given.

Thus, for instance, the expression

[forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} wedge{scriptsize B}] leftrightarrow(T{scriptsize A} wedge T{scriptsize B}))]

is a single sentence of the language (mathcal{L}_T) saying that a conjunction ofsentences of (mathcal{L}_T) is true ifand only if both sentences are true. In contrast,

[Tulcorner phi wedge psi urcorner leftrightarrow (Tulcorner phi urcorner wedge Tulcorner phi urcorner)]

is only a schema. That is, it stands for the set of all sentencesthat are obtained from the above expression by substituting sentencesof (mathcal{L}_T) for the Greekletters (phi) and (psi). The single sentence(forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} wedge{scriptsize B}] leftrightarrow (T{scriptsize A} wedge T{scriptsize B}))) implies all sentenceswhich are instances of the schema, but the instances of the schema donot imply the single universally quantified sentence. In general, thequantified versions are stronger than the corresponding schemata.

3. Typed theories of truth

In typed theories of truth, only the truth of sentences notcontaining the same truth predicate is provable, thus avoiding theparadoxes by observing Tarski’s distinction between object andmetalanguage.

3.1 Definable truth predicates

Certain truth predicates can be defined within the language ofarithmetic. Predicates suitable as truth predicates for sublanguagesof the language of arithmetic can be defined within the language ofarithmetic, as long as the quantificational complexity of the formulasin the sublanguage is restricted. In particular, there is aformula (Tr_0 (x)) that expressesthat (x) is a true atomic sentence of the language ofarithmetic, that is, a sentence of the form (n=k),where (k) and (n) are identical numerals. For furtherinformation on partial truth predicates see, for instance,Hájek and Pudlak (1993), Kaye (1991) and Takeuti (1987).

The definable truth predicates are truly redundant, because theyare expressible in PA; therefore there is no need to introduce themaxiomatically. All truth predicates in the following are not definablein the language of arithmetic, and therefore not redundant at least inthe sense that they are not definable.

3.2 The (T)-sentences

The typed (T)-sentences are all equivalences of theform (Tulcorner phi urcorner leftrightarrow phi), where (phi) is a sentence not containing the truthpredicate. Tarski (1935) called any theory proving these equivalences‘materially adequate’. Tarski (1935) criticised anaxiomatization of truth relying only on the (T)-sentences, notbecause he aimed at a definition rather than an axiomatization oftruth, but because such a theory seemed too weak. Thus although thetheory is materially adequate, Tarski thought thatthe (T)-sentences are deductively too weak. He observed, inparticular, that the (T)-sentences do not prove the principleof completeness, that is, the sentence(forall{scriptsize A}(T{scriptsize A}vee T[neg{scriptsize A})])where the quantifier (forall{scriptsize A}) is restrictedto sentences not containing T.

Theories of truth based on the (T)-sentences, and theirformal properties, have also recently been a focus of interest in thecontext of so-called deflationary theories oftruth. The (T)-sentences (Tulcorner phi urcorner leftrightarrow phi) (where (phi) does not contain (T)) are notconservative over first-order logic with identity, that is, they provea sentence not containing (T) that is not logically valid. Forthe (T)-sentences prove that the sentences (0=0) and (neg 0=0) aredifferent and that therefore at least two objects exist. In otherwords, the (T)-sentences are not conservative over the emptybase theory. If the (T)-sentences are added to PA, theresulting theory is conservative over PA. This means that the theorydoes not prove (T)-free sentences that are not already provablein PA. This result even holds if in addition tothe (T)-sentences also all induction axioms containing thetruth predicate are added. This may be shown by appealing to theCompactness Theorem.

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In the form outlined above, T-sentences express the equivalence between (Tulcorner phi urcorner) and (phi) only when (phi) is a sentence.In order to capture the equivalence for properties ((x) has property P iff ‘P’ is true of (x)) one must generalise the T-sentences. The result are usually referred to as the uniform T-senences and are formalised by the equivalences (forall x(Tulcorner phi(underline{x})urcorner leftrightarrow phi(x))) for each open formula (phi(v)) with at most (v) free in (phi).Underlining the variable indicates it is bound from the outside.More precisely, (ulcorner phi(underline{x})urcorner) stands for the result of replacing the variable (v)in (ulcorner phi(v)urcorner) by the numeralof (x).

3.3 Compositional truth

As was observed already by Tarski (1935), certain desirablegeneralizations don’t follow from the T-sentences. For instance,together with reasonable base theories they don’t imply that aconjunction is true if both conjuncts are true.

In order to obtain systems that also prove universally quantifiedtruth-theoretic principles, one can turn the inductive clauses ofTarski’s definition of truth into axioms. In the followingaxioms, (AtomSent_{PA}(ulcorner{scriptsize A}urcorner))expresses that ({scriptsize A}) is an atomic sentence of thelanguage ofarithmetic, (Sent_{PA}(ulcorner{scriptsize A}urcorner))expresses that ({scriptsize A}) is a sentence of the languageof arithmetic.

  1. (forall{scriptsize A}(AtomSent_{PA}({scriptsize A}) rightarrow(T{scriptsize A} leftrightarrow Tr_0 ({scriptsize A}))))
  2. (forall{scriptsize A}(Sent_{PA}({scriptsize A}) rightarrow(T[neg{scriptsize A}] leftrightarrow neg T{scriptsize A})))
  3. (forall{scriptsize A}forall{scriptsize B}(Sent_{PA}({scriptsize A}) wedge Sent_{PA}({scriptsize B}) rightarrow (T[{scriptsize A} wedge{scriptsize B}] leftrightarrow(T{scriptsize A} wedge T{scriptsize B}))))
  4. (forall{scriptsize A}forall{scriptsize B}(Sent_{PA}({scriptsize A}) wedge Sent_{PA}({scriptsize B}) rightarrow (T[{scriptsize A} vee{scriptsize B}] leftrightarrow (T{scriptsize A} vee T{scriptsize B}))))
  5. (forall{scriptsize A}(v)(Sent_{PA}(forall v{scriptsize A}) rightarrow(T[forall v{scriptsize A}(v)] leftrightarrow forall xT[{scriptsize A}(underline{x}))]))
  6. (forall{scriptsize A}(v)(Sent_{PA}(forall v{scriptsize A}) rightarrow(T[exists v{scriptsize A}(v)] leftrightarrow exists xT[{scriptsize A}(underline{x}))]))

Axiom 1 says that an atomic sentence of the language of Peanoarithmetic is true if and only if it is true according to thearithmetical truth predicate for this language((Tr_0) was defined in Section3.1). Axioms 2–6 claim that truth commutes with allconnectives and quantifiers. Axiom 5 says that a universallyquantified sentence of the language of arithmetic is true if and onlyif all its numerical instances are true.(Sent_{PA}(forall v{scriptsize A}))says that ({scriptsize A}(v)) is a formula with atmost (v) free (because(forall v{scriptsize A}(v)) is asentence).

If these axioms are to be formulated for a language like set theorythat lacks names for all objects, then axioms 5 and 6 require the useof a satisfaction relation rather than a unary truth predicate.

Axioms in the style of 1–6 above played a central rolein Donald Davidson‘s theory of meaning andin several deflationist approaches to truth.

The theory given by all axioms of PA and Axioms 1–6 but withinduction only for (T)-free formulae is conservative over PA,that is, it doesn’t prove any new (T)-free theorems thatnot already provable in PA. However, not all models of PA can beexpanded to models of PA + axioms 1–6. This follows from aresult due to Lachlan (1981). Kotlarski, Krajewski, and Lachlan (1981)proved the conservativeness very similar to PA + axioms 1–6 bymodel-theoretic means. Although several authors claimed that this result is also finitarily provable, no suchproof was available until Enayat & Visser (2015) and Leigh(2015). Moreover, the theory given by PA + axioms 1–6 is relatively interpretable in PA. However, this result is sensitive to the choice of the base theory: it fails for finitely axiomatized theories (Heck 2015, Nicolai 2016). These proof-theoretic results have been used extensively in the discussion of truth-theoretic deflationism (see Cieśliński 2017).

Of course PA + axioms 1–6 is restrictive insofar as it does notcontain the induction axioms in the language with the truthpredicate. There are various labels for the system that is obtained byadding all induction axioms involving the truth predicate to thesystem PA + axioms 1–6: T(PA), CT, PA(S) or PA + ‘there is afull inductive satisfaction class’. This theory is no longerconservative over its base theory PA. For instance one can formalisethe soundness theorem or global reflection principle for PA, that is,the claim that all sentences provable in PA are true. The globalreflection principle for PA in turn implies the consistency of PA,which is not provable in pure PA byGödel’s Second Incompleteness Theorem. Thus T(PA) is not conservative over PA. T(PA) is muchstronger than the mere consistency statement for PA: T(PA) isequivalent to the second-order system ACA of arithmeticalcomprehension (see Takeuti 1987 and Feferman 1991). More precisely,T(PA) and ACA are intertranslatable in a way that preserves allarithmetical sentences. ACA is given by the axioms of PA with fullinduction in the second-order language and the following comprehensionprinciple:

[exists Xforall y(yin X leftrightarrow phi(x))]

where (phi(x)) is any formula (in which (x) may or maynot be free) that does not contain any second-order quantifiers, butpossibly free second-order variables. In T(PA), quantificationover sets can be defined as quantification over formulas withone free variable and membership as the truth of the formula asapplied to a number.

As the global reflection principle entails formal consistency, the conservativeness result for PA + axioms 1–6 implies that the global reflection principle for Peano arithmetic is not derivable in the typed compositional theory without expanding the induction axioms.In fact, this theory proves neither the statement that all logical validities are true (global reflection for pure first-order logic) nor that all the Peano axioms of arithmetic are true.Perhaps surprisingly, of these two unprovable statements it is the former that is the stronger.The latter can be added as an axiom and the theory remains conservative over PA (Enayat and Visser 2015, Leigh 2015).In contrast, over PA + axioms 1–6, the global reflection principle for first-order logic is equivalent to global reflection for Peano arithmetic (Cieśliński 2010), and these two theories have the same arithmetic consequences as adding the axiom of induction for bounded ((Delta_0)) formulas containing the truth predicate (Wcisło and Łełyk 2017).

The transition from PA to T(PA) can be imagined as an act of reflection on the truth of (mathcal{L})-sentences in PA. Similarly, the step from the typed (T)-sentences to the compositional axioms is also tied to a reflection principle, specifically the uniform reflection principle over the typed uniform (T)-sentences.This is the collection of sentences (forall x, Bew_S (ulcorner phi(underline{x})urcorner) rightarrow phi)(x) where (phi) ranges over formulas in (mathcal{L}_T) with one free variable and S is the theory of the uniform typed T-sentences.Uniform reflection exactly captures the difference between the two theories: the reflection principle is both derivable in T(PA) and suffices to derive the six compositional axioms (Halbach 2001).Moreover, the equivalence extends to iterations of uniform reflection, in that for any ordinal (alpha , 1 + alpha) iterations of uniform reflection over the typed (T)-sentences coincides with T(PA) extended by transfinite induction up to the ordinal (varepsilon_{alpha}), namely the (alpha)-th ordinal with the property that (omega^{alpha} = alpha) (Leigh 2016).

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Much stronger fragments of second-order arithmetic can beinterpreted by type-free truth systems, that is, by theories of truththat prove not only the truth of arithmetical sentences but also thetruth of sentences of the language (mathcal{L}_T) with the truthpredicate; see Section 4 below.

3.4 Hierarchical theories

The above mentioned theories of truth can be iterated byintroducing indexed truth predicates. One adds to the language of PAtruth predicates indexed by ordinals (or ordinal notations) or oneadds a binary truth predicate that applies to ordinal notations andsentences. In this respect the hierarchical approach does not fit theframework outlined in Section 2, because the languagedoes not feature a single unary truth predicate applying to sentencesbut rather many unary truth predicates or a single binary truthpredicate (or even a single unary truth predicate applying to pairs ofordinal notations and sentences).

In such a language an axiomatization of Tarski’s hierarchy of truthpredicates can be formulated. On the proof-theoretic side iteratingtruth theories in the style of T(PA) corresponds to iteratingelementary comprehension, that is, to iterating ACA. The system ofiterated truth theories corresponds to the system of ramified analysis(see Feferman 1991).

Visser (1989) has studied non-wellfoundedhierarchies of languages and axiomatizations thereof. If one addsthe (T)-sentences (T_nulcorner phi urcorner leftrightarrow phi) to the language of arithmetic where (phi) contains onlytruth predicates (T_k)with (kgt n) to PA, a theory isobtained that does not have a standard ((omega)-)model.

4. Type-free truth

The truth predicates in natural languages do not come with anyouvert type restriction. Therefore typed theories of truth (axiomaticas well as semantic theories) have been thought to be inadequate foranalysing the truth predicate of natural language, although recentlyhierarchical theories have been advocated by Glanzberg (forthcoming)and others. This is one motive for investigating type-free theories oftruth, that is, systems of truth that allow one to prove the truth ofsentences involving the truth predicate. Some type-free theories oftruth have much higher expressive power than the typed theories thathave been surveyed in the previous section (at least as long asindexed truth predicates are avoided). Therefore type-free theories oftruth are much more powerful tools in the reduction of other theories(for instance, second-order ones).

4.1 Type-free (T)-sentences

The set ofall (T)-sentences (Tulcorner phi urcorner leftrightarrow phi), where (phi) is any sentence of thelanguage (mathcal{L}_T), thatis, where (phi) may contain (T), is inconsistent with PA (orany theory that proves the diagonal lemma) because ofthe Liar paradox. Therefore one mighttry to drop from the set of all (T)-sentences only those thatlead to an inconsistency. In other words, one may consider maximalconsistent sets of (T)-sentences. McGee (1992) showed thatthere are uncountably many maximal sets of (T)-sentences thatare consistent with PA. So the strategy does not lead to a singletheory. Even worse, given an arithmetical sentence (i.e., a sentencenot containing (T)) that can neither be proved nor disproved inPA, one can find a consistent (T)-sentence that decides thissentence (McGee 1992). This implies that many consistent setsof (T)-sentences prove false arithmetical statements. Thus thestrategy to drop just the (T)-sentences that yield aninconsistency is doomed.

A set of (T)-sentences that does not imply any falsearithmetical statement may be obtained by allowing only those (phi)in (T)-sentences (Tulcorner phi urcorner leftrightarrow phi) that contain (T) only positively, that is, in thescope of an even number of negation symbols. Like the typed theoryin Section 3.2 this theory does not provecertain generalizations but proves the same T-free sentences as thestrong type-free compositional Kripke-Feferman theory below (Halbach2009). Schindler (2015) obtained a deductively very strong truth theory based on stratified disquotational principles.

4.2 Compositionality

Besides the disquotational feature of truth, one would also like tocapture the compositional features of truth and generalize the axiomsof typed compositional truth to the type-free case. To this end,axioms or rules concerning the truth of atomic sentences with thetruth predicate will have to be added and the restrictionto (T)-free sentences in the compositional axioms will have tobe lifted. In order to treat truth like other predicates, one will addthe axiom(forall{scriptsize A}(T[T{scriptsize A}]leftrightarrow T{scriptsize A})) (where(forall{scriptsize A}) ranges over all sentences). If thetype restriction of the typed compositional axiom fornegation is removed, the axiom(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})) is obtained.

However, the axioms(forall{scriptsize A}(T[T{scriptsize A}]leftrightarrow T{scriptsize A})) and(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})) are inconsistent overweak theories of syntax, so one of them has to be given up. If(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})) is retained, one willhave to find weaker axioms or rules for truth iteration, but truthremains a classical concept in the sense that(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})) implies the law ofexcluded middle (for any sentence either the sentence itself or itsnegation is true) and the law of noncontradiction (for no sentence thesentence itself and its negation are true). If, in contrast,(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})) is rejected and(forall{scriptsize A}(T[T{scriptsize A}]leftrightarrow T{scriptsize A})) retained, then it willbecome provable that either some sentences are true together withtheir negations or that for some sentences neither they nor theirnegations are true, and thus systems of non-classical truth areobtained, although the systems themselves are still formulated inclassical logic. In the next two sections we overview the mostprominent system of each kind.

4.3 The Friedman–Sheard theory and revision semantics

The system FS, named after Friedman and Sheard (1987), retains thenegation axiom(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})). The furthercompositional axioms are obtained by lifting the type restriction totheir untyped counterparts:

  1. (forall{scriptsize A}(AtomSent_{PA}({scriptsize A}) rightarrow(T{scriptsize A} leftrightarrow Tr_0 ({scriptsize A}))))
  2. (forall{scriptsize A}(T[neg{scriptsize A}] leftrightarrow neg T{scriptsize A}))
  3. (forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} wedge{scriptsize B}] leftrightarrow(T{scriptsize A} wedge T{scriptsize B})))
  4. (forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} vee{scriptsize B}] leftrightarrow(T{scriptsize A} vee T{scriptsize B})))
  5. (forall{scriptsize A}(v)(Sent(forall v{scriptsize A}) rightarrow(T[forall v{scriptsize A}(v)] leftrightarrow forall xT[{scriptsize A}(underline{x}))])
  6. (forall{scriptsize A}(v)(Sent(forall v{scriptsize A}) rightarrow(T[exists v{scriptsize A}(v)] leftrightarrow exists xT[{scriptsize A}(underline{x}))]))
These axioms are added to PA formulated in thelanguage (mathcal{L}_T). As thetruth iteration axiom(forall{scriptsize A}(T[T{scriptsize A}]leftrightarrow T{scriptsize A})) is inconsistent, only thefollowing two rules are added:

Truth Or Dare Questions

If (phi) is a theorem, one may infer (Tulcorner phi urcorner), and conversely, if (Tulcorner phi urcorner) is a theorem, one may infer (phi).

It follows from results due to McGee (1985) that FS is(omega)-inconsistent, that is, FS proves(exists xneg phi(x)), but proves also (phi)(0),(phi)(1), (phi)(2), … for some formula (phi(x))of (mathcal{L}_T). Thearithmetical theorems of FS, however, are all correct.

In FS one can define all finite levels of the classical Tarskianhierarchy, but FS isn’t strong enough to allow one to recoverany of its transfinite levels. Indeed, Halbach (1994) determined itsproof-theoretic strength to be precisely that of the theory oframified truth for all finite levels (i.e., finitely iterated T(PA);see Section 3.4) or, equivalently, the theory oframified analysis for all finite levels. If either direction of therule is dropped but the other kept, FS retains its proof-theoreticstrength (Sheard 2001).

It is a virtue of FS that it is thoroughly classical: It isformulated in classical logic; if a sentence is provably true in FS,then the sentence itself is provable in FS; and conversely if asentence is provable, then it is also provably true. Its drawback isits (omega)-inconsistency. FS may be seen as an axiomatization ofrule-of-revision semantics for all finite levels (see the entry onthe revision theory of truth).

4.4 The Kripke–Feferman theory

The Kripke–Feferman theory retains the truth iteration axiom(forall{scriptsize A}(T[T{scriptsize A}]leftrightarrow T{scriptsize A})), but the notion of truthaxiomatized is no longer classical because the negation axiom(forall{scriptsize A}(T[neg{scriptsize A}]leftrightarrow neg T{scriptsize A})) is dropped.

The semantical construction captured by this theory is ageneralization of the Tarskian typed inductive definition of truthcaptured by T(PA). In the generalized definition one starts with thetrue atomic sentence of the arithmetical language and then onedeclares true the complex sentences depending on whether itscomponents are true or not. For instance, as in the typed case, if(phi) and (psi) are true, their conjunction (phi wedge psi) will betrue as well. In the case of the quantified sentences their truthvalue is determined by the truth values of their instances (one couldrender the quantifier clauses purely compositional by using asatisfaction predicate); for instance, a universally quantifiedsentence will be declared true if and only if all its instances aretrue. One can now extend this inductive definition of truth to thelanguage (mathcal{L}_T) bydeclaring a sentence of theform (Tulcorner phi urcorner) trueif (phi) is already true. Moreover one will declare(neg Tulcorner phi urcorner) true if(neg phi) is true. By making this idea precise, one obtains a variantof Kripke’s (1975) theory of truth with the so called Strong Kleenevaluation scheme (see the entryon many-valued logic). Ifaxiomatized it leads to the following system, which is known as KF(‘Kripke–Feferman’), of which several variantsappear in the literature:

  1. (forall{scriptsize A}(AtomSent_{PA}({scriptsize A}) rightarrow(T{scriptsize A} leftrightarrow Tr_0 ({scriptsize A}))))
  2. (forall{scriptsize A}(AtomSent_{PA}({scriptsize A}) rightarrow(T[neg{scriptsize A}] leftrightarrow neg Tr_0 ({scriptsize A}))))
  3. (forall{scriptsize A}(T[T{scriptsize A}] leftrightarrow T{scriptsize A}))
  4. (forall{scriptsize A}(T[neg T{scriptsize A}] leftrightarrow T[neg{scriptsize A})])
  5. (forall{scriptsize A}(T[neg neg{scriptsize A}] leftrightarrow T{scriptsize A}))
  6. (forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} wedge{scriptsize B}] leftrightarrow(T{scriptsize A} wedge T{scriptsize B})))
  7. (forall{scriptsize A}forall{scriptsize B}(T[neg({scriptsize A} wedge{scriptsize B})] leftrightarrow (T[neg{scriptsize A}] vee T[neg{scriptsize B})]))
  8. (forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} vee{scriptsize B}] leftrightarrow(T{scriptsize A} vee T{scriptsize B})))
  9. (forall{scriptsize A}forall{scriptsize B}(T[neg({scriptsize A} vee{scriptsize B})] leftrightarrow (T[neg{scriptsize A}] wedge T[neg{scriptsize B})]))
  10. (forall{scriptsize A}(v)(Sent(forall v{scriptsize A}) rightarrow(T[forall v{scriptsize A}(v)] leftrightarrow forall xT[{scriptsize A}(underline{x}))])
  11. (forall{scriptsize A}(v)(Sent(forall v{scriptsize A}) rightarrow(T[neg forall v{scriptsize A}(v)] leftrightarrow exists xT[neg{scriptsize A}(underline{x}))])
  12. (forall{scriptsize A}(v)(Sent(forall v{scriptsize A}) rightarrow(T[exists v{scriptsize A}(v)] leftrightarrow exists xT[{scriptsize A}(underline{x}))]))
  13. (forall{scriptsize A}(v)(Sent(forall v{scriptsize A}) rightarrow(T[neg exists v{scriptsize A}(v)] leftrightarrow forall xT[neg{scriptsize A}(underline{x}))]))

Apart from the truth-theoretic axioms, KF comprises all axioms of PAand all induction axioms involving the truth predicate. The system iscredited to Feferman on the basis of two lectures for the Associationof Symbolic Logic, one in 1979 and the second in 1983, as well as insubsequent manuscripts. Feferman published his version of the systemunder the label Ref(PA) (‘weak reflective closure of PA’)only in 1991, after several other versions of KF had already appearedin print (e.g., Reinhardt 1986, Cantini 1989, who both refer to thisunpublished work by Feferman).

KF itself is formulated in classical logic, but it describes anon-classical notion of truth. For instance, one canprove (Tulcorner Lurcorner leftrightarrow Tulcornerneg Lurcorner)if (L) is the Liar sentence. Thus KF proves that either boththe liar sentence and its negation are true or that neither istrue. So either is the notion of truth paraconsistent (a sentence istrue together with its negation) or paracomplete (neither istrue). Some authors have augmented KF with an axiom ruling outtruth-value gluts, which makes KF sound for Kripke’s modelconstruction, because Kripke had ruled out truth-value gluts.

Feferman (1991) showed that KF is proof-theoretically equivalent tothe theory of ramified analysis through all levels below(varepsilon_0), the limit of the sequence (omega ,omega^{omega},omega^{omega^{ omega} },ldots), or a theory oframified truth through the same ordinals. This result shows that in KFexactly (varepsilon_0) many levels of the classical Tarskianhierarchy in its axiomatized form can be recovered. Thus KF is farstronger than FS, let alone T(PA). Feferman (1991) devised also astrengthening of KF that is as strong as full predicative analysis,that is ramified analysis or truth up to the ordinal(Gamma_0).

Just as with the typed truth predicate, the theory KF (more precisely, a common variant of it) can be obtained via an act of reflection on a system of untyped (T)-sentences. The system of (T)-sentences in question is the extension of the uniform positive untyped (T)-sentences by a primitive falsity predicate, that is, the theory features two unary predicates (T) and (F) and axioms

[begin{align*}&forall x(Tulcorner phi(underline{x})urcorner leftrightarrow phi(x)) & forall x(Fulcorner phi(underline{x})urcorner leftrightarrow phi '(x))end{align*}]

for every formula (phi(v)) positive in both (T) and (F), where (phi ') represents the De Morgan dual of (phi) (exchanging (T) for (F) and vice versa).From an application of uniform reflection over this disquotational theory, the truth axioms for the corresponding two predicate version of KF are derivable (Horsten and Leigh, 2016). The converse also holds, as does the generalisation to finite and transfinite iterations of reflection (Leigh, 2017).

4.5 Capturing the minimal fixed point

As remarked above, if KFproves (Tulcorner phi urcorner) for somesentence (phi) then (phi) holds in all Kripke fixed point models. Inparticular, there are (2^{aleph_0}) fixedpoints that form a model of the internal theory of KF. Thus from theperspective of KF, the least fixed point (from which Kripke’s theoryis defined) is not singled out. Burgess (forthcoming) provides anexpansion of KF, named (mu)KF, that attempts to capture the minimalKripkean fixed point. KF is expanded by additional axioms that expressthat the internal theory of KF is the smallest class closed under thedefining axioms for Kripkean truth. This can be formulated as a singleaxiom schema that states, for each open formula (phi),

If (phi) satisfies the same axioms of KF as the predicate (T) then (phi) holds of every true sentence.

From a proof-theoretic perspective (mu)KF is significantly strongerthan KF. The single axiom schema expressing the minimality of thetruth predicate allows one to embed into (mu)KF the systemID(_1) of one arithmetical inductive definition, animpredicative theory. While intuitively plausible, (mu)KF suffers thesame expressive incompleteness as KF: Since the minimal Kripkean fixedpoint forms a complete (Pi^{1}_1) set and theinternal theory of (mu)KF remains recursively enumerable, there arestandard models of the theory in which the interpretation of the truthpredicate is not actually the minimal fixed point. At present therelacks a thorough analysis of the models of (mu)KF.

4.6 Axiomatisations of Kripke’s theory with supervaluations

KF is intended to be an axiomatization of Kripke’s (1975)semantical theory. This theory is based on partial logic with theStrong Kleene evaluation scheme. In Strong Kleene logic not everysentence (phi vee neg phi) is a theorem; in particular, thisdisjunction is not true if (phi) lacks a truthvalue. Consequently (Tulcorner Lvee neg Lurcorner)(where (L) is the Liar sentence) is not a theorem of KF and itsnegation is even provable. Cantini (1990) has proposed a system VFthat is inspired by the supervaluations scheme. In VF all classicaltautologies are provably trueand (Tulcorner L vee neg Lurcorner), for instance, is a theorem ofVF. VF can be formulatedin (mathcal{L}_T) and usesclassical logic. It is no longer a compositional theory oftruth, for the following is not a theorem of VF:

[forall{scriptsize A}forall{scriptsize B}(T[{scriptsize A} vee{scriptsize B}] leftrightarrow(T{scriptsize A} vee T{scriptsize B})).]

Not only is this principle inconsistent with the other axioms ofVF, it does not fit the supervaluationist model for itimplies (Tulcorner Lurcorner vee Tulcorner neg Lurcorner),which of course is not correct because according to the intendedsemantics neither the liar sentence nor its negation is true: bothlack a truth value.

Extending a result due to Friedman and Sheard (1987), Cantinishowed that VF is much stronger than KF: VF is proof-theoreticallyequivalent to the theory ID(_1) of non-iterated inductivedefinitions, which is not predicative.

5. Non-classical approaches to self-reference

The theories of truth discussed thus far are all axiomatized inclassical logic. Some authors have also looked into axiomatic theoriesof truth based on non-classical logic (see, for example, Field 2008,Halbach and Horsten 2006, Leigh and Rathjen 2012). There are a numberof reasons why a logic weaker than classical logic may bepreferred. The most obvious is that by weakening the logic, somecollections of axioms of truth that were previously inconsistentbecome consistent. Another common reason is that the axiomatic theoryin question intends to capture a particular non-classical semantics oftruth, for which a classical background theory may prove unsound.

There is also a large number of approaches that employ paraconsistent or substructural logics. In most cases these approaches do not employ an axiomatic base theory such as Peano arithmetic and therefore deviate form the setting considered here, although there is no technical obstacle in applying paraconsistent or substructural logics to truth theories over such base theories. Here we cover only accounts that are close to the setting considered above. For further information on the application of substructural and paraconsistent logics to the truth-theoretic paradoxes see the relevant section in the entry on the liar paradox.

5.1 The truth predicate in intuitionistic logic

The inconsistency of the (T)-sentences does not rely onclassical reasoning. It is also inconsistent over much weaker logicssuch as minimal logic and partial logic. However, classical logic doesplay a role in restricting the free use of principles of truth. Forinstance, over a classical base theory, the compositional axiom forimplication ((rightarrow)) is equivalent to the principle of completeness,(forall{scriptsize A}(T[{scriptsize A}] vee T[neg{scriptsize A})]). If the logic underthe truth predicate is classical, completeness is equivalent to the compositional axiom for disjunction. Without the law ofexcluded middle, FS can be formulated as a fully compositional theorywhile not proving the truth-completeness principle (Leigh & Rathjen2012). In addition, classical logic has an effect on attempts tocombine compositional and self-applicable axioms of truth. If, forexample, one drops the axiom of truth-consistency from FS (theleft-to-right direction of axiom 2 in Section 4.3)as well as the law of excluded middle for the truth predicate, it ispossible to add consistently the truth-iteration axiom(forall{scriptsize A}(T[{scriptsize A}]rightarrow T[T{scriptsize A}])).The resulting theorystill bears a strong resemblance to FS in that the constructiveversion of the rule-of-revision semantics for all finite levelsprovides a natural model of the theory, and the two theories share the same (Pi^{0}_2) consequences (Leigh & Rathjen 2012; Leigh, 2013). This result should be contrasted with KF which, if formulatedwithout the law of excluded middle, remains maximally consistent withrespect to its choice of truth axioms but is a conservative extension ofHeyting arithmetic.

5.2 Axiomatising Kripke’s theory

Kripke’s (1975) theory in its different guises is based on partiallogic. In order to obtain models for a theory in classical logic, theextension of the truth predicate in the partial model is used again asthe extension of truth in the classical model. In the classical modelfalse sentences and those without a truth value in the partial modelare declared not true. KF is sound with respect to these classicalmodels and thus incorporates two distinct logics. The first is the‘internal’ logic of statements under the truth predicateand is formulated with the Strong Kleene valuation schema. The secondis the ‘external’ logic which is full classical logic. Aneffect of formulating KF in classical logic is that the theory cannotbe consistently closed under the truth-introduction rule

If (phi) is a theorem of KF, so is (Tulcorner phi urcorner).

A second effect of classical logic is the statement of the excludedmiddle for the liar sentence. Neither the Liar sentence nor itsnegation obtains a truth value in Kripke’s theory, so the disjunctionof the two is not valid. The upshot is that KF, if viewed as anaxiomatisation of Kripke’s theory, is not sound with respect to itsintended semantics. For this reason Halbach and Horsten (2006) andHorsten (2011) explore an axiomatization of Kripke’s theory withpartial logic as inner and outer logic. Their suggestion, atheory labelled PKF (‘partial KF’), can be axiomatised asa Gentzen-style two-sided sequent calculus based on Strong Kleenelogic (see the entry on many-valuedlogic). PKF is formed by adding to this calculus thePeano–Dedekind axioms of arithmetic including full induction andthe compositional and truth-iteration rules for the truth predicate asproscribed by Kripke’s theory. The result is a theory of truth that issound with respect to Kripke’s theory.

Halbach and Horsten show that this axiomatization of Kripke’stheory is significantly weaker than it’s classical cousin KF. Theresult demonstrates that restricting logic only for sentences with thetruth predicate can hamper also the derivation of truth-free theorems.

5.3 Adding a conditional

Field (2008) and others criticised theories based on partial logicfor the absence of a ‘proper’ conditional andbi-conditional. Various authors have proposed conditionals andbi-conditionals that are not definable in terms of (neg , vee) and(wedge). Field (2008) aims at an axiomatic theory of truth notdissimilar to PKF but with a new conditional. Feferman (1984) alsointroduced a bi-conditional to a theory in non-classical logic. UnlikeField’s and his own 1984 theory, Feferman’s (2008) theory DT isformulated in classical logic, but it’s internal logic is again apartial logic with a strong conditional.

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