Formal Theories of Truth

INTRODUCTION

Volker Halbach

New College, Oxford

Trinity 2015

Why formal theories of truth?

Solutions to the liar paradox and other paradoxesTruth and consequence: What can one actually do with atruth predicate conceived, e.g., as a device of disquotation?In particular, is truth a substantial notion that allows onenew insights into non-semantic issues?Is truth always founded in non-semantic facts?Reduction: parts of mathematics can be reduced to a theoryof truth; the proof-theoretic strength of truth theoriesTruth and reflection: truth as a device for stating verystrong reflection principlesTruth in other areas of philosophy, e.g., in the definition ofknowledge

Philosophical decisions

Truth may be taken to be applying to any kind of objects(sentences as types or tokens, propositions. . . ) as long as theseobjects have the structure of sentence types.

I will take truth to apply to sentences of a fixed formallanguage only, but not to ‘foreign’ sentences.

I’ll emphasize the axiomatic approach – at least in thebeginning. This is not to say that truth cannot be defined interms of correspondence etc.

Formal Theories of Truth I

THE THEORY OF SYNTAX ANDDIAGONALISATION

Volker Halbach

New College, Oxford

Trinity 2015

There is something fishy about the liar paradox:

(1) (1) is not true.

Somehow the sentence ‘says’ something about itself, and whenpeople are confronted with the paradox for the first time, theyusually think that this feature is the source of the paradox.

Self-reference

However, there are many self-referential sentence that arecompletely unproblematic:

(2) (2) contains 5 occurences of the letter ‘c’.

If (1) is illegitimate because of its self-referentiality, then (2)must be illegitimate as well. Moreover, the effect that isachieved via the label ‘(1)’ can be achieved without this device.At the same time one can dispense with demonstratives like‘this’ that might be used to formulate the liar sentence:

This sentence is not true.

In fact, the effect can be achieved using weak arithmeticalaxioms only. And the axioms employed are beyond any(serious) doubt. This was shown by Gödel.

Arithmetic

The approach via arithmetic is indirect. Arithmetic talks aboutnumbers, not about sentences. Coding sentences andexpressions by numbers allows to talk about the numericalcodes of sentences and therefore arithmetic is indirectly aboutsentences.

My approach here avoids this detour via numbers. I present atheory of expressions that is given by some (as I hope) obviousaxioms on expressions. The trick (diagonalization) that is thenused for obtaining a self-referential sentence is the same as inthe case of arithmetic.

Self-reference

Self-reference might look like a pathological phenomenoncausing work for philosophical logicians only, which bears littleinterest elsewhere. Self-reference, however, has not only‘detrimental’ applications as in the liar paradox, but alsomathematical useful applications as in the proof of theRecursion Theorem (see, e.g., [39]).

Therefore most logicians agree nowadays that it is notself-reference that causes trouble in (1), but rather the truthpredicate. The initial impression was wrong: not self-referenceis to be blamed for the paradox but rather our concept of truth.

The alphabet

In the following I describe a language L. An expression of L isan arbitrary finite string of the following symbols. Such stringsare also called expressions of L.

DefinitionThe symbols of L are:

1 infinitely many variable symbols v, v1, v2, v3,. . .2 predicate symbols = and T ,3 function symbols q,a and sub,4 the connectives ¬, → and the quantifier symbol ∀,5 auxiliary symbols ( and ),6 possibly finitely many further function and predicate

symbols, and7 If e is a string of symbols then e is also a symbol. e is

called a quotation constant.All the mentioned symbols are pairwise different.

Notational conventions

In the following I shall use x, y and z as (meta-)variables forvariables. Thus x may stand for any symbol v, v1, v2, . . . It isalso assumed that x, y etc stand for different variables.Moreover, it is always presupposed variable clashes are avoidedby renaming variables in a suitable way.

It is important that a is a single symbol and not a string of morethan one symbols even if a itself is a string built from severalsymbols.

A string of symbols of L is any string of the above symbols.Usually I suppress mention of L. The empty string is also astring.

Terms

We shall now define the notions of a term and of a formula ofL.

DefinitionThe L-terms are defined as follows:

1 All variables are terms.2 If e is a string of symbols, then e is a term.3 If t, r and s are terms, then q(t), (sat), sub(r, s, t) are terms,

and similarly for all further function symbols

The empty string

Since the empty string is a string of symbols is a term. Sincelooks so odd, I shall write 0 for . From an ontologically pointof view the empty string is a weird thing. One might beinclined to say that it is not anything. I have only a pragmaticexcuse for assuming the empty string: it is useful, though notindispensable.

What the empty string is for the expressions is the number zerofor the natural numbers. It is not hard to see that 0 is useful innumber theory.

Formulæ

Formulæ, sentences, free and bound occurrences of variablesare defined in the usual way.

Example

1 ∀v3(v3 = ∧∀ ∧ Tv3) is a sentence.2 v12 = ¬T¬ is a sentence, i.e., the formula does not feature a

free variable.

Non-linearity

RemarkThe reason for overlining expressions e rather than to includethem in quotation marks like this peq is not a stylistic one.pvqpvq may be parsed in the following two ways:

︷︸︸︷pvq

︷︸︸︷pvq︸ ︷︷ ︸

On the first reading indicated by the parentheses above, theexpression if of the form sat; on the second reading it is anatomic term p. . .q. If quotations are marked by overlining thequoted strings, this ambiguity does not arise.

A theory of expressions

The theory A which will be described in this section isdesigned in order to obtain smooth proofs in the following. Ihave not aimed at a particularly elegant axiomatization.

A simple intended model of the theory has all expressions of Las its domain. The intended interpretation of the functionsymbols will become clear from the axioms A1–A4 except forthe interpretation of sub. I shall return to sub below.

The axioms

All instances of the following schemata and rules are axioms ofthe theory A:

Definition

A1 all axioms and rules of first-order predicate logic includingthe identity axioms.

A2 aab = ab, where a and b are arbitrary strings of symbols.

A3 q(a) = a

A4 sub(a, b, c) = d, where a and c are arbitrary strings ofsymbols, b is a single symbol (or, equivalently, a string ofsymbols of length 1), and d is the string of symbolsobtained from a by replacing all occurrences of the symbolb by the strings c.

The axioms

Definition (additional axioms)

A5 ∀x∀y∀z((xay)az) = (xa(yaz))

A6 ∀x∀y(xay = 0→ x = 0 ∧ y = 0)

A7 ∀x∀y(xay = x ↔ y = 0) ∧ ∀x∀y(yax = x ↔ y = 0)

A8 ∀x1∀x2∀y∀z(sub(x1, y, z)asub(x2, y, z)) = sub(x1ax2, y, z)

A9 ¬a = b, if a and b are distinct expressions.

Comments

A1-A4 describe the functions of concatenation, quotation andsubstitution by providing function values for specific entries.From these axioms one cannot derive (non-trivial) universallyquantified principles and therefore axioms like the associativelaw fora A5 are not derivable from A1–A4.

Comments

The concatenation of two expressions e1 and e2 is simply theexpression e1 followed by e2. For instance, ¬¬v is theconcatenation of ¬ and ¬v.

Therefore ¬¬v = ¬a¬v is an instance of A2 as well as¬¬v = ¬¬av.

Concatenating the empty string with any expression e givesagain the same expression e. Therefore we have, for instance,∀a0 = ∀ as an instance of A2.

Comments

An instance of A3 is the sentence qv¬ = v¬. Thus q describesthe function that takes an expression and returns its quotationconstant.

Comments

In A4 I have imposed the restriction that b must be a singlesymbol. This does not imply that the substitution functioncannot be applied to complex expressions; just A4 does not sayanything about the result of substituting a complex expression.

The reason for this restriction is that the result of substitution ofa complex strings may be not unique. For instance, the result ofsubstituting ¬ for ∧∧ in ∧ ∧ ∧ might be either ∧¬ or ¬∧. Theproblem can be fixed in several ways, but I do not need tosubstitute complex expressions in the following. Therefore I donot ‘solve’ the problem but avoid it by the restriction of b to asingle symbol.

Comments

A1-A4 are already sufficient for proving the diagonalizationTheorem 13.

A5 simplifies the reasoning with strings a great deal. SinceA ` (xay)az = xa(yaz), that is,a is associative by A5, I shallsimply write xayaz. for the sake of definiteness we canstipulate that xayaz is short for (xay)az and similarly formore applications ofa.

Comments

A6–A8 will only be invoked in the section on arithmetic.

A6 tells us that is either a or b is not the empty expression thenthe concatenation of a and b is non-empty as well.

A7 postulates that only the empty string does not change anobject if it is concatenated with this object.

Comments

In the following I shall need only the following instantiation ofA8:

∀x1∀x2∀z(sub(x1,v, z)asub(x2,v, z)) = sub(x1ax2,v, z)

I just thought that the universally quantified version looks lessad hoc.

The variable y is not restricted to single symbols, mostly. So A8claims that if the problem of the uniqueness of the result ofsubstituting complex expressions is solved, then the followingholds: The result of concatenating the result of substituting cfor b in a1 with the result of substituting c for b in a2 is thesame as the result of substituting c for b in a1

aa2.

Comments

I write A ` ϕ if and only if the formula ϕ is a logicalconsequence of the theory A.

Example

A ` sub(¬¬,¬,¬¬¬) = ¬¬¬¬¬¬

Example

A ` sub(v = v ∧ v = v,v,v2) = v2 = v2 ∧ v = v

Comments

These axioms suffice for proving Gödel’s celebrateddiagonalization lemma.

RemarkOf course, there is no such cheap way to Gödel’s theorems.Gödel showed that the functions sub and q (and furtheroperations) can be defined in an arithmetical theory fornumerical codes of expressions. To this end he proved that allrecursive functions can be represented in a fixed arihmeticalsystem. And then he proved that the operation of substitutionetc. are recursive. This requires some work and ideas.

Diagonalization

The diagonalization function dia is defined in the followingway:

Definitiondia(x) = sub(x,v,q(x))

RemarkThere are at least two ways to understand the syntactical statusof dia. It may be considered an additional unary functionof L,and the above equation is then an additional axiom of A.Alternatively, one can conceive dia as a metalinguisticabbreviation, which does not form part of the language L, butwhich is just short notation for a more complex expression. Thissituation will encountered in the following frequently.

A lemma

LemmaAssume ϕ(v) is a formula not containing bound occurrences of v.Then the following holds:

A ` dia(ϕ(dia(v))) = ϕ(dia(ϕ(dia(v))))

Proof.In A the following equations can be proved::

dia(ϕ(dia(v))) = sub(ϕ(dia(v)),v,q(ϕ(dia(v))))

= sub(ϕ(dia(v)),v, ϕ(dia(v)))

= ϕ(dia(ϕ(dia(v))))

a

The diagonal lemma

Theorem (diagonalization)

If ϕ(v) is a formula of L with no bound occurrences of v, then onecan find a formula γ such that the following holds:

A ` γ ↔ ϕ(γ)

Proof.

Choose as γ the formula ϕ(dia(ϕ(dia(v))). Then one has by theprevious Lemma:

A ` ϕ(dia(ϕ(dia(v)))︸ ︷︷ ︸γ

↔ ϕ(ϕ(dia(ϕ(dia(v))))︸ ︷︷ ︸γ

)

a

Formal Theories of Truth II

THE T-SENTENCES

Volker Halbach

New College, Oxford

Trinity 2015

Inconsistency

I shall prove the inconsistency of some theories with the theoryA. ‘inconsistent’ always means ‘inconsistent with A’.

Since I did not fix the axioms of A and admitted further axiomsin A, inconsistency results can be formulated in two ways. Onecan either say ‘A is inconsistent if it contains the sentence ψ’ orone says ‘ψ is inconsistent with A’.

The T-scheme

The first inconsistency result is the famous liar paradox. It isplausible to assume that a truth predicate T for the language Lsatisfies the T-scheme

(3) Tψ ↔ ψ

for all sentences ψ of L. This scheme corresponds to the scheme

‘A’ is true if and only if A,

where A is any English declarative sentence.

The liar in A

Theorem (liar paradox)

The T-scheme Tψ ↔ ψ for all sentences ψ of L is inconsistent.

Proof.Apply the diagonalization theorem 13 to the formula ¬Tv. Thentheorem 13 implies the existence of a sentence γ such that thefollowing holds: A ` γ ↔ ¬Tγ . Together with the instanceTγ ↔ γ of the T-scheme this yields an inconsistency. γ is calledthe ‘liar sentence’. a

Tarski’s theorem

Since the scheme is inconsistent such a truth predicate cannotbe defined in A, unless A itself is inconsistent.

Corollary (Tarski’s theorem on the undefinability of truth)

There is no formula τ(v) such that τ(ψ)↔ ψ can be derived in A forall sentences ψ of L, if A is consistent.

Proof.Apply the diagonalization theorem 13 to τ(v) as above. If τ(v)contains bound occurrences of v they can be renamed such thatthere are no bound occurrences of v. a

The scope of Tarski’s theorem

It is not so much surprising that the axioms listed explicitly inDefinition 5 do not allow for a definition of such truth predciateτ(v). According to Definition 5, however, A may containarbitrary additional axioms. Thus Tarski’s Theorem says thatadding axioms to A that allow for a truth definition renders Ainconsistent.

Extending the language

Nevertheless one can add a new predicate symbol which is notin L, and add True ψ ↔ ψ as an axiom scheme for all sentencesof L. In this case ϕ cannot contain the symbol True and thediagonalization theorem 13 does not apply to True v because itapplies only to formulæ ϕ(v) of L.

TheoremAssume that the language L is expanded by a new predicate symbolTrue and all sentences True ψ ↔ ψ (for ψ a sentence of L) are addedto A. The resulting theory is consistent if A is consistent.

The theory of disquotation

Call the theory A plus all these equivalences TB. Thus TB isgiven by the following set of axioms:

A ∪ {True ψ ↔ ψ : ψ a sentence of L}

The proof

The idea for the proof is due to Tarski.

I shall show that a given proof of a contradiction ⊥ in the theoryTB can be transformed into a proof of ⊥ in A. In the givenproof only finitely many axioms with True can occur; let

True ψ0 ↔ ψ0,True ψ1 ↔ ψ1, . . .True ψn ↔ ψn

be these axioms. τ(v) is the following formula of the languageL:

(v = ψ0 ∧ ψ0) ∨ (v = ψ1 ∧ ψ1) ∨ . . . (v = ψn ∧ ψn)Obviously one has

τ(ψ0)↔ ψ0

and similarly for all ψk (k ≤ n).

Now replace everywhere in the given proof any formula True t,where t is any arbitrary term, by τ(t) and add above any formeraxiom True ψk ↔ ψk a proof of τ(ψk)↔ ψk, respectively. Theresulting structure is a proof in A of the contradiction ⊥.

Conservativity

The proof establishes a stronger result: Adding the T-sentences

True ψ ↔ ψ

(ψ a sentence without True ) to A yields a conservativeextension of A:

TheoremTB is conservative over A. That is, If ϕ is a sentence without Truethat is provable in TB, then ϕ is already provable in A only.

Proof.Just replace ⊥ by ϕ in the proof above. a

The proof shows that these T-sentences do not allow to proveany new ‘substantial’ insights. Works also with full induction.

Conservativity over logic

The T-sentences are not conservative over pure logic. TheT-sentences prove that there are at least two different objects:

True ∀ = ∀ ↔ ∀ = ∀ T-sentence

∀ = ∀ tautology

True ∀ = ∀ two preceding lines

True ¬∀ = ∀ ↔ ¬∀ = ∀ T-sentence

¬True ¬∀ = ∀

∀ = ∀ 6= ¬∀ = ∀

Tarski on TB

Tarski [43, p. 256] proves the consistency result for TB, and oneshould expect that the resulting theory is attractive because itsatisfies Convention T, but Tarski says:

The value of the result obtained is considerably diminished by the factthat the axioms mentioned in Th. III have a very restricted deductivepower. A theory of truth founded on them would be a highlyincomplete system, which would lack the most important and mostfruitful general theorems. Let us show this in more detail by a concreteexample.

Tarski on TB

Consider the sentential function ‘x∈Tr or x∈Tr’. [“∈ Tr” is thetruth predicate, “∈Tr” the negated truth predicate; “x” designates thenegation of “x”.] If in this function we substitute for the variable ‘x’structural-descriptive names of sentences, we obtain an infinitenumber of theorems, the proof of which on the basis of the axiomsobtained from the convention T presents not the slightest difficulty.

Tarski on TB

But the situation changes fundamentally as soon as we pass to thegeneralization of this sentential function, i.e. to the general principleof contradiction. From the intuitive standpoint the truth of all thosetheorems is itself already a proof of the general principle; this principlerepresents, so to speak, an ‘infinite logical product’ of those specialtheorems. But this does not at all mean that we can actually derive theprinciple of contradiction from the axioms or theorems mentioned bymeans of the normal modes of inference usually employed. On thecontrary, by a slight modification of Th. III it can be shown that theprinciple of contradiction is not a consequence (at least in the existingsense of the word) of the axiom system described.

Tarski on TB

It seems Tarski admits here that Convention T is highlyincomplete: a good theory of truth should not only yield theT-sentences, it should also yield the general principle ofcontradiction ‘For any sentence (without True ) either thesentence or its negation is true.’

Perhaps Tarski thought that definitions of truth always yield thegeneralisations, if they satisfy Convention T. But TB triviallyprovides a definition of truth satisfying Convention T: onedefines truth as True .

Later serious examples were proved: BG defines truth for thelanguage of ZF. The truth definition satisfies Convention T, butit does not yield the generalisation Tarski expects from a gooddefinition of truth.

More notation

In order to prove the result that TB doesn’t prove thosegeneralisations, I need more axioms.

Sent(x) is a unary predicate, ¬. a unary function symbol. Iassume that Sent(x) represents the property of being a sentenceof L, ¬. represents the function that takes a sentence and returnsits negation:

Additional AxiomA ` Sent(ϕ) iff ϕ is a sentence of L.

Additional AxiomA ` ¬.ϕ = ¬ϕ

No generalisations

I can now formulate and proof Tarski’s (and Gupta’s)complaint:

TheoremTB 6` ∀x(Sent(x)→ (True x ∨ True ¬.x)) (assuming that A isconsistent).

Proof

Assume otherwise. Then there is a proof of∀x(Sent(x)→ (True x ∨ True ¬.x)) in from a finite subtheory Sof TB. Only finitely many T-sentences can be in S. Let

True ψ0 ↔ ψ0,True ψ1 ↔ ψ1, . . . ,True ψn ↔ ψn

be these T-sentences. τ(v) is the following formula of thelanguage L:((v = ψ0∧ψ0)∨(v = ψ1∧ψ1)∨. . . (v = ψn∧ψn)

)∧(v = ψ1∧. . .∧v = ψn)

As above, True v can be interpreted as τ(v).

If χ is none of the ψ0, . . . , ψn, we have A ` ¬τ(χ) ∧ ¬τ(¬.χ).

Finite axiomatisability

Davidson made a big fuss about the following observation.

TheoremTB is not finitely axiomatisiable over A.

That is, there is no sentence σ such that A + σ proves the samesentences as TB.

Let a proof of σ in TB be given. Replace the truth predicateagain in that proof by the partial truth predicate(

(v = ψ0 ∧ ψ0) ∨ (v = ψ1 ∧ ψ1) ∨ . . . (v = ψn ∧ ψn))

This shows that the truth predicate of A + σ is definable in Aalready. But by assumption A + σ proves all T-sentences, and byTarski’s theorem this truth predicate cannot be A-definable.

Summary

Adding a new truth predicate to A and axiomatising it bytyped T-sentences yields a conservative extension of A.The resulting theory TB does not prove generalisation suchas

∀x(Sent(x)→ (True x ∨ True ¬.x)) or∀x∀y

(Sent(x) ∧ Sent(y)→ (True (x∧. y)↔ (True x ∧ True y))

)

TB is not finitely axiomatisable.According to Tarski, a decent theory of truth should notonly yield the T-sentences (and satisfy Convention T), butalso prove those generalisations.

Philosophical moral

The truth predicate of TB may have its merits: it allows one toaxiomatise certain generalisations finitely (Horwich [21]Halbach [15]). But it doesn’t prove the generalisations Tarskiexpected from a decent theory of truth.

Moreover, TB has been criticised, because theobject-/metalanguage distinction seems to restrictive.

Weakening the T-sentences

Forget about the ‘new’ truth predicate True . I return to our oldlanguage L again, which contains already a truth predicate T .

The next theorem is a strengthening of Theorem 14. For notionslike necessity one will not postulate axioms of the form Tψ ↔ ψbut only that. However, one would expect a rule ofnecessitation to hold: once ψ has been derived, one mayconclude Tψ. With these weakened assumptions on T the liarparadox can still be derived.

Montague’s paradox

Theorem (Montague’s paradox [36])

The schema Tψ → ψ is inconsistent with the rule ψTψ

.

The rule ψTψ

is called NEC in the following.

Proof.

γ ↔ ¬Tγ diagonalizationTγ ↔ ¬γTγ → γ axiom¬Tγ logicγ first lineTγ NEC

a

Another formulation of the T-sentences

Often the T-sentences ar stated in the following way:

Tψ ↔ ψ

where ψ must not contain T . It’s thought that this is safe. But Idon’t trust that formulation anymore.

How not to state the T-sentences

TheoremAssume L contains a unary predicate symbol N (for necessity of somekind, let’s say), and assume further:

T Tϕ↔ ϕ for all sentences ϕ of L not containing T .N1 Nϕ→ ϕ for all sentences ϕ of L not containing N.N2 Whenever A ` ϕ, then also A `Nϕ for all sentences ϕ of L not

containing N.Then A is inconsistent.

Proof

γ ↔ ¬TNγ diagonalisation

TNγ ↔ ¬γNγ → ¬γNγ → γ (N1)¬Nγ two previous lines

¬TNγ Tγ first and last lineNγ (N2)

How not to state the T-sentences

Usually it is thought that typing is a remedy to the paradoxes.The example shows that this works only as long as typing is notapplied to more than one predicate.

The result is the first of various paradoxes (vulgoinconsistencies) that arise from the interaction of predicates.

Summary

Adding a new truth predicate to A and axiomatising it bytyped T-sentences yields a conservative extension of A.The resulting theory TB does not prove generalisation suchas

∀x(Sent(x)→ (True x ∨ True ¬.x)) or∀x∀y

(Sent(x) ∧ Sent(y)→ (True (x∧. y)↔ (True x ∧ True y))

)

TB is not finitely axiomatisable.According to Tarski, a decent theory of truth should notonly yield the T-sentences (and satisfy Convention T), butalso prove those generalisations.‘Mixing’ the T-sentences with axiomatisations of othernotions such as necessity can lead to inconsistencies. Sotype restrictions don’t solve all problems.

Liberalising the type restriction

There have been various proposals to lift the type restrictions onthe T-sentences.

Motives:Eg the following T-sentence looks ok:

‘‘Grass is red’ is not true’ is true iff ‘Grass is red’ isnot true.

A more liberal approach might help to regain deductivepower.

However, one seems to be caught between Scylla andCharybdis: the typed truth predicate of TB is too weak, whilethe full unrestricted T-schema is too strong.

It seems reasonable to steer between the two extremes in themiddle. . .

But there are other creatures as horrifying as deductiveweakness and inconsistency, as McGee [34] has demonstrated.

Horwich’s proposal

[. . . ] we must conclude that permissible instantiations of theequivalence schema are restricted in some way so as to avoidparadoxical results. [. . . ] Given our purposes it suffices for us toconcede that certain instances of the equivalence schema are not to beincluded as axioms of the minimal theory, and to note that theprinciples governing our selection of excluded instances are, in orderof priority: (a) that the minimal theory not engender ‘liar-type’contradictions; (b) that the set of excluded instances be as small aspossible; and—perhaps just as important as (b)—(c) that there be aconstructive specification of the excluded instances that is as simple aspossible. Horwich 1990 p. 41f

Maximal consistent instances of schema T

So the aim is to find a set of sentences Tϕ↔ ϕ such thatThe set is consistent.The set is maximal, ie no further sentences of the formTϕ↔ ϕ can be consistently added over A.The set is recursively enumerable (?).

Maximal consistent instances of schema T

Theorem (McGee)

Let ϕ be some sentence, then there is a sentence γ such that

A ` ϕ↔ (Tγ ↔ γ)

Proof.

A `γ ↔ (Tγ ↔ ϕ) diagonalisationA `ϕ↔ (Tγ ↔ γ) propositional logic

a

Maximal consistent instances of schema T

McGee’s observation spells disaster for Horwich’s proposal.

Theorem (McGee)

If a consistent set of T-sentences is recursive, it’s not maximal: byGödel’s first incompleteness theorem there will be an undecidablesentence ϕ, which is equivalent to a T-sentence.Maximal sets are too complicated. They can’t be Π0

1 or Σ01.

There are many, in fact uncountably many different maximalconsistent sets of T-sentences (if A is consistent).Consistent sets of T-sentences can prove horrible results worsethan any inconsistency.

Strong instances of schema T

McGee’s observation has its destructive uses, but it also has aneglected constructive side.

It’s often assumed that an axiomatisation of truth byT-sentences is either inacceptably weak or inconsistent. McGee’stheorem shows that this view is incorrect.

Assume you have a favourite axiomatisation of truth (say the KFaxioms or the like). Let χ the conjunction of these axioms. ThenMcGee’s theorem implies the existence of a T-sentence such that

A ` χ↔ (True γ ↔ γ)

Thus Davidson’s theory, KF and so on can be finitelyaxiomatised by a single T-sentence.

The problem remains to tell a story why one should acceptTγ ↔ γ . If one justifies the acceptance of that T-sentence byappeal to your favourite theory, we have given updisquotationalism.

Strong instances of schema T

Gut feeling

Tarski’s way of blocking the paradoxes is less damaging to the‘inductive’ definition of truth than to the T-sentences as axioms.The T-sentences are as good as any axiomatic theory of truth, ifthe paradoxes are blocked in an appropriate way.We need to come up with a better method for sorting the goodinstances from the bad instances of schema T.

To me it’s still unclear whether it might be possible to defend atheory based on T-sentence which is not deductively weak.

Missing: maximal conservative sets of instances of schema TCieslinski [6].‘Uniform’ T-sentences and positive T-sentences. Idon’t have the tools available for treating them now. But thereare well motivated and stron theories based on T-sentences.

Formal Theories of Truth IV

MORE BEASTS AND DRAGONS

Volker Halbach

New College, Oxford

Trinity 2015

How things can go wrong

Paradox is not the same as mere inconsistency: there are manyways things can go wrong:

The theory is inconsistent.The theory cannot be combined with another plausibletheory. If a theory of future cannot be combined with theanalogous theory of past truth, something is wrong.The theory is internally inconsistent: the theory proves thateverything is true.The theory proves a false claim in the base language (ie inthe language without the truth predicate).The theory has trivial models, eg, truth can be interpretedby the empty set.The theory is ω-inconsistent.

Generally, consistency proofs are good, but a fullproof-theoretic analysis is better. Only such an analysis canprove that the theory doesn’t contain any hidden paradoxes.

The constructive applications

On the next couple of slides I sketch some classical applicationsof diagonalisation.

Many of them can be turned into ‘paradoxes’.

Gödel’s first theorem

Ok, it isn’t Gödel’s incompleteness theorem, but it’s verysimilar in structure:

Theorem (Gödel’s first theorem)

Assume A ` ϕ if and only if A ` Tϕ holds for all sentences. Thenthere is a sentence γ , such that neither γ itself nor its negation isderivable in A except that A itself is already inconsistent.

Proof

A `γ ↔ ¬Tγ diagonalisation(4)A `γ assumption(5)A `Tγ NEC(6)A `¬Tγ (4)(7)A `¬γ assumption(8)A `Tγ (4)(9)A `γ CONEC(10)

The liar again

TheoremAssume A ` ϕ if and only if A ` Tϕ holds for all sentences. Then theliar sentence is undecidable in A, if A is consistent.

Thus if the T-schema is weakened to a rule, the liar sentencemust be undecidable. Thus theories (such as KF) containingNEC and deciding the liar sentence, cannot have CONEC.

The real incompleteness theorem

Gödel showed that a provability predicate Bew(v) can bedefined in a certain system of arithmetic corresponding to ourtheory A. more precisely, he defined a formula Bew(v)

A ` ψ if and only if A ` Bew(ψ)

holds for all formulæ ψ of L if A is ω-consistent. ω-consistencyis a stronger condition than pure consistency.

A look at the second incompleteness theorem

The ‘modal’ reasoning leading to the second incompletenesstheorem can be paraphrased in A.

The second incompleteness theorem and Löb’s theorem havebeen used to derive further paradoxes. I believe that mostparadoxes involving self-reference can be reduced to Löb’stheorem.

In particular, the incompleteness theorems yield moreinformation on weakenings of the T-scheme and ways to blockMontague’s paradox.

Weaker reflection

Theorem

The scheme TTϕ→ ϕ is inconsistent with NEC. The same holds for

TTTϕ→ ϕ etc.

Proof.

A `γ ↔ ¬TTγA `TTγ → γ assumption

A `¬TTγ two preceding linesA `γ first lineA `Tγ NEC

A `TTγ 4

a

Internal inconsistency

Plain inconsistency is not the only way a system can fail toacceptable. “Internal” inconsistency is almost as startling. Let ⊥be some fixed logical contradiction, e.g., ¬ 6= ¬. A theory is saidto be internally inconsistent (with respect to T ) if and only ifA ` T⊥.

Theorem (Thomason [45])

The schema TTϕ→ ϕ is internally inconsistent with NEC.

Proof.One runs the proof of Montague’s theorem in the scope of T . a

The Löb derivability conditions

Let K be the following scheme:

(K) Tϕ→ ψ → (Tϕ→ Tψ)

4 is the following scheme:

(4) Tϕ→ TTϕ

K4 contains NEC, K, 4 and all axioms of A.K4 has been thoughtto be adequate for necessity and, in some cases, for truth.

RemarkOne can show that Gödel’s provability predicate satisfies K4.NEC, K, 4 formulated for the provability predicate are knownas Löb’s derivability conditions. See [3] for more information.

Löb’s theorem

Now I want to generalise the question: for which sentences canwe have Tϕ→ ϕ?

Theorem (Löb’s theorem)

K4 ` TTϕ→ ϕ→ Tϕ

The corresponding rule follows as well:

TheoremIf K4 ` Tϕ→ ϕ, then K4 ` ϕ

Thus in the context of K4 adding Tϕ→ ϕ makes ϕ itselfprovable.

Löb’s theorem: the proof

Proof.

γ ↔ (Tγ → ϕ) diagonalization

Tγ → TTγ → ϕ K

Tγ → TTγ → Tϕ K and NECTγ → Tϕ 4

(Tϕ→ ϕ)→ (Tγ → ϕ)

(Tϕ→ ϕ)→ γ first line

T(Tϕ→ ϕ)→ γ NEC

T(Tϕ→ ϕ)→ Tγ K

T(Tϕ→ ϕ)→ Tϕ line 4

a

Gödel’s second theorem

Now fix a contradiction, for instance 0 6= 0 and call it ⊥.

Theorem (Gödel’s second theorem)

K4 is inconsistent with ¬T⊥. Thus K4 6` ¬T⊥ if K4 is consistent.

There is also a formalized version of Gödel’s secondincompleteness theorem, which can easily be derived fromLöb’s theorem.

Theorem (Gödel’s second theorem formalized)

K4 ` T⊥ ∨ ¬T¬T⊥.

The dark side again

Now here is another paradox. I didn’t know where else itshould be put.

Very much like the paradox on how not to formalise theT-sentences is arises from the interaction of two predicates, viztwo truth predicates: future and past truth.

Horsten and Leitgeb call it the ‘no future’ paradox.

No future: the language

Assume L contains four predicates G, H, F and P. The intendedreading of Gx is “x always will be the case”, while Hx shouldbe read as “x always has been the case”. Similarly Fx is to beread as “x will be the case at some point (in the future)”; finallyPx stands for “x has been the case at some point (in the past)”.G and H can easily be defined from F and P, respectively (oralso vice versa). The four predicates correspond to the wellknown operators from temporal logic, the difference being thatG and H are here predicates rather than operators.

No future: the axioms

The system K∗t is given by the following axiom schemes for allsentences ϕ and ψ of the language L. This means in particularthat ϕ and ψ may contain the predicates G and H.G1 Gϕ→ ψ → (Gϕ→ Gψ)H1 Hϕ→ ψ → (Hϕ→ Hψ)1

G2 ϕ→ HFϕ

H2 ϕ→ GPϕG3 Gϕ↔ ¬F¬ϕH3 Hϕ↔ ¬P¬ϕ

Nϕ

Gϕand

ϕ

Hϕfor all sentences ϕ.

1In [19] there is a typo in the formulation of this axiom: the last occurrenceof H is a G in the original paper.

No future: the inconcistency

These axioms are analogues of axioms from temporal logic.

K∗t is consistent (see [19]), but “internally” inconsistent, i.e., K∗tproves that there is no future and no past.

Theorem (no future paradox, [19])

K∗t ` H⊥ ∧G⊥.

Thus K∗t claims that at all moments in the future ⊥ will hold.Since ⊥ is a contradiction, there cannot be any moment in thefuture. Therefore there is no future. Analogously, but lessdramatically, there also has never been a moment in the past.

No future: the proof

I shall only prove that there is no future, i.e., K∗t ` G⊥.

K∗t ` γ ↔ GP¬γ diagonalisation(11)

K∗t ` ¬γ ↔ ¬GP¬γK∗t ` ¬γ → γ H2

K∗t ` γ(12)

K∗t ` GP¬γ from (??) and previous line(13)K∗t ` Hγ N and (12)K∗t ` ¬P¬γ H3

K∗t ` G¬P¬γ N(14)

K∗t ` G⊥ (13), (14) and G1(15)

The last line follows because we have Gϕ→ (¬ϕ→ ⊥) for all ϕand, in particular, for P¬γ , by N.

No future: an inconsistency

In this framework one can assert that there is a future by sayingthat if ϕ will always be the case then ϕ will be the case at sometime:

No future: an inconsistency

(FUT) Gϕ→ Fϕ

Corollary ([19])

H2, G3, H3, N and FUT together are inconsistent.

Proof.One proves (13) and (14) as in he preceding Theorem andapplies FUT to the latter in order to obtain F¬P¬γ , whichimplies in turn ¬GP¬γ by G3 and is therefore inconsistent with(13). a

Actually Horsten and Leitgeb [19] have proved the dual of thiscorollary.

Formal Theories of Truth V

TARSKIAN TRUTH AXIOMATIZED

Volker Halbach

New College, Oxford

Trinity 2015

Schemata and universally quantified axioms

The axiomatic theories I have considered so far, are based onschemata as axioms, eg, on Tϕ↔ ϕ.

However, I have already mentioned that one would like to haveuniversally quantified theorems as consequences of our theorysuch as

∀x(Sent(x)→ (True x ∨ True ¬.x))

Thus it’s only natural to look at theories containing suchsentences as axioms. Examples are Tarski’s definition of truthturned into an axiomatic theory, or the Kripke-Feferman theoryof truth.

To formulate such axioms I look back again at our theory A ofsyntax.

Quantifying-in

Quantifying into quotational context is notoriously problematic.Occurrences of variables within quotation marks cannot bebound from ‘outside’. For instance, the quantifier ∀v is idling inthe sentence ∀vTv = v. In some cases, however, it is possible tobind quoted variables in a sense to be explained. As long as ourexpressions are assumed to range over expressions there is away to bind variables in a quoted expression.

Quantifying-in

Assume we want to say that it is true that every expression isidentical with itself, we cannot do this by ∀vTv = v, but bysaying the following:

For all expressions e: If we replace in the formula v = vevery occurrence of v by the quotational constant for e, thenthe resulting sentence is true.

This can be formalized by the following expression:

∀v Tsub(qv,v,v = v)

From this we can derive, for instance, T¬ = ¬ in A.

Quantifying-in

The trick can be generalized. Assume ϕ(x) is a formula with nobound occurrences of the variable x, then we abbreviate by

ϕ(•x) the complex term

sub(qx, x, ϕ(x))

Perhaps, with your permission, can we say that sub replacesonly free occurrences of variables? I am pretty confident that Ican define a new sub′ function that does exactly this. Or Idefine sub in this way from the beginning.

We have for every expression its quotational constant as itsstandard name; moreover, the q describes a function assigningto each expression its quotational constant. This allows toreplace the quoted variable by a name for the object for that thevariable stands.

Limitations of the method

So far, A a theory of expressions and, possibly, of furtherobjects.

For the function symbol q I have employed the following axiom:

q(a) = a

There are no axioms that tell us what happens when q isapplied to other objects.

In the best case it gives us a name of that object, if there is one.

The method described above for quantifying into quotedcontexts works only if q gives us a name for each object. In atheory of expressions only this condition is satisfied.

If q doesn’t give us a name for every object, we would have touse satisfaction (instead of unary truth) or de re-necessity(instead of unary de dicto-necessity).

The uniform T-sentences

Now I strengthen the T-sentences a little bit by adding‘uniformity’.

The theory UTB is very much like TB:

DefinitionCall the theory A plus all the following equivalences UTB:

∀x(True ϕ(•x)↔ ϕ(x))

where ϕ(x) is a formula of L with at most x free.

The uniform T-sentences

I am tempted to say that

TheoremUTB is conservative over A.

The claim is true at least for reasonable A, such as the minimalA containing only the axioms I have mentioned so far.

Proof: One can define partial truth predicates applying tosentences up to a certain complexity, if A is strong enough.Then use a compactness argument. Alternatively use partialinductive satisfaction classes, see Kaye [22].

The uniform T-sentences and disquotationalism

I guess UTB is a disquotationalist theory – perhaps not of truthbut of satisfaction.

UTB doesn’t prove generalisations such as

∀x(Sent(x)→ (True x ∨ True ¬.x)).

We can even chooses PA as our base theory A and add allinduction axioms including those with T . The resulting theoryis still conservative over PA.

Defining membership from truth

I have defined ϕ(•x) as the complex term sub(qx, x, ϕ(x)).

DefinitionDefine x ∈ y as sub(qx,v, y), and define Set(y) asSent(sub(¬,v, y)) ∧ ¬Sent(y).

Thus Sent(y) says that y itself isn’t a sentence, but the result ofreplacing all free occurrences of v is, ie, y is a formula withexactly v free.

Moreover, I write ∀X for ∀x(Set(x)→ . . . etc.

How much second-order quantification do we get in UTB?

Comprehension in UTB

The following is the first of a couple of results that show thatone can mimic quantification over sets in certain truth-theoretictheories.

Theorem

UTB ` ∃X∀z(z ∈ X ↔ ϕ(z)

), where ϕ(z) is some formula not

containing True .

Proof.

UTB ` ∀z(True ϕ(z)↔ ϕ(z)

)(16)

UTB ` ∃X∀z(z ∈ X ↔ ϕ(z)

)(17)

a

Comprehension in UTB

Thus uniform T-sentences and comprehension principles areclosely related.

For those interested in conservativeness and the case A = PA:All this holds even in the presence of full induction. Thusparameter-free ACA0 is conservative over PA.

Later I’ll mention more results of this kind. Strong second-ordersystems of arithmetic such as RA<Γ0 or the like are reducible totruth theories. Strong ontological assumptions are reduced tosemantical assumptions.

The need for stronger theories

So we still need to look for a stronger theory.

Davidson’s proposal: turn Tarski’s definition of truth intoaxioms.At first I present Tarski’s theory in my framework (you mayfind that there is little resemblance).

Defining truth

Definition‘is true’ is defined by induction on the complexity ofL-sentences:

1 A sentence s = t is true iff the value of s is the value of t,where s and t are closed terms of the language; and so onfor further predicates of L

2 A negated L-sentence ¬ϕ is true iff ϕ is not true.3 A conditional ϕ→ ψ is true iff ϕ is false or and ψ is true (ϕ

and ψ are sentences of L)4 A universally quantified L-sentence ∀xϕ(x) is true iff ϕ(e)

for all objects e.

For the last axiom it’s assumed that there are only expressionsin the domain of our standard model (or that e is the standardname for e, whatever e is).

The mathematics needed for defining truth

Ok, this definition is not very precise. If ZF is out mathematicaltheory we can prove that there is a set of sentences containingexactly those sentences claimed to be true in the definition onthe previous slide.

To that end one can prove that inductive definitions determineunique sets – under certain conditions. If one applies thedefinition of truth to the language of set theory, we knowalready from Tarski’s theorem on the undefinability of truththat there is no set containing the sentences declared true bythat definition.

Before formalising the definition of truth for L-sentences I needmore expressive power in A.

Values

I assume from now on that the language of A contains also aunary function symbol val and that A proves the followingequations:

Additional Axiomval(t) = e if and only if t denotes a term denoting the expression

denoted by e in the standard model.

val represents the function that gives applied to a term of thelanguage it’s value, ie, he object denoted by that term.

Example

A ` val¬ = ¬ translated into the metalanguage:‘the value of ‘¬’ is ‘¬’’.A ` val q∀¬ = ∀¬, so val disquotes terms (but notsentences).A ` val q(¬a∀∀) = ¬∀∀

More predicates

I need a predicate expressing that an object is a sentence of L:

Additional AxiomA ` Sent(t) if and only if t is a term denoting a sentence in the

standard model.

Example

A ` Sent(∀v v = v)

A ` Sent(val∀v v = v)

More predicates

I need a predicate expressing that an object is closed term of L:

Additional AxiomA ` ClT(t) if and only if t is a term denoting a closed term in thestandard model.

Example

A ` ClT(∀)A ` ClT(qa∀v v = v)

More dots

In the following I’ll use Feferman’s dot notation extensively. Iwant to express in A ‘the negation of’, ‘the conjunction of . . .and . . . ’, ‘the universal quantification of . . . with respect tovariable . . . ’.

These function expressions can be introduced as new axiom orthey can be defined, eg:

Definition

x=. ydef= xa=ay

¬.xdef= ¬ax

x→. ydef= (axa→aya)

∀.xydef= ∀axay

Turning the clauses of the definition into axioms

DefinitionThe theory D is given by all axioms of A and the followingaxioms:

1 ∀x∀y(ClT(x)∧ClT(y)→(True (x=. y)↔val(x)=val(y))

)A sentence s = t is true iff the value of s is the value of t,where s and t are closed terms of the language.

2 ∀x(ClT(x)→ (True (Sent(x). )↔ Sent(val(x)))

)and so on for further predicates of L. . .

3 ∀x(Sent(x)→ (True ¬.x ↔ ¬True x)

)A negated L-sentence ¬ϕ is true iff ϕ is not true.

4 ∀x∀y(Sent(x)∧Sent(y)→ (True(x→. y)↔(Truex→Truey)

)A conditional ϕ→ ψ is true iff ϕ is false or and ψ is true (ϕand ψ are sentences of L)

5 ∀x∀y(Sent(∀.xy)→ (True (∀.xy)↔ ∀zTrue sub(qz, x, y))

)A universally quantified L-sentence ∀xϕ(x) is true iff ϕ(e)for all objects e.

Turning the clauses of the definition into axioms

The grey comments on the previous slide are merely themetatheoretic counterparts of the axioms; here is the pureversion:

DefinitionThe theory D is given by all axioms of A and the followingaxioms:

1 ∀x∀y(ClT(x)∧ClT(y)→(True (x=. y)↔val(x)=val(y))

)2 ∀x

(ClT(x)→ (True (Sent(x). )↔ Sent(val(x)))

). . .

3 ∀x∀y(Sent(x)∧Sent(y)→ (True(x→. y)↔(Truex→Truey)

)4 ∀x∀y

(Sent(x) ∧ Sent(y)→ (True (x→. y)↔ (True x →

True y)))

5 ∀x∀y(Sent(∀.xy)→ (True (∀.xy)↔ ∀zTrue sub(qz, x, y))

)

Comments on the axioms of D

True is not a symbol of L: any axiom schemata of Acontains only substitution instances from L, that is,without True .The last axiom makes use of ‘quantifying in’.Additional axioms for every predicate symbol of L have tobe added. Here I should say something about schematictheories and list definitions. . .The axioms capture a compositional conception of truth.I suppose that these are the axioms for truth Davidsonalluded to when talking about turning the definitionalclauses of truth into axioms, although there are some openquestions. . .D proves many of the desired generalisations such as∀x(Sent(x)→ (True x ∨ True ¬.x))

Deflationism and conservativeness

Some deflationists wrt truth are keen on a theory of truth thatproves generalisations but that is at the same time ‘insubstantial’in the sense that it is still conservative over the base theory A,that is, it doesn’t prove any new sentences in the originallanguage L (ie, the language without True ) (cf Shapiro [40],Field [9], Halbach [15], Ketland [23]).

So we would like to show for all sentences ϕ of L:

If D ` ϕ, then A ` ϕ.

Models

Several people have claimed that this can be shown by provingthat any model of A can be extended to a model of D.

It follows from a result by Lachlan [27] that this isn’t feasible:

Theorem (Lachlan [27])

Take A to be PA. IfM is a model of A that can be be extended to amodel of D, thenM is recursively saturated or the standard model.

A set of of formulae is recursively saturated iff every recursive finitelysatisfiable set of finitely many variables is globally satisfiable.

Discuss nonstandard models of A; existence of nonstandardsentences.

This result dooms all attempts to prove conservativeness byextending any given model of A to a model of D.

Conservativeness

Theorem (Kotlarski, Krajewski, and Lachlan [24])

Every countable recursively saturated model of PA can be extended toa model of D.

That is sufficient for establishing conservativeness:

TheoremIf D ` ϕ, then A ` ϕ.

Remark (Smith [42]?)

Countability of the model is required.

Conservativeness via cut elimination

McGee [35] complained that the model-theoretic proof ofconservativeness is of little use to the deflationist: he ought tobe able to prove conservativeness in his theory A. At least forsufficiently strong A we get:

Theorem (Leigh [28], Enayat & Visser [7], but not Halbach [14])

If D ` ϕ, then A ` ϕ. Moreover, this implication can be proved inPRA.

The theory is still fairly weak. It doesn’t prove that all sentence¬ = ¬ ∧ ¬ = ¬ ∧ ¬ = ¬ . . . are true.

Global reflection (cf Kreisel and Lévy [25])

We can use the truth predicate to state soundness of theories.For instance, we might try to prove

∀x(Sent(x) ∧ BewA(x)→ True x

)The obvious strategy is to prove first that all axioms of A aretrue and that all rules of inference are true preserving; then byinduction on the length of proofs in A, one concludes that alltheorems of A are true.

What we lack now is some induction principle. . .

Arithmetic in A

Strings vvv . . . can be used as natural numbers, and I callexpressions vvv . . . numerals because they act as constants fornumbers. I shall write n for v . . .v︸︷︷︸

n

and call it the numeral for n.

For instance, 4 is the numeral for 4 and stands for vvvv.

DefinitionNat(x) is defined as sub(x,v, 0) = 0.

The idea is that substituting the empty string for v in a string ofv’s gives the empty string. If the original string had containedsome symbol different from v, the symbol would be left. Theempty string is the empty string of v’s, so it is a natural numberand this is provable in A.

Arithmetic in A

LemmaA ` Nat(n) for all natural numbers n.

Proof.sub(0,v, 0) = 0 is an instance of A4.

Then proceed inductively using A8. a

I write ∀nϕ(n) for ∀x(Nat(x)↔ ϕ(x)) and similarly ∃nϕ(n) for∃x(Nat(x) ∧ ϕ(x)).

Addition and multiplication

Addition and multiplication can be mimicked in L byconcatenation and substitution, respectively.

Lemma

Assume n, k, n + k and n · k are numerals for n, k, n + k and n · k,respectively. Then the following holds:

1 A ` nak = n + k2 A ` sub(n, v, k) = n · k

In particular we have A ` na1 = n + 1.

Induction

Now we can add induction in the full language with True to D:

DefinitionD+ is the theory D plus all of the following axioms:

ϕ(0) ∧ ∀n(ϕ(n)→ ϕ(n + 1))→ ∀nϕ(n)

Actually I am adding to A also the axioms for a predicatesymbol Cons(x) expressing that x is a quotational constant.

That will allow to define notions such as ‘x is a proof’.

Proving consistency

Now with a suitable definition of BewA and some additionalaxioms in A we are able to prove the following in D+:

∀x(Sent(x) ∧ BewA(x)→ True x

)In particular, we’ll be able to prove

Sent(⊥) ∧ BewA(⊥)→ True ⊥

where ⊥ is some contradiction, and therefore

¬BewA(⊥)

which isn’t derivable in A by Gödel’s second incompletenesstheorem, at least if BewA is well behaved.

How much can one do in D+?

We now that D+ is not conservative over A. But one can domuch more than just proving consistency.

TheoremD+ proves ∃X∀n(n ∈ X ↔ ϕ(n)), where ϕ(n) is a formula possiblycontaining subformulae of the form z ∈ Y where Y is free and wherethere are no further occurrences of the truth predicate in ϕ(n).

In the proof the ‘Tarski clauses’ are needed to handle theparameters Y .

‘Tarskian truth is as strong as arithmetical comprehension.’

Thus is does not only prove global reflection but much more.

Formal Theories of Truth VI

KRIPKE’S THEORY OF TRUTH

Volker Halbach

New College, Oxford

Trinity 2015

Type restrictions

The theories TB, UTB, and D are typed theories in the sensethat there is no sentence ϕ containing the predicate True suchthat, eg, D ` True ϕ.

TB, UTB, and D don’t say anything about sentences containingTrue , so they are not incompatible with a theory proving claimsabout the truth of sentences containing True . This is anadvantage of the axiomatic approach, because on the semanticaccount one would have to decide for any sentence whetherthey are in the extension of True or not.

Hierarchical solutions

Typed truth theories can be iterated, ie, one can introduce newpredicates True 1, True 2, . . . , True ω, True ω+1, . . . , True ε0 as faras you can count. . . At any level one could use axiomsanalogous to TB, UTB, and D or pursue a semantic approachusing Tarski’s definition of truth.

I did it up to ωCK1 , that is, as far one can count recursively.

Beyond that point the languages become non-recursive, andpushing the hierarchy further becomes an exercise in countingrather than in the theory of truth.

Kripke proposed to look at ill-founded hierarchies of truthpredicates, eg, True n applies to all sentences containing truthpredicates True i with i > n. Visser [46] axiomatised truth insuch a language by axioms analogous to TB and showed thatthe resulting theory is ω-inconsistent.

Iterating truth without typing

Anyway, we would like to say more about the truth of sentencescontaining the very same truth predicate. Therefore I return tothe type-free truth predicate T contained in the language L.Clearly there are sentences that ought to be true, although theycontain T :

T∀x x = x

TT∀x x = x

TTaq ∀x x = x

Declaring these sentences true seems unproblematic, as they arenot self-referential in any way.

Defining type-free truth

Assume we have a modelM for the language without T ; we tryto define type-free truth based on that model.

Of course we know what to do with sentences not containing T :they should get into the extension of T iff they are true in M;otherwise their negation should be in. Moreover, if ϕ and ψ arealready in, they are ‘safe’ and their conjunction ϕ ∧ ψ should bethrown into the extension S and so on.

So assume we have a modelM for the language without T . Iproceed in the style of Tarski (with a certain twist), but add thetruth predicate itself to the object language.

Defining type-free truth

DefinitionA set S of L-sentences is a Kripke fixed-point iff for allsentences ϕ and ψ the following holds:

1 s = t is true inM iff (s = t) ∈ S. Similarly for all otheratomic sentences without T : ϕ is true inM iff ϕ ∈ S.

2 s = t is false inM iff (¬s = t) ∈ S. Similarly for all otheratomic sentences without T : ¬ϕ is true inM iff ¬ϕ ∈ S.

3 (ϕ ∈ S and ψ ∈ S) iff (ϕ ∧ ψ) ∈ S.4 (¬ϕ ∈ S or ¬ψ ∈ S) iff (¬(ϕ ∧ ψ)) ∈ S.5 ϕ ∈ S iff(¬¬ϕ) ∈ S.6 ϕ(e) ∈ S for all objects e , iff ∀xϕ(x) ∈ S.7 ¬ϕ(e) ∈ S for some object e , iff (¬∀xϕ(x)) ∈ S.8 If ϕ ∈ S and the value of the term t is ϕ, iff (Tt) ∈ S.9 If ¬ϕ ∈ S and the value of the term t is ϕ, iff (¬Tt) ∈ S.

Comments on the definition

Up to item 7 this is a ‘Tarskian’ definition of truth.8 and 9 imply the following

ϕ ∈ S iff Tϕ ∈ S.If ¬ϕ ∈ S iff ¬Tϕ ∈ S.

But why having different clauses for ϕ ∧ ψ and ¬(ϕ ∧ ψ)?to answer that question we need to look at the proof for theexistence of Kripke fixed-points.

Existence of fixed points

TheoremFor anyM satisfying the base theory there are Kripke fixed-points.

I don’t know who proved this first. It seems Kripke [26], Martinand Woodruff [31], and Cantini (mid 70ies) all proved similarresults independently. The most notable contribution of Kripkeconsists in a systematic investigation into the structure of thefixed points (minimal, intrinsic, and maximal fixed points) aswell as in the application to various evaluation schemata.

Existence of fixed points: the proof

Proof.LetM be given. Start with S = Ø and expand S successively byapplying the clauses in the definition read from left to rightonly, eg,

If s = t is true inM then (s = t) ∈ S.If s = t is false inM then (¬s = t) ∈ S.If ϕ ∈ S and ψ ∈ S then (ϕ ∧ ψ) ∈ S.If ¬ϕ ∈ S or ¬ψ ∈ S then (¬(ϕ ∧ ψ)) ∈ S.If ϕ ∈ S and the value of the term t is ϕ, then (Tt) ∈ S.

a

Existence of fixed points: the proof

Proof.For cardinality reason there must be as stage where thisprocedure reaches a fixed point – given that our languageincluding all constants has some cardinality, ie, given that ourlanguage together with constants for all objects isn’t a properclass.

This yields a procedure for generating an S satisfying all theclauses of the definition.

The fixed point obtained in this way is minimal. a

Starting with other sets

One could also start with any other set of L-sentences and thenclose it in the way described above.

In some cases this will not yields a Kripke fixed-point, eg if asentence false inM is in the starting set.

Starting, eg with the truth teller yields another Kripke fixedpoint. The truth teller is a sentence τ such that A ` τ ↔ ¬ττ

Fixed points

But why should we have different clauses for ϕ ∧ ψ and¬(ϕ ∧ ψ)? why don’t we have a simple clause for ¬?If we had a clause

If ϕ 6∈ S, then ¬ϕ ∈ S.

we would have to add ¬ϕ in the first step and then, once ϕhad been added we would have to remove ¬ϕ again. Sosentences might flip in and out of S.Because S becomes larger and larger by applying theclauses, there must be a stage where all the sentences thatcan be added have been added.

Properties of fixed points

As a surrogate for the T-sentences we obtain the followingmetatheoretic equivalence:

Theoremϕ ∈ S iff Tϕ ∈ S.

This is a consequence of clause 8.

The liar

TheoremKripke fixed-points are not closed under classical logic.

Proof.The liar sentence is a sentence of the form ¬Tt where the valueof t is ¬Tt itself. I distinguish two cases:

1 Tt ∈ S. Then ¬Tt ∈ S by clause 9, ie S contains a sentencetogether with its negation.

2 Tt 6∈ S. Then ¬Tt 6∈ S by clause 9, ie S contains neither theliar nor its negation.

a

The logic of Kripke fixed-points

So any Kripke fixed-point S either contains the liar sentencetogether with its negation (truth value glut) or it containsneither the liar or its negation (truth value gap).

The minimal fixed-point contains neither the liar or its negation.If one starts with the singleton of the liar sentence and closes offunder the Kripke clauses, then one obtains a fixed pointcontaining the liar sentence together with its negation.

In general, so far nothing rules out ‘inconsistent’ fixed points, ieKripke fixed-points S such that there is a sentence ϕ with ϕ ∈ Sand ¬ϕ ∈ S (truth value ‘gluts’). One can describe Kripke fixedpoints as sets closed under a four-valued logic (true, false, gap,and glut). See Visser [46].

Kripke [26] focused on consistent fixed points, ie Kripkefixed-points without truth-value gluts.

The logic of consistent Kripke fixed-points

As shown above, Kripke fixed-points are not closed underclassical logic. Consistent Kripke fixed-points are closed undera three-valued or partial logic called Strong Kleene logic: somesentences are true, some are false, some lack a truth value.

Here are the truth tables for ∧ and ¬:

ϕ ψ (ϕ ∧ ψ)T T TT F FT - -F T FF F FF - F- T -- F F- - -

ϕ ¬ϕT FF T- -

Other consistent Kripke fixed-points

A consistent Kripke fixed-point S is maximal iff there is noKripke fixed-point S′ $ S. There is more than one maximalKripke fixed-point.

A consistent Kripke fixed-point S is intrinsic iff any ϕ ∈ S iscontained in any Kripke maximal fixed-point. There are variousintrinsic Kripke-fixed points.

The consistent Kripke fixed-point S is maximal intrinsic iff it allintrinsic Kripke fixed-points. There is only one such fixed point.

Consistent Kripke fixed-points

These different consistent Kripke fixed-points can be used toclassify sentences:

The liar isn’t in any consistent Kripke fixed-point. suchsentences are called paradoxical.The truth teller is contained in some consistent Kripkefixed-points. It’s not an element of an intrinsic consistentKripke fixed-point.By the diagonal lemma, there is a sentence such thatA ` τ ↔ (Tτ ∨ ¬Tτ). Such a τ is an element of themaximal intrinsic consistent Kripke fixed-point, but it’s notan element of the minimal fixed-point.

Other evaluation schemata

Kripke [26] presented his theory in a different way whichallows one to change the logic of the fixed-points to other logics,eg supervaluations and Weak Kleene logic.

Supervaluations will make λ ∨ ¬λ (λ the liar true), but onelooses compositionality. The disjunction will be true althoughboth disjuncts lack truth values (and such disjunction ‘usually’do not have a truth value).

Complexity considerations

Closing a set under the operations 1-9 may take manysteps. In the case of the standard model of arithmetic or asimple structure of expressions it takes ωCK

1 many steps.S is defined by an inductive definition, but any suchdefinition (positive inductive definition) may be reduced tothe definition of truth (Π1

1 sets).These complexity issues is fairly independent from theevaluation scheme. For the complexity considerations seeBurgess [4], [5], McGee [33].There is a straightforward translation between the Tarskianhierarchy up to ωCK

1 and the minimal Kripke fixed point(Halbach [13]).

Formal Theories of Truth VII

THE KRIPKE-FEFERMAN THEORY OFTRUTH

Volker Halbach

New College, Oxford

Trinity 2015

What has happened so far

Last time I have shown how to define a set of sentences havingcertain nice properties (Kripke fixed-point).

The definition of the set starts from a given modelM of Ainterpreting all function and predicate symbols of L except forT .

Here is the definition again:

Defining type-free truth

DefinitionA set S of L-sentences is a Kripke fixed-point iff for allsentences ϕ and ψ the following holds:

1 s = t is true inM iff (s = t) ∈ S. Similarly for all otheratomic sentences without T : ϕ is true inM iff ϕ ∈ S.

2 s = t is false inM iff (¬s = t) ∈ S. Similarly for all otheratomic sentences without T : ¬ϕ is true inM iff ¬ϕ ∈ S.

3 (ϕ ∈ S and ψ ∈ S) iff (ϕ ∧ ψ) ∈ S.4 (¬ϕ ∈ S or ¬ψ ∈ S) iff (¬(ϕ ∧ ψ)) ∈ S.5 ϕ ∈ S iff (¬¬ϕ) ∈ S.6 ϕ(e) ∈ S for all objects e , iff ∀xϕ(x) ∈ S.7 ¬ϕ(e) ∈ S for some object e , iff (¬∀xϕ(x)) ∈ S.8 ϕ ∈ S iff (Tt) ∈ S, if the value of the term t is ϕ.9 ¬ϕ ∈ S iff (¬Tt) ∈ S, if the value of the term t is ϕ.

interpreting L

AssumeM is a model of the language L without T , andassume that S is a subset of the domain ofM. Then (M, S) isthe classical model of L interpreting all symbols in the sameway asM and assigning to T the set S as extension.

Remark(M, S) |= Tϕ iff ϕ ∈ S.

Generally the problem is to find a ‘neat’ extension S for thetruth predicate.

Closing off

AssumeM is the intended model of A (without aninterpretation of T ) and assume further that S is some Kripkefixed-point. Then (M, S) ought to be a well behaved model forL.

Assume (M, S) is such a model of L, then the following willhold:

(M, S) |= Tϕ↔ TTϕ

(M, S) |= Tϕ ∧ ψ ↔ (Tϕ ∧ Tψ)(M, S) |= ¬Tλ if S is a consistent Kripke fixed-point and ifλ is the liar sentence. In this case (M, S) |= λ.(M, S) 6|= Tλ ∨ ¬λ if S is a consistent Kripke fixed-point, butof course (M, S) |= λ ∨ ¬λ as (M, S) is a classical model.

Again: why axioms?

When one tries to apply Kripke’s theory to our overall theory,eg Zermelo–Fraenkel set theory or the like, one is confrontedwith the difficulty that we don’t have a starting model, i.e. ‘thestandard model of ZF’.

Of course one can claim that this is quite ok: ZF is our ‘topmost’theory: everything sensible – including semantics – must bedefinable in it.

My proposal: The topmost theory ought to contain semantics aswell. This semantic theory is not reduced to set theory again,but given axiomatically. We may hope that axioms for truth thatwork for Peano arithmetic as base theory work also for ZFC asbase theory.

The reflective closure

In [8] Feferman proposed to define the reflective closure of atheory (such as PA or ZF). This is obtained from the theory byadding axioms describing a Kripke fixed-point.

Feferman takes schematic theories as his starting point, which arethen applied to the language including the truth predicate.Thus the reflective closure of a theory contains always allinstances of the induction scheme (induction, replacement,separation,. . . ).

The reflective closure

Feferman thought of the reflective closure as a way of makingexplicit all assumptions or commitments implicit in theacceptance of a theory.

If we accept a theory S, then we are also committed to theconsistency of S. We are also committed to the local reflectionprinciple for S

BewS(ϕ)→ ϕ

for all sentences ϕ, and iterations of these reflection principles.

The reflective closure

The strongest reflection principle for a theory S is the Globalreflection principle

∀x(Sent(x) ∧ BewS(x)→ Tx

)There are a couple of problems with the exact formulation ofthe Global reflection principle, but you get the idea. . .

Thus rather than adding proof-theoretic reflection principles(consistency statement, local or uniform reflection), one adds atruth theory proving Global reflection and iterations thereof.

To this end one can add iterations of Global reflection usingtyped truth predicates – or, much more elegantly, axiomsdescribing the definition of Kripke fixed-points.

History of the system

Feferman proposed as system similar to the one above in 1979

in a talk, which resulted in his 1991 paper [8]. The firstpublished version of KF appeared in Reinhardt [38]; Cantini’sKF has weaker induction; McGee’s [33] version is similar toReinhardt’s. Occasionally the Consistency axiom is omitted.

Axiomatising Kripke’s theory

DefinitionThe theory KF is given by all axioms of A including allinduction axioms and the following axioms:

1 ∀x∀y(ClT(x)∧ClT(y)→(T(x=. y)↔val(x)=val(y))

)s = t is true inM iff (s = t) ∈ S

2 ∀x∀y(ClT(x)∧ClT(y)→(T¬. (x=. y)↔val(x) 6=val(y))

)s = t is false inM iff (¬s = t) ∈ S

3 . . . and so on for all predicates other than = and T .4 ∀x∀y

(Sent(x)∧Sent(y)→ (T(x∧. y)↔(Tx∧Ty)

)(ϕ ∈ S and ψ ∈ S) iff (ϕ ∧ ψ) ∈ S.

5 ∀x∀y(Sent(x)∧Sent(y)→ (T¬. (x∧. y)↔(T¬.x∨T¬.y)

)(¬ϕ ∈ S or ¬ψ ∈ S) iff (¬(ϕ ∧ ψ)) ∈ S

6 ∀x(Sent(x)→ (T¬.¬.x↔Tx

)ϕ ∈ S iff (¬¬ϕ) ∈ S.

Axiomatising Kripke’s theory

Definition7 ∀x∀y

(Sent(∀.xy)→ (T(∀.xy)↔ ∀zTsub(qz, x, y))

)ϕ(e) ∈ S for all objects e , iff ∀xϕ(x) ∈ S

8 ∀x∀y(Sent(∀.xy)→ (T(¬. ∀.xy)↔ ∃zT¬. sub(qz, x, y))

)¬ϕ(e) ∈ S for some object e , iff (¬∀xϕ(x)) ∈ S

9 ∀x(ClT(x)→(TT. x↔Tval(x))

)ϕ ∈ S iff (Tt) ∈ S, if the value of the term t is ϕ

10 ∀x(ClT(x)→(T¬. T. x↔T¬. val(x))

)¬ϕ ∈ S iff (¬Tt) ∈ S, if the value of the term t is ϕ, .

11 (Consistency axiom)∀x(Sent(x)→(T¬.x→¬Tx)

)

The KF axioms on one page . . .

1 ∀x∀y(ClT(x)∧ClT(y)→(T(x=. y)↔val(x)=val(y))

)2 ∀x∀y

(ClT(x)∧ClT(y)→(T¬. (x=. y)↔val(x) 6=val(y))

)3 . . . and so on for all predicates other than = and T .4 ∀x∀y

(Sent(x)∧Sent(y)→ (T(x∧. y)↔(Tx∧Ty)

)5 ∀x∀y

(Sent(x)∧Sent(y)→ (T¬. (x∧. y)↔(T¬.x∨T¬.y)

)6 ∀x

(Sent(x)→ (T¬.¬.x↔Tx

)7 ∀x∀y

(Sent(∀.xy)→ (T(∀.xy)↔ ∀zTsubsub(qz, x, y))

)8 ∀x∀y

(Sent(∀.xy)→ (T(¬. ∀.xy)↔ ∃zT¬. sub(qz, x, y))

)9 ∀x

(ClT(x)→(TT. x↔Tval(x))

)10 ∀x

(ClT(x)→(T¬. T. x↔T¬. val(x))

)11 (Consistency axiom)∀x(Sent(x)→(T¬.x→¬Tx)

)

The mathematical content of KF

TheoremKF with PA as base theory A proves the same arithmetical theorems as

ε0 = ωω...ω

-iterated arithmetical comprehension RA<ε0 or ε0-timesiterated ‘Tarskian’ truth.

Feferman has tried to characterise the reflective closure of PA isvarious ways. KF fits into the picture.

Anyway, KF allows one to carry out reasoning that wouldotherwise require quite a lot of typed truth predicates.

But is it a nice theory?

Consistency

Over KF without the Consistency axiom, the following areequivalent:

(i) CONS, that is, ∀x(Sent(x)→ ¬(True x ∧ True ¬.x)

)(ii) ∀x

(Sent(x)→ (True ¬.x → ¬True x)

)(iii) the schema ∀~x(True ϕ( ~x)→ ϕ(~x)) for all formulas ϕ(~x); ~x

stands here for a string x1, . . . , xn of variables

(iv) the schema ∀x(True ϕ(x)→ ϕ(x)

)for all formulas ϕ(x);

this schema allows only one free variable in the respectiveinstantiating formula.

(v) ∀x(True ¬True x → ¬True x)

The liar in KF

In particular KF ` Tϕ→ ϕ (previous page (iii))Since KF is formulated in classical logic we have a sentence λs.t.: KF ` λ↔ ¬TλLemmaKF proves the liar sentence, ie KF ` λ.

Given the intended semantics, this shouldn’t be too surprising;see 143.

Proof.

KF ` Tλ→ λ previous page (iii)

KF ` Tλ→ ¬λ diagonal lemma

KF ` ¬TλKF ` λ diagonal lemma

a

Worries about the consistency axiom

If the consistency axiom

∀x(Sent(x)→(T¬.x→¬Tx)

)is dropped, then the liar is no longer provable.

KF without the consistency axiom can be seen as anaxiomatisation of arbitrary Kripke fixed-points, ie, fixed-pointswith truth-value gluts as well as gaps, while the original KFaxiomatises consistent Kripke fixed-points (note that neitherforces the minimal fixed point).

The consistency axiom

Over KF without the consistency axiom the following axiomsand axiom schemata are equivalent:

(i) The consistency axiom:∀x(Sent(x)→(T¬.x→¬Tx)

)(ii) ∀x

(Sent(x)→¬(T¬.x∧T¬.x)

)(iii) the schema ∀x

(Tϕ(x)→ ϕ(x)

)for all formulas ϕ(x) of L

(iv) ∀x(Sent(x)→(T¬Tx → ¬Tx)

)

Internal and external logic

Still, even after dropping the consistency axiom KF describes anon-classical notion of truth in classical logic.

The internal logic {ϕ : KF ` Tϕ} of KF and its external logic,that is, the set of all KF-provable sentences are highly disparate.

Example

KF ` λ ∨ ¬λ, but KF 6` Tλ ∨ ¬λ (also for KF withoutconsistency axiom)KF ` τ ∨ ¬τ , but KF 6` Tτ ∨ ¬τ (also for KF withoutconsistency axiom; τ the truth teller)

KF ` λ but KF ` ¬Tλ (this requires the consistency axiom)

Internal and external logic

Thus KF with and without consistency axiom is a theory inclassical logic of non-classical truth.

Isn’t that against the spirit of Kripke’s theory? Isn’t what we arereally interested an axiomatisation of S rather than of (M, S)?Shouldn’t we just stick to non-classical logic when adoptingand axiomatising a non-classical notion of truth?

Feferman (to some extent) and Reinhardt seem to agree: whatwe are really interested in is a system allowing us to derivesentences that are elements of all Kripke fixed points S.Reinhardt thought one shouldn’t ‘trust’ a classical metatheory,but that one should try to show that it is safe because it’sdispensable.

But at the same time they agree that one shouldn’t try to thinkin Strong Kleene logic, because nothing like sustained ordinaryreasoning can be carried out in such non-classical logics.

Axiomatising Kripke fixed-points

And what happened when your parents told you that youshouldn’t try to do something?

So Leon and I [16] tried to write down a theory directlyaxiomatising the conditions on Kripke fixed-points in threevalued logic.

Thus our intended model is not (M, S) but we want to generateall sentences in all Kripke fixed-points (over the standardmodel), ie, all sentences in the minimal Kripke fixed point.

We tried . . . and perished.

Axiomatising Kripke fixed-points

First Leon and I tried to prove that the system PKF inStrong-Kleene logic is as strong as the classical system KF, iethat both systems prove the same T-free sentences. After allFeferman’s [8] analysis of KF makes use of the ‘grounded’sentences only.

When we tried to carry out Feferman’s proof in PKF we didn’texperience problems with the derivation of the truth-theoreticsentences (eg it wasn’t a problem that KF proves the liar, whileit’s gappy in PKF). The problem was the proof of amathematical schema: transfinite induction up to ε0.

TheoremPKF proves the same arithmetical sentences as RAωω or ωω-timesiterated ‘Tarskian’ truth D+.

Using alternative evaluation schemata

I have focused on Kripke’s theory with the Strong-Kleeneevaluation scheme for handling truth value gaps.

Kripke showed how to carry out his construction for otherpolicies on truth-value gaps. There have been proposals to giveKF-like axiomatisations of these theories.

Conclusions

the use of classical logic isn’t a matter of mere convenience:doing everything in Strong-Kleene logic cripples ourmathematical reasoning.Reinhardt’s hopes to justify the use of classical logic and ofKF by an ‘instrumentalist’ interpretation are doomed.In order to evaluate an axiomatic theory of truth innon-classical logic it’s not sufficient to look at the liarsentence and other truth-theoretic problems. One shouldcheck to what extent one has to give up classical patternsof reasoning.

Supervaluations

Cantini proposed a theory VF axiomatising Kripke’s theorywith supervaluations.

TheoremVF is proof-theoretically equivalent to ID1 and thus much strongerthan KF.

Weak Kleene

Feferman [8] mentioned also that KF with Weak-Kleene logicshould have the same proof-theoretic strength as KF.

Example

KF ` Tλ ∨ 0 = 0 butWKF 6` Tλ ∨ 0 = 0

Theorem (Fujimoto)

KF and WKF are proof-theoretically equivalent.

Classical logic again

I doubt more and more that KF-like systems are the way to go.

These systems describe a non-classical notion of truth, and if wego for such a notion we should revise our informal metatheoryaccordingly and start to think in non-classical logic. I tried itonce and it proved to be a traumatic experience.

Thus I favour very much systems that are formulated inclassical logic and describe a classical notion of truth. . .

Formal Theories of Truth VIII

Classical Symmetric Truth

Volker Halbach

New College, Oxford

Trinity 2015

What has happened so far

I discussed the Kripke-Feferman theory and its variants. KF hasthe following drawbacks:

Internal and external logic are incompatible. That is, wedon’t have the following for all sentences ϕ and ψ:

KF ` Tϕ iff KF ` ϕ

Call a system symmetric iff internal and external logic areidentical.The internal logic of KF is non-classical. If we force KF tobe symmetric, we have to axiomatise KF in partial logicand abandon classical logic completely.

There may be good reasons to think about a partial notion oftruth in classical logic. In fact, that’s the typical situation insemantic theories of truth: people describe all kinds of funnynon-classical systems by using classical systems as metatheories.

Symmetry

If we want object- and metatheory to be identical or at leastcompatible, then Kripke’s theory forces us to abandon classicallogic. In the end it’s a non-classical theory.

But can we have a system based on classical logic that issymmetric? That is, is there a system that is fully classical? Isthere is system with a classical internal and external logic?

Such a theory would have to satisfy two criteria:1 The system is formulated in classical logic.2 Internal and external logic should coincide.

Symmetry

These requirements are easily satisfied: We formulate ourtheory in classical logic (as I am always doing in these lectures)and then close the system under the following two rules:

NECϕ

Tϕ

Tϕ

ϕCONEC

NEC forces the internal logic to contain the external logic, whileCONEC forces the external logic to contain the internal logic,that is:

{ϕ : A ` ϕ} = {ϕ : A ` Tϕ}

Obviously then internal and external logic are closed underclassical logic.

Symmetry and the liar

LemmaIf A is consistent and closed under NEC and CONEC, neither the liarnor its negation is provable in A.

Proof.

A ` ` λ↔ ¬λ fixed point lemmaA ` ` λ assumption

A ` ` ¬Tλ previous line

A ` ` Tλ NECso λ cannot be provableA ` ` ¬λ assumption

A ` ` Tλ first lineA ` ` λ CONEC

so ¬λ cannot be provable

a

Symmetry and the liar

Thus classical symmetric systems (ie classical systems closedunder NEC and CONEC) don’t decide the liar.

Don’t get worried: I don’t claim that neither λ nor ¬λ can beconsistently added to A, but after adding the liar sentence thesystem cannot be closed again under NEC and CONEC.

I take the undecidability of the liar as a nice property ofsymmetric systems: its provability in KF is an odd feature of KF.

Symmetry

One can easily show that closing A under NEC and CONECyields a consistent theory (if A isn’t crazily behaved), but theresulting system will be weak.

Basically closing under NEC and CONEC means adding the‘rule version’ of the T-sentences.

I want a stronger theory, eg a type-free version of ‘Tarskian’truth, that is, the truth predicate ought to commute with theconnectives and quantifiers. So I write down the D axiomswithout type restriction. . .

The Friedman-Sheard theory

DefinitionThe theory FS is given by all axioms of A including allinduction axioms and the following axioms:

1 ∀x∀y(ClT(x)∧ClT(y)→(T(x=. y)↔val(x)=val(y))

)2 . . . and so on for all predicates other than = and T .3 ∀x

(Sent(x)→ (T¬.x↔¬Tx

)4 ∀x∀y

(Sent(x)∧Sent(y)→ (T(x∧. y)↔(Tx∧Ty)

)5 ∀x∀y

(Sent(∀.xy)→ (T(∀.xy)↔ ∀zTsub(qz, x, y))

)6 FS is closed under NEC and CONEC.

History of the Friedman-Sheard theory

Friedman & Sheard studied the theory in their 1987 paper [10]under a slightly different axiomatisation. They proved theconsistency of the theory.

Before its actual birth (in publication) McGee damaged itsreputation in [32].

Duality axioms

For FS we don’t need the ‘dual’ axioms such as

∀x∀y(Sent(x)∧Sent(y)→ (T¬. (x∧. y)↔(T¬.x∨T¬.y)

)They are already consequences of FS.

Complete and consistent truth

LemmaFS proves the following theorems:

1 ∀x(Sent(x)→ ¬(Tx ∧ T¬.x)

)(consistency)

2 ∀x(Sent(x)→ (Tx ∨ T¬.x)

)(completeness)

Thus FS explicitly denies the existence of truth value gluts andgaps.

The consistency axiom drops out as a consequence of Axiom 3,while it caused so much trouble in KF. Generally KF-likesystems are incompatible with Axiom 3 which excludes truthvalue gluts and gaps.

This shows that FS describes a thoroughly classical notion oftruth.

Truth clauses

In KF we had the following axiom:

∀x(ClT(x)→(TT. x↔Tval(x))

)Such an axiom would nicely complement Axiom 1. But becauseof Axiom 3, this axiom also allows one to derive the fullT-sentences, which makes adding such axioms impossible:

Lemma

The schema TTϕ↔ Tϕ for all sentences is inconsistent with FS.

Proof-theoretic analysis

Theorem (Halbach [12])

FS is proof-theoretically equivalent to ω-iterated Tarskian truth, ie toRA<ω.

Thus FS is much weaker than KF (or KF in partial logic), which

are equivalent to ε0 = ωω...ω

and ωω-iterated truth.

Basically an application of NEC adds an other level of Tarskiantruth (although CONEC proved to be powerful on its own –Sheard [41] 2001).

Iterating truth

The proof-theoretical analysis reveals that FS is not good atiterations of truth. The axiom

∀x(ClT(x)→(TT. x↔Tval(x))

)of KF allows one to prove (by induction) that long iterations ofapplication of the truth predicate are true.

In FS we can only add one truth predicate at a time by applyingNEC.

Iterating truth into the transfinite

It would be nice if we had a way of proving iterations of thetruth predicate into the transfinite, so we are able to show in FSthat eg all sentences

T0 = 0, TT0 = 0, TTT0 = 0, . . .

are true.

One can try to replace NEC by reflection principles.

Iterating truth into the transfinite

Define FS0 as the system FS but without NEC or CONEC.

FSn+1 is the system FSn with global reflection for FSn, ie with

∀x(Sent(x)→ (BewFSn(x)→ Tx)

)FSω is then the system

⋃n∈ω FSn.

FSω+1 is then FSω plus global reflection for FSω:

∀x(Sent(x)→ (BewFSω(x)→ Tx)

)

Iterating truth into the transfinite

TheoremFSω+1 is inconsistent.

Thus one cannot consistently iterate truth beyond all finitelevels in FS. It’s possible to reformulate NEC to allow for morethan finitely many applications in a sense, but doing it leads toan inconsistent theory.

That’s strange: a theory that cannot consistently reflect on itself!

TheoremFS plus global reflection for FS

∀x(Sent(x)→ (BewFS(x)→ Tx)

)is inconsistent.

McGee’s theorem on ω-inconsistency

The inconsistency results are a consequence of a theorem due toMcGee [32].

So far all paradoxes have been purely ‘modal’, in the sense thatthey did not involve quantification. McGe’s paradox involves aquantifier.

McGee’s theorem on ω-inconsistency

Theorem (McGee [32])

Assume that A proves the following schemes for all sentences ϕ and ψand all formulas χ(v) having at most v free.

(i) NEC(ii) Tϕ→ ψ → (Tϕ→ Tψ)

(iii) T¬ϕ→ ¬Tϕ

(iv) ∀xTχ( •x)→ T∀xχ(x)Then A is ω-inconsistent, that is, A ` ζ(n) for all n ∈ ω andA ` ¬∀nζ(n)

FS proves all these sentences and is closed under NEC.

McGee’s theorem on ω-inconsistency: the proof

I assume throughout this section that A satisfies all axiomslisted on the previous page. I shall apply the diagonalizationtheorem twice; this is the first instance:(18)

A ` γ(n, x, y)↔ ∃k(n = ka1∧x = ∀x(γ(•k, x,

•y)→ Tx))∨(n = 0∧x = y)

The free variables n, x, y do not present a problem becausenothing prevents free variables from occurring in thediagonalized formula. The relativization to number variables nis unproblematic, as well.

McGee’s theorem on ω-inconsistency: the proof

Roughly speaking, the use of γ is the following. Assume asentence σ has been fixed. Then γ(0, x, σ) holds, if x is thesentence σ. γ(1, x, σ) holds if x is the sentence∀x(γ(0, x, σ)→ Tx), that is, x is a sentence equivalent to Tσ.Similarly, γ(2, x, σ) says that x is a certain sentence equivalentto TTσ.

McGee’s theorem on ω-inconsistency: the proof

The second application of the diagonalization theorem 13 isstraightforward:

(19) σ ↔ ¬∀n∀x(γ(n, x, σ)→ Tx)

McGee’s theorem on ω-inconsistency: the proof

The following lemma holds also for arbitrary sentences in placeof σ.

Lemma

(i) A ` γ(0, x, σ)↔ x = σ

(ii) A ` γ(na1, x, σ)↔ x = ∀x(γ( •n, x, σ)→ Tx)

McGee’s theorem on ω-inconsistency: the proof

Proof.

From A6 we we derive n = 0→ ¬∃k n = ka1, which in turnimplies the following:

n = 0→ (γ(n, x, y)↔ (n = 0 ∧ x = y))

This yields (i). (ii) is proved similarly. a

McGee’s theorem on ω-inconsistency: the proof

From (19) we obtain the following:

A ` ¬σ ↔ ∀n∀x(γ(n, x, σ)→ Tx)

→ ∀x(γ(0, x, σ)→ Tx)

→ Tσ Lemma 83 (i)(20)

McGee’s theorem on ω-inconsistency: the proof

An application of NEC and axiom (ii) of theorem 82 to (19)yields the following equivalence:

A ` Tσ → T¬∀n∀x(γ(n, x, σ)→ Tx)

→ ¬T∀n∀x(γ(n, x, σ)→ Tx) axiom (iii)

→ ¬∀nT∀x(γ( •n, x, σ)→ Tx) axiom (iv)

→ ¬∀n∀x(x = ∀x(γ( •n, x, σ)→ Tx)→ Tx)

→ ¬∀n∀x(γ(na1, x, σ)→ Tx) Lemma 83 (ii)→ ¬∀n∀x(γ(n, x, σ)→ Tx) Lemma ??

→ σ(21)

McGee’s theorem on ω-inconsistency: the proof

Taken together, (20) and (21) imply A ` σ, which in turnimplies by (19)

(22) A ` ¬∀n∀x(γ(n, x, σ)→ Tx)

From A ` σ, however, we have also by NEC:

McGee’s theorem on ω-inconsistency: the proof

A `TσA `∀x(γ(0, x, σ)→ Tx) Lemma 83 (i)(23)

A `T∀x(γ(0, x, σ)→ Tx) NEC

A `∀x(x = ∀x(γ(0, x, σ)→ Tx)→ Tx)

A `∀x(γ(0a1, x, σ)→ Tx) Lemma 83 (ii)

A `∀x(γ(1, x, σ)→ Tx) axiom A2(24)

A `T∀x(γ(1, x, σ)→ Tx) NEC

A `∀x(x = ∀x(γ(1, x, σ)→ Tx)→ Tx)

A `∀x(γ(1a1, x, σ)→ Tx) Lemma 83 (ii)

A `∀x(γ(2, x, σ)→ Tx) axiom A2(25)...

McGee’s theorem on ω-inconsistency: the proof

(22) and the sequence of lines continuing (23), (24) and (25)establish the ω-inconsistency of A according to definition ??.

More ω-inconsistencies

Further ω-inconsistencies of truth theories have beenestablished by [47], [48] and [29].

Formal Theories of Truth IX

THE REAL CULPRIT

Volker Halbach

New College, Oxford

Trinity 2015

The cases

In the lectures we have encountered various paradoxes andinconsistencies:

The T-sentencesMontague’s paradoxGödel’s second incompleteness theorem (inconsistency of¬T⊥)McGee’s paradox

and there are many more paradoxes (Yablo’s paradox, alsovarious inconsistencies in Friedman & Sheard [10]).

The victims

Various theories have been proved to be inconsistent. Theparadoxes do not only concern truth but also various forms ofnecessity, temporal notions, deontic notions, if treated asdiagonalisable predicates.

The unrestricted T-sentences, the predicate version of the modalsystem T didn’t survive. FS got away scarred for life.

The suspects

The liar? He is certainly involved in Montague’s paradoxand the liar paradox. But is he pulling the strings?McGee? The liar is hardly sophisticated enough to haveeliminated FS via ω-inconsistency. McGee’s fixed pointlooks more sophisticated.Martin Löb? He is a good citizen; after all he is onlyinvolved in the Gödel business not in the paradoxes.

The possible worlds perspective

Many of the paradoxes can be visualised by possible worldssemantics.

Truth-theoretic axioms relate to modal principles, eg, inMontague’s paradox. The liar paradox is just one of manymodal paradoxes.

How can one develop possible worlds semantics, if necessity istreated as a predicate rather than as a modal operator?

The following is a survey of Halbach, Leitgeb, Welch [17]

Frames

Frames are defined as in operator modal logic (ie the usualmodal logic):

DefinitionA frame is an ordered pair 〈W,R〉 where

W 6= Ø (the set of worlds),R ⊆W ×W (the accessibility relation).

T should behave like the box in modal logic: Tϕ should be trueat a world w iff ϕ is true at all worlds that can be ‘seen’ from w.

PW-models

DefinitionA PW-model is a triple 〈W,R, V 〉 s.t.

〈W,R〉 is a frame,V assigns to any w∈W a set V(w) of L-sentences s.t.V (w) = {ϕ∈L: ∀v(wRv→V (v)|= ϕ)}

V (w) is the extension assigned to T by V at world w.

DefinitionI write V (w) |= ϕ iff the sentence ϕ is true when all vocabularyother than T is interpreted in the standard way and T is giventhe extension V (w).

Remarkϕ ∈ V (w) iff V (w) |= Tϕ.

PW-models and frames

DefinitionA frame 〈W,R〉 admits a valuation iff there is a V such that〈W,R, V 〉 is a PW-model.

In standard operator modal logic one can define a suitablevaluation for any given frame. This is not possible for modalpredicates.

Normality

Proposition

Assume 〈W,R, V 〉 is a PW-model. Then the following holds:For all w ∈W :V (w) |= Tϕ→ ψ → (Tϕ→ Tψ)

If V (w) |= ϕ for all w ∈W , then V (w) |= Tϕ for all w ∈W .

RemarkIf ϕ is a true sentence without T , then V (w) |= ϕ holds for allw ∈W of a PW-model 〈W,R, V 〉.

Truth = necessity with one accessible world

Imagine w sees only one other world u. Then Tϕ is true in w iffϕ is true in u.

LemmaIf there is exactly one u such that wRu, then

V (w) |= Tϕ iff V (u) |= ϕ

and w is a ‘metaworld’ of u.

The step from V (u) to V (w) corresponds to an application ofthe rule of revision in rule-of-revision semantics in the sense of[11], [18], [2].

If u = w in the above lemma we would have

V (w) |= Tϕ iff V (w) |= ϕ

and thus V (w) |= Tϕ↔ ϕ, because we are in classical logic.

The liar in PWS

Proposition (liar paradox)

The following frame (‘one world sees itself’) does not admit avaluation:

•��

Proof.Use the liar fixed point ϕ↔ ¬Tϕ. a

This is just a special case of a more general observation:

Montague in PWS

Proposition (Montague)

If 〈W,R〉 admits a valuation, then 〈W,R〉 is not reflexive.

Proof.Assume 〈W,R, V 〉 is a PW-model with R reflexive. ϕ is the liarsentence, i.e., PA ` ϕ↔ ¬Tϕ. For arbitrary w ∈W we have:

V (w) |=Tϕ→ ¬ϕ(26)V (w) |=Tϕ→ ϕ(27)V (w) |=¬Tϕ(28)V (w) |=ϕ(29)

By normality V (w) |=Tϕ. Contradiction! a

Montague’s paradox is more general than the liar paradox. Butthere are even more general results.

‘two worlds see one other’

Proposition

The following frame (‘two worlds see one other’) does not admit avaluation:

• ** •jj

Proof.

fixed point ϕ↔ ¬TTϕ. a

Proposition

The following frame does not admit a valuation:

• ++ •jj

•

??__

Proof.

ϕ↔ ¬TTϕ ∧ ¬Tϕ a

Proposition

The following frame does not admit a valuation:

•99 // •

Proof.

w144// w2

Assume that this frame admits a valuation V .By diagonalization there is a sentence ϕ such that

PA` ϕ↔ (Tϕ→ T¬ϕ)

Therefore V (w) |= ϕ↔ (Tϕ→ T¬ϕ) for all worlds w ofPW-models 〈W,R, V 〉.1. Case: V (w1) |= ϕ

��1ϕ, 2¬T¬ϕ3Tϕ→ T¬ϕ

4¬Tϕ//

5¬ϕ6¬(Tϕ→ T¬ϕ)

7¬T¬ϕ

2. Case: V (w1) |= ¬ϕ

¬ϕ, Tϕ��

// w2

a

Gödel’s second theorem

Proposition (Gödel)

In a transitive frame every world must either see a dead end or be adead end.

Gödel’s second theorem says that either PA is inconsistent orthe consistency of PA is not provable, ie:

Bew⊥ ∨ ¬BewCon

which is Bew⊥ ∨ ¬Bew¬Bew⊥Proof.

V (w) |= T⊥ ∨ ¬T¬T⊥ a

McGee’s theorem

Proposition (McGee)

The frames 〈ω, <〉 and 〈ω, Pre〉 do not admit valuations.

0

��1

��2

��...

Proof.McGee’s fixed point. a

The Characterization Problem

Which frames admit valuations?

We know already, e.g., that reflexive frames and transitiveframes without dead ends do not admit valuations.

An answer to this question would generalize all mentionedresults on the paradoxes.

RemarkIn operator modal logic the analogous question does not arisebecause models can be built on every frame.

Converse wellfoundedness

DefinitionA world w ∈W in converse wellfounded (in 〈W,R〉) iff everysubset of {v ∈W : wRv} has an R-maximal element(iff there is no infinite chain w1, w2, w3, . . . of worlds such thatwRw1Rw2Rw3R . . . holds.)

DefinitionA frame 〈W,R〉 is converse wellfounded iff all w ∈W areconverse wellfounded.

��γ + 1

��

V (γ + 1)(T) = V (γ)(T)

||

γ

��

V (γ)(T) =⋂α∈γ

V (α)(T)

|⋂

ω

��

V (ω)(T) =⋂n∈ω

V (n)(T)

|⋂

3

��

V (3)(T) = V (2)(T) ∩ {ϕ∈L: V (2)|= ϕ}

|⋂

2

��

V (2)(T) = V (1)(T) ∩ {ϕ∈L: V (1)|= ϕ}

|⋂

1

��

V (1)(T) = {ϕ∈L: V (0)|= ϕ}

|⋂

0 V (0)(T) = L

A sufficient condition

Definition

The depth of a world w is defined to be the rank of R−1

restricted to the converse wellfounded part of {v ∈W : wRv}.

Let γ be least so that Lγ has a transitive Σ1-end extension.

Proposition (Aczel [1])

γ is the least ordinal such such that Lγ is first order-reflecting, that is,if ϕ is any formula (with parameters from Lγ ) in the language of settheory, then

Lγ |= ϕ =⇒ ∃α<γ Lα |= ϕ.

Remark

γ is an admissible that is much larger than ωCK1 .

A sufficient condition

Theorem

Assume 〈W,R〉 is transitive and every converse illfounded world in〈W,R〉 has depth at least γ . Then 〈W,R〉 admits a valuation.

A necessary condition

DefinitionADM∗ is the class of all admissible ordinals without ω togetherwith all of its limit points.

Theorem

If 〈W,R〉 admits a valuation, then in the transitive closure 〈W,R∗〉of 〈W,R〉 every converse illfounded world in W has depth α withα ∈ ADM∗ or α ≥ γ .

This is proved via Löb’s theorem.

RemarkThere are PW-models with converse illfounded worlds of depthωCK

1 .

A necessary condition

TheoremIf 〈W,R〉 admits a valuation, then in the transitive closure 〈W,R∗〉of 〈W,R〉 every converse illfounded world in W has depth α withα ∈ ADM∗ or α ≥ γ .

This theorem implies all other limitative results mentioned sofar: the liar paradox, Montague’s paradox, McGee’s paradox,Löb’s theorem, Gödel’s second theorem, ‘two worlds see oneanother’,. . .

Transitive frames

TheoremIf a transitive frame 〈W,R〉 admits a valuation, then:

1 All worlds w with depth smaller than ωCK1 have to be converse

wellfounded.2 For worlds with depth greater or equal to γ there are no

restrictions.3 All converse illfounded worlds with depth between ωCK

1 and γmust have depth in ADM∗. Only for those worlds our resultsdon’t provide full information.

General frames

Theorem 105 cannot be generalized to non-transitive frames:

Proposition

There are frames that do not admit a valuation, although theirtransitive closure does.

Proposition

If 〈W,R〉 is converse wellfounded, then 〈W,R〉 admits a valuation.

Conclusion

Converse wellfoundedness is related to all normalpredicates, not only to provability.Possible worlds semantics can shed some light on theparadoxes, much like provability logic and its possibleworlds semantics shed some light on provability.Löb’s theorem is the real culprit and the source of allparadoxes. I.e. all paradoxes can be reduced to Löb’stheorem because Theorem 107 is proved via Löb’s theoremand all possible worlds versions of the paradoxes followfrom Theorem 107. We have reduced all known paradoxesto failures of converse wellfoundedness below ωCK

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