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Truth and Paradox

Leon HorstenUniversity of

Bristol

Truth and the Liar

Definitions,Models, Axioms

TypedDisquotationalTheories

TypedCompositionalTheories

Type-freeDisquotationalTheories

Type-freecompositionaltheories

The RevisionTheory

SelectedReferences

Truth and Paradox

Leon HorstenUniversity of Bristol

Summer School onSet Theory and Higher-Order Logic

London, 1–6 August 2011

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Structure of the Tutorial

1. Introduction: Truth and the Liar

2. Definition, Models, Axioms

3. Typed Disquotational Theories

4. Typed Compositional Theories

5. Type-free Disquotational Theories

6. Type-free Compositional Theories

7. The Revision Theory of Truth

8. Selected References

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Tarski-biconditionals

The disquotational intuition: supposing φ is tantamount tosupposing the truth of φ, and conversely.

A plausible axiom scheme for truth:

‘φ’ is true if and only if φ.

I This scheme has infinitely many instances

I Instances of this scheme are called Tarski-biconditionals.

(after the logician Alfred Tarski)

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Paradoxical sentences

The liar sentence L:

This sentence is not true.

The truth teller:

This sentence is true.

I L will be our paradigmatic test case

I but we should be aware that it is not the onlyparadoxical sentence

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Semantic paradox

The argument of the liar paradox:

Proof.It is an instance of the Tarski biconditional scheme that L istrue if and only if L. But L if and only if L is not true —forthis is what L says of itself. So L is true if and only if L isnot true: a short truth table calculation convinces us that wehave lapsed into inconsistency.

How credible is this argument?

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The framework

For simplicity, we want a toy model for English.

Language: The language of truth (LT ): the language ofarithmetic plus a truth predicate T .

Background theory: First-order Peano arithmetic (PAT ).[The truth predicate T is allowed in the induction scheme.]

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The arithmetical background

Why arithmetic?

Coding

I PA can serve as a theory of syntax (via coding);

I PA allows the formation of self-referential sentences (viacoding).

Note: We will be sloppy about notation.

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Defining truth

Tarski showed us how to give a definition of truth for aformal language in purely logical and mathematical terms.

material adequacy condition: a definition of truth fora language L should imply all the Tarski-biconditionals forsentences of L.

⇒ Metawissenschaft

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Defining first-order arithmetical truth insecond-order arithmetic

There is a first-order formula val+ that defines the atomicarithmetical truths.Consider the condition Ψ(X , φ):

φ atomic closed→ [X (φ)↔ val+(φ)] ∧∃ψ, λ : φ = ψ ∧ λ→ [X (φ)↔ X (ψ) ∧ X (λ)] ∧ . . .

Definitiontrue(φ) ≡ ∀X : ∀λΨ(X , λ)→ X (φ).

The second-order arithmetical predicate true(x) satisfiesTarski’s material adequacy condition.

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Models for truth

A theory of truth for a language L should describe a class ofintended models for L.

[This thesis was popular in the 1970s and 1980s.]

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Universality

We want a theory of truth for our language:

We want a truth theory for English in English.

Metalanguage = Objectlanguage

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Definitions and universality

Problem for the definitional approach: A truthdefinition for L (objectlanguage) can only be given in alanguage (metalanguage) L′ that is essentially richer than L:

Theorem (Undefinability theorem I)

There is no first-order arithmetical formula that definesfirst-order arithmetical truth.

But perhaps a theory of truth for English as it now is canonly be given in future English . . .

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Models and universality

A class of models for a language L is described in anessentially richer metalanguage (incompleteness theorem).

Additional problem for the model-theoreticapproach: the domain of a model is a set

The domain of discourse of English does not form a set.

But perhaps the techniques can be adapted to proper classdomains . . .

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Axioms for Truth

ThesisA theory of truth for a language L should posit rules ofinference and / or axioms for L.

The problems of the definitional and the model-theoreticapproach are not applicable.

⇒ there is no immediately apparent obstacle to our universalambitions

DefinitionLet LT be LPA plus a primitive truth predicate T .

Aim: Formulate a truth theory for LT in LT .

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The role of models and definitions

Models can have a strong heuristic force: they give pictures.

We can give truth definitions for fragments of our language.

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Soundness and completeness

Desiderata for axiomatic truth theories:

I soundness

I truth theoretic completeness

More specific desiderata will be discussed at the end of thelectures.

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The naive theory of truth

Recall the disquotational intuition.

⇒ Take all the Tarski-biconditionals as your theory of truth.

The formal theory NT :

1. PAT

2. T (φ)↔ φ for all φ ∈ LT

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Godel’s diagonal lemma

Theorem (Godel)

For each formula φ(x) ∈ LT , there is a sentence λ ∈ LT

such that PAT proves

λ↔ φ(λ).

This sentence λ is self-referential.

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Tarski’s undefinability theorem

Theorem (Undefinability theorem 2)

No consistent extension S of PAT proves T (φ)↔ φ for allφ ∈ LT .

Proof.Use the diagonal lemma to produce a (liar) sentence λ:

PAT ` λ↔ ¬T (λ).

If the theory S in question indeed proves T (φ)↔ φ for allφ ∈ LT , then in particular S proves T (λ)↔ λ. Puttingthese equivalences together, we obtain a contradiction in S:

S ` T (λ)↔ ¬T (λ).

Consequence: NT is inconsistent.

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The typed disquotational theory

Tarski’s diagnosis: the root of the disease lies inallowing the Tarski-biconditionals to regulate thetruth-conditions of sentences that themselves contain thetruth predicate (such as L).

⇒ typed truth theories

The disquotational theory DT :

DT1 PAT ;

DT2 T (φ)↔ φ for all φ ∈ LPA.

Tarski’s strictures are respected.

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Soundness

Proposition

DT has a nice model.

Proof.Consider the model

M =: 〈N, {φ | φ ∈ LPA ∧ N |= φ}〉,

i.e., the model in which as the extension of the truthpredicate we take all arithmetical truths. An induction onthe length of proofs in DT verifies that M |= DT .

Consequence: DT is arithmetically sound.

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The Tarskian hierarchy 1

Conversation:

A: It is true that 0 = 0.B: What you have just said is true.

Proposition

DT 6` T (T (0 = 0))

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The theory DT1:

1. PAT ,T1 ;

2. T (φ)↔ φ for all φ ∈ LPA;

3. T1(φ)↔ φ for all φ ∈ LT .

TheoremDT1 has a nice model.

Proposition

DT1 ` T1(T (0 = 0))

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Extended conversation:

A: It is true that 0 = 0.B: What you have just said is true.C: Yes, B, that is very true.

⇒ the Tarskian hierarchy DT , DT1, DT2,. . .

⇒ the notion of truth is irrevocably fragmented

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Contextualist theories

(Burge, Gaifman, Barwise & Perry, Glanzberg, . . . )

ThesisTruth is a uniform but indexical concept.

Sentence L is not true0.Sentence L is true1.

Because of the indexical shift in extension of the truthpredicate between context 0 and context 1, this is not acontradiction.

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The strengthened liar paradox

(S) Sentence S is not true in any context.

I this is called a strengthened liar sentence

I it is not hard to figure out that there is no context inwhich S can be coherently evaluated.

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The weakness of disquotationalism

The compositional intuition: truth distributes over thelogical connectives.

Proposition

For all φ ∈ LPA: DT ` T (φ) ∨ T (¬φ)

Proof.Already propositional logic alone proves φ ∨ ¬φ. Tworestricted Tarski-biconditionals are T (φ)↔ φ andT (¬φ)↔ ¬φ. Combining these facts yields the result.

Proposition (Tarski)

DT 0 ∀φ ∈ LPA : T (φ) ∨ T (¬φ)

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The typed compositional theory

(Davidson...)

The compositional theory TC :

TC1 PAT ;

TC2 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);

TC3 ∀φ ∈ LPA : T (¬φ)↔ ¬T (φ);

TC4 ∀φ, ψ ∈ LPA : T (φ ∧ ψ)↔ (T (φ) ∧ T (ψ));

TC5 ∀φ(x) ∈ LPA : T (∀xφ(x))↔ ∀xT (φ(x)).

TheoremTC has nice models, and DT ⊆ TC .

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Substitution

For a moment we have to be sticklers for notation...

A sentence of the form ∃xT (φ(x)) is really expressed alongthe following lines:

“There is a number x such that when the standardnumeral for x is substituted for the variable x inφ(x), a true sentence results.”

Thus a substitution function (expressible in the language ofarithmetic) appears in formulae such as TC5, but also in allthe other compositional axioms.

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True and true of

What if we do not have standard names for all elements ofthe language of discourse (R,. . . )?

⇒ we work with a satisfaction relation (‘true of’) and definethe truth predicate in terms of it.

The axiom TC5 then becomes (roughly):

∀x ,∀φ(y) : Sat(x ,∀yφ(y))↔ ∀zSat(z , φ(y)))

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Deflationism and conservativeness

Deflationism: The concept of truth does not play asubstantial role in philosophical, mathematical, scientificdebates.

DefinitionA theory of truth S is arithmetically conservative over PA iffor every sentence φ ∈ LPA, if S ` φ, then already PA ` φ.

Proposition

DT is arithmetically conservative over PA.

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The non-conservativeness of truth 1

TheoremTC ` ∀φ ∈ LPA : BewPA (φ)→ T (φ), where BewPA(...) isan arithmetical predicate that expresses provability in Peanoarithmetic in a natural way.

Proof.This is proved by an induction, inside TC , on the length ofproofs. [Here we need that T is allowed in the inductionscheme.]For the basis case, we have to prove that all the axioms oftrue. [For the subcase of mathematical induction we againneed that T is allowed in the induction scheme.]

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Corollary

TC ` ¬BewPA(0 = 1).

Proof.This follows from the previous theorem and the fact thatDT ⊆ TC by instantiating 0 = 1 for φ.

So by Godel’s second incompleteness theorem, TC is notconservative over PA.

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More precise information:

DefinitionThe second-order system ACA contains full second-orderinduction but only those instances of Comprehension

∃X∀y : X (y)↔ φ(y)

where φ(y) contains no bound second-order quantifiers (anddoes not contain X free).

TheoremThe first-order arithmetical strength of TC is exactly that ofthe second-order system ACA.

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Back to deflationism

Thesis (deflationism)

The concept of truth is not substantial but provides extraconceptual power.

⇒ An axiomatic theory of truth must be conservative butnoninterpretable.

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A deflationist truth theory 1

Fischer proposed a minimally adequate truth theory PT−

Definitiontot(φ(x)) ≡ ∀y : T (φ(y) ∨ T (¬φ(y))

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PT-1 PAT without the induction axiom;

PT-2 ∀φ(x) ∈ LT : [tot(φ(x)) ∧ T (φ(0)) ∧ ∀y(T (φ(y))→T (φ(y + 1))]→ ∀xT (φ(x));

PT-3 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);

PT-4 ∀ atomic φ ∈ LPA : T (¬φ)↔ ¬val+(φ);

PT-5 ∀φ, ψ ∈ LPA : T (φ ∧ ψ)↔ (T (φ) ∧ T (ψ));

PT-6 ∀φ, ψ ∈ LPA : T (¬(φ ∧ ψ))↔ (T (¬φ) ∨ T (¬ψ));

PT-7 ∀φ(x) ∈ LPA : T (∀xφ(x))↔ ∀xT (φ(x));

PT-8 ∀φ(x) ∈ LPA : T (¬∀xφ(x))↔ ∃xT (¬φ(x));

PT-9 ∀φ ∈ LPA : T (¬¬φ)↔ T (φ).

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Observation: PT− is still highly compositional, but not ascompositional as TC .

TheoremPT− is conservative over PA but is not interpretable in PA.

PT− seems sufficient for capturing the technical use of theconcept of truth that is made in mathematics (such asmodel theory).

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Type-free truth

I Truth theories which are formulated in languages thatcontain truth predicates of different levels and whichprove iterated truth ascriptions only if the hierarchyconstraints are satisfied, are called typed theories oftruth.

I There also exist truth systems which contain a singletruth predicate but which do validate sentences of theform T (T (0 = 0)). These systems are called type-freetheories of truth (reflexive, semantically closed theories).

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The strength of type-free disquotation

Theorem (McGee)

Any theory extending PA can be reaxiomatized by theaxioms of PA and a set of Tarski-biconditionals.

Proof.Consider an axiom ψ of S . Using the diagonal lemma, wecan find a sentence such that

λ↔ (T (λ)↔ ψ)

is provable in PA. This equivalence is logically equivalent to:

ψ ↔ (T (λ)↔ λ).

So ψ is PA-provably equivalent to the Tarski-biconditionalT (λ)↔ λ.

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Positive disquotation

(Halbach)

The theory PUTB is the theory given by PAT and the set ofall sentences

∀x (T (φ(x))↔ φ(x)),

where in the formula φ(x), T must not occur in the scope ofan odd number of negation symbols in the formula φ.

TheoremPUTB ` T (T (0 = 0))

But: PUTB 6` ∀φ ∈ LPA : T (¬φ)↔ ¬T (φ)

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Going partial

(Kripke)

Structure of the argument of the liar paradox:

1. L ∨ ¬L

2. L⇒ ⊥

3. ¬L⇒ ⊥

So, ⊥

Moral: Do not assert the law of excluded third.

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Partial models

I The aim is build a model for the language LT in stages

I The arithmetical vocabulary is interpreted throughoutas in the standard model N.

I The truth predicate T will be the only partiallyinterpreted symbol: it will receive, at each ordinal stage,an extension E and an anti-extension A.

M = (E ,A)

Note:

1. E ∩ A = ∅2. E ∪ A does not typically exhaust the domain, for

otherwise T would be a total predicate.

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A chain of partial models 1

M0 = (E0,A0) =: (∅, ∅) ;

Eα+1 =: {φ ∈ LT |Mα |=sk φ} and

Aα+1 =: {φ ∈ LT |Mα |=sk ¬φ} ;

For λ limit ordinal, we set:

Eλ =:⋃κ<λ

Eκ,

Aλ =:⋃κ<λ

Aκ.

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A chain of partial models 2

This inductive definition gives rise to a transfinite chain ofpartial models:

(E0,A0), (E1,A1), . . . , (Eω,Aω), . . .

Theorem (monotonicity)

For any two partial models (Ea,Aa), (Eb,Ab), if Ea ⊆ Eb andAa ⊆ Ab, then

{φ | (Ea,Aa) |=sk φ} ⊆ {φ | (Eb,Ab) |=sk φ}.

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The minimal fixed point model

A consequence of this is that:

Corollary

For all α, β with α < β, we have:

{φ | (Eα,Aα) |=sk φ} ⊆ {φ | (Eβ,Aβ) |=sk φ}.

Proposition

For some ordinal ρ, Eρ = Eρ+1 and Aρ = Aρ+1.

This ordinal ρ is called the (Strong Kleene) minimal fixedpoint, and (Eρ,Aρ) is called the minimal fixed point model.

Theorem (Kripke)

The minimal Strong Kleene fixed point is a Π11-complete set.

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Almost having it all

The minimal fixed point almost makes the unrestrictedTarski-biconditionals true:

TheoremFor all sentences φ in LT :

Mρ |=sk φ⇔Mρ |=sk T (φ).

Proof.First, suppose Mρ |=sk φ. Then by the definition of thesequence of partial models, Mρ+1 |=sk T (φ). But since Mρ

is a fixed point, we have Mρ+1 = Mρ. So Mρ |=sk T (φ).Second, suppose Mρ |=sk T (φ). Then there must be anordinal α < ρ such that Mα |=sk φ. And therefore, bymonotonicity, Mρ |=sk φ.

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Paradoxical sentences

DefinitionClosing off a partial model means letting its anti-extensionspill over into all of the complement of its extension.

TheoremMρ 2sk T (L) and Mρ 2sk ¬T (L).

Proof.By closing off the model Mρ we obtain a classical modelMc

ρ. Suppose that Mρ |=sk T (L), and thereby Mρ |=sk L.Then by monotonicity, also Mc

ρ |=sk T (L) and Mcρ |=sk L.

But since Mcρ is just a classical model, the diagonal lemma

holds in it. So we have

Mcρ |=sk L↔ ¬T (L).

But putting these three facts together gives us acontradiction. So we deny our supposition and conclude thatMρ 2sk T (L).

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Other valuation schemes

So far we have been using the Strong Kleene valuationscheme...

I Weak Kleene: “if a component is gappy, then the wholeis gappy.” (Feferman)

I Paraconsistent: “Truth gluts instead of truth gaps ”(Priest)

I . . .

Formally, it does not make a lot of difference . . .

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Supervaluation

In the supervaluation approach, a formula φ ∈ LT isregarded as true in a partial model M = (E ,A) if and only ifφ is true in all total (or classical) models Mc = 〈N, C〉 forwhich the interpretation C of the truth predicate is such thatE ⊆ C and A ⊆ N\C.

Similarly, we say that a formula φ ∈ LT is regarded as falsein a partial model M = (E ,A) if and only if φ is false in alltotal (or classical) models Mc = 〈N, C〉 for which theinterpretation C of the truth predicate is such that E ⊆ Cand A ⊆ N\C.

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Properties of supervaluation fixed points

The laws of classical logic are supervaluation-true in theminimal fixed point of the supervaluation scheme.

The supervaluation fixed point is not compositional.

TheoremThe supervaluation fixed point is a complete Π1

1 set.

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Axiomatising Strong Kleene Fixed Points

The theory KF :

KF1 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);

KF2 ∀ atomic φ ∈ LPA : T (¬φ)↔ val−(φ);

KF3 ∀φ ∈ LT : T (¬¬φ)↔ T (φ);

KF4 ∀φ, ψ ∈ LT : T (φ ∧ ψ)↔ (T (φ) ∧ T (ψ));

KF5 ∀φ, ψ ∈ LT : T (¬ (φ ∧ ψ))↔ (T (¬φ) ∨ T (¬ψ));

KF6 ∀φ (x) ∈ LT : T (∀xφ (x))↔ ∀yT (φ (y));

KF7 ∀φ (x) ∈ LT : T (¬∀xφ (x))↔ ∃yT (¬φ (y));

KF8 ∀φ ∈ LT : T (T (φ))↔ T (φ);

KF9 ∀φ ∈ LT : T (¬T (φ))↔ T (¬φ);

KF10 ∀φ ∈ LT : ¬(T (φ) ∧ T (¬φ)).

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Inner logic and outer logic

TheoremKF has nice models.

Proof.The closed off minimal fixed point model.

DefinitionThe inner logic of KF is the collection of sentences φ ∈ LT

such that KF ` T (φ).

Proposition

KF ` L ∧ ¬T (L), where L is the liar sentence.

So the inner logic of KF does not coincide with the externallogic of KF .

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Ramified Analysis

Let us add new second-order variables X 1,Y 1, . . . to thelanguage of second-order arithmetic. Add a newcomprehension principle

∃X 1∀y : X 1(y)↔ φ(y)

where φ(y) contains no bound second-order quantifiers ofthe new kind (and does not contain X 1 free).

The resulting system is called ACA1.

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The strength of KF

We can go on, and construct ACAn, . . . , ACA<ω...

Definitionε0 is the least upper bound of {ω, ωω, ωωω

, . . .}.

Theorem (Feferman)

The first order arithmetical theorems of KF are exactly thoseof ACAε0 .

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Truth and Classes

Definitionx ∈ y ≡ y ∈ LT ∧ T (y(x))

Then KF can be seen as a theory of definable or predicativeclasses (Feferman).

⇒ KF as a way of ascending from PA to predicative analysis.

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A way out?

What is the source of the divergence between inner andouter logic?

Motivation of Kripke’s theory = partial

Logic of KF = classical

⇒ Mismatch

Solution: Formalise Kripke’s theory in partial logic!

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Logic and mathematics

Restricted conditionalisation:

T (φ) ∨ T (¬φ) φ (Hyp)...ψ (Hyp)

φ→ ψ

Rule of induction:

φ(0) φ(x) (Hyp)...

φ(s(x)) (Hyp)

∀xφ(x)

Here φ(x) ranges over all formulae of LT .

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The system PKF

PKF1val+(t1 = t2)

T (t1 = t2)

T (t1 = t2)

val+(t1 = t2)

PKF2T (φ) ∧ T (ψ)

T (φ ∧ ψ)

T (φ ∧ ψ)

T (φ) ∧ T (ψ)

PKF3T (φ) ∨ T (ψ)

T (φ ∨ ψ)

T (φ ∨ ψ)

T (φ) ∨ T (ψ)

PKF4∀xT (φ(x))

T (∀xφ(x))

T (∀xφ(x))

∀xT (φ(x))

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The system PKF Ctd

PKF5∃xT (φ(x))

T (∃xφ(x))

T (∃xφ(x))

∃xT (φ(x))

PKF6T (φ)

T (T (φ))

T (T (φ))

T (φ)

PKF7¬T (φ)

T (¬φ)

T (¬φ)

¬T (φ)

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Properties of PKF

TheoremInner logic of PKF = Outer logic of PKF

TheoremPKF holds in all fixed point models of the Strong Kleeneconstruction.

TheoremThe first-order arithmetical theorems of PKF are those ofACAωω .

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A chain of classical models

M0 =: 〈N, ∅〉

Mα+1 =: 〈N, {φ ∈ LT |Mα |= φ}〉.

For λ a limit ordinal:

Mλ =: 〈N, {φ ∈ LT | ∃β∀γ : (γ ≥ β ∧ γ < λ)⇒Mγ |= φ}〉.

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Stable truth

Stable truth:A sentence φ ∈ LT is said to be stably true if at someordinal stage α, φ enters in the extension of the truthpredicate of Mα and stays in the extension of the truthpredicate in all later models.

Stable falsehood:A sentence φ ∈ LT is said to be stably false if at someordinal stage α, φ is outside the extension of the truthpredicate of Mα and stays out forever thereafter.

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Nearly stable truth

Nearly stable truth:A sentence φ ∈ LT is said to be nearly stably true if forevery stage α after some stage β, there is a natural numbern such that for all natural numbers m ≥ n, φ is in theextension of the truth predicate of Mα+m.

Nearly stable falsehood:A sentence φ ∈ LT is said to be nearly stably false if forevery stage α after some stage β, there is a natural numbern such that for all natural numbers m ≥ n, φ is outside theextension of the truth predicate of Mα+m.

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Properties of the (nearly) stable truths

The liar sentence is neither (nearly) stably true nor (nearly)stably false.

Proposition

The chain of revision models is eventually periodic.

TheoremThe (nearly) stable truths form a set that is morecomplicated than a complete Π1

1 set.

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The Friedman-Sheard theory 1

The theory FS :

FS1 PAT ;

FS2 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);

FS3 ∀φ ∈ LT : T (¬φ)↔ ¬T (φ);

FS4 ∀φ, ψ ∈ LT : T (φ ∧ ψ)↔ T (φ) ∧ T (ψ);

FS5 ∀φ(x) ∈ LT : T (∀xφ(x))↔ ∀xT (φ(x)).

NEC From a proof of φ, infer T (φ);

CNEC From a proof of T (φ), infer φ.

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The Friedman-Sheard theory 2

Proposition

Inner logic of FS = outer logic of FS

Proposition

FS is not stably true.

Proof.FS3 is not stably true: both the liar sentence and itsnegation are false at all limit ordinals

Proposition

FS is nearly stably true.

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The strength of FS

FS only proves finite truth-iterations.

TheoremThe first-order arithmetical consequences of FS are exactlythose of ACAω.

Corollary

FS is arithmetically sound.

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Omega-inconsistency

Even though FS is consistent and indeed even arithmeticallysound, it is in some sense “almost inconsistent”:

DefinitionAn arithmetical theory T is ω-inconsistent if for someformula φ(x), the theory T proves ∃xφ(x) while at the sametime for every n ∈ N, T proves ¬φ(n)

Theorem (McGee)

For some formula φ (x) ∈ LT : FS ` ∃xφ (x) andFS ` ¬φ (n) for all n ∈ N.

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A fork in the road

DT

TC

FS-like KF-like

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Desiderata for axiomatic truth theories

I Coherence

I Tarski-biconditionals

I Compositionality

I Sustaining ordinary reasoning

I Strength

I Capturing a picture

Fulfillment of these norms is a matter of degree.

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References

I Cantini, A. Logical Frameworks for Truth andAbstraction. North-Holland, 1996.

I Feferman, S. Reflecting on Incompleteness. Journal ofSymbolic Logic 56(1991), p. 1–49.

I Gupta, A. & Belnap, N. The Revision Theory of Truth.MIT Press, 1993.

I Halbach, V. Axiomatic Theories of Truth. CambridgeUniversity Press, 2011.

I Horsten L. The Tarskian Turn. Deflationism andaxiomatic truth. MIT Press, 2011.

I McGee, V. Truth, Vagueness and Paradox. An essay onthe logic of truth. Hackett, 1991.

I Visser, A. Semantics and the Liar Paradox. In: D.Gabbay et al (eds) Handbook of Philosophical Logic.Volume 4. Reidel, 1984, p. 617–706.

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