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Stanford Encyclopedia ofPhilosophy

The Revision Theory of Truth First published Fri Dec 15, 1995; substantive revision Fri Jul 28, 2006

Consider the following sentence:

(1) is not true. (1)

It has long been known that the sentence, (1), produces a paradox, the so-called liar's paradox: it

seems impossible consistently to maintain that (1) is true, and impossible consistently to maintain

that (1) is not true. (For details, see Section 1, below.) Given such a paradox, one might besceptical of the notion of truth, or at least of the prospects of giving a scientifically respectable

account of truth. Alfred Tarski's great accomplishment was to show how to give — contra this

scepticism — a formal definition of truth for a wide class of formalized languages. Tarski did not ,

however, show how to give a definition of truth for languages (such as English) that contain their

own truth predicates. He thought that this could not be done, precisely because of the liar's

paradox. He reckoned that any language with its own truth predicate would be inconsistent, as

long as it obeyed the rules of standard classical logic, and had the ability to refer to its own

sentences.

Given the close connection between meaning and truth, it is widely held that any semantics for a

language L, i.e., any theory of meaning for L, will be closely related to a theory of truth for L:indeed, it is commonly held that something like a Tarskian theory of truth for L will be a central

part of a semantics for L. Thus, the impossibility of giving a Tarskian theory of truth for languages

with their own truth predicates threatens the project of giving a semantics for languages with their

own truth predicates.

We had to wait until the work of Kripke 1975 and of Martin & Woodruff 1975 for a systematic

formal proposal of a semantics for languages with their own truth predicates. The basic thought is

simple: take the offending sentences, such as (1), to be neither true nor false. Kripke, in particular,

shows how to implement this thought for a wide variety of languages, in effect employing a

semantics with three values, true, false and neither .[1] It is safe to say that Kripkean approaches

have replaced Tarskian pessimism as the new orthodoxy concerning languages with their owntruth predicates.

One of the main rivals to the three-valued semantics is the Revision Theory of Truth, or RTT,

independently conceived by Hans Herzberger and Anil Gupta, and first presented in publication in

Herzberger 1982a and 1982b, Gupta 1982 and Belnap 1982 — the first monographs on the topic

are Yaqū b 1993 and the locus classicus, Gupta & Belnap 1993. The RTT is designed to model the

kind of reasoning that the liar sentence leads to, within a two-valued context . The central idea is

the idea of a revision process: a process by which we revise hypotheses about the truth-value of

one or more sentences. The present article's purpose is to outline the Revision Theory of Truth.

We proceed as follows:

• 1. Semiformal introduction

• 2. Framing the problem

of 23The Revision Theory of Truth (Stanford Encyclopedia of Philosophy)

23.09.2014http://plato.stanford.edu/entries/truth-revision/



◦ 2.1 Truth languages

◦ 2.2 Ground models

◦ 2.3 The liar's paradox (again)

• 3. Basic notions of the RTT

◦ 3.1 Revision rules

◦ 3.2 Revision sequences

• 4. Interpreting the formalism◦ 4.1 The signification of T

◦ 4.2 The ‘iff’ in the T-biconditionals

◦ 4.3 The paradoxical reasoning

◦ 4.4 The signification thesis

◦ 4.5 The supervenience of semantics

◦ 4.6 Yaqū b's interpretation of the formalism

• 5. Further issues

◦ 5.1 Three-valued semantics

◦ 5.2 Amendments to the RTT

◦ 5.3 Revision theory for circularly defined concepts

◦ 5.5 Applications◦ 5.5 An open question

• Bibliography

• Academic Tools

• Other Internet Resources

• Related Entries

1. Semiformal introduction

Let's take a closer look at the sentence (1), given above:


It will be useful to make the paradoxical reasoning explicit. First, suppose that


It seems an intuitive principle concerning truth that, for any sentence p, we have the so-called

T-biconditional

‘ p’ is true iff p. (3)

(Here we are using ‘iff’ as an abbreviation for ‘if and only if’.) In particular, we should have

‘(1) is not true’ is true iff (1) is not true. (4)

Thus, from (2) and (4), we get

‘(1) is not true’ is true. (5)

Then we can apply the identity,

(1) = ‘(1) is not true.’ (6)





to conclude that (1) is true. This all shows that if (1) is not true, then (1) is true. Similarly, we can

also argue that if (1) is true then (1) is not true. So (1) seems to be both true and not true: hence the

paradox. As stated above, the three-valued approach to the paradox takes the liar sentence, (1), to

be neither true nor false. Exactly how, or even whether, this move blocks the above reasoning is a

matter for debate. The RTT is not designed to block reasoning of the above kind, but to model it-

or most of it.[2] As stated above, the central idea is the idea of a revision process: a process by

which we revise hypotheses about the truth-value of one or more sentences.

Consider the reasoning regarding the liar sentence, (1) above. Suppose that we hypothesize that (1)

is not true. Then, with an application of the relevant T-biconditional, we might revise our

hypothesis as follows:

Hypothesis: (1) is not true.

T-biconditional: ‘(1) is not true’ is true iff (1) is not true.

Therefore: ‘(1) is not true’ is true.

Known identity: (1) = ‘(1) is not true’.

Conclusion: (1) is true. New revised hypothesis: (1) is true.

We could continue the revision process, by revising our hypothesis once again, as follows:

New hypothesis: (1) is true.

T-biconditional: ‘(1) is not true’ is true iff (1) is not true.

Therefore: ‘(1) is not true’ is not true.

Known identity: (1) = ‘(1) is not true’.

Conclusion: (1) is not true.

New new revised hypothesis: (1) is not true.

As the revision process continues, we flip back and forth between taking the liar sentence to be

true and not true.

Example 1.1

It is worth seeing how this kind of revision reasoning works in a case with several

sentences. Let's apply the revision idea to the following three sentences:

(8) is true or (9) is true. (7)

(7) is true. (8)(7) is not true. (9)

Informally, we might reason as follows. Either (7) is true or (7) is not true. Thus,

either (8) is true or (9) is true. Thus, (7) is true. Thus (8) is true and (9) is not true, and

(7) is still true. Iterating the process once again, we once again get (8) is true, (9) is

not true, and (7) is true. More formally, consider any initial hypothesis, h0, about the

truth values of (7), (8) and (9). Either h0 says that (7) is true or h0 says that (7) is not

true. In either case, we can use the T-biconditional to generate our revised hypothesis

h1: if h0 says that (7) is true, then h1 says that ‘(7) is true’ is true, i.e. that (8) is true;

and if h0 says that (7) is not true, then h1 says that ‘(7) is not true’ is true, i.e. that (9)

is true. So h1 says that either (8) is true or (9) is true. So h2 says that ‘(8) is true or (9)

is true’ is true. In other words, h2 says that (7) is true. So no matter what hypothesis h0

we start with, two iterations of the revision process lead to a hypothesis that (7) is





true. Similarly, three or more iterations of the revision process, lead to the hypothesis

that (7) is true, (8) is true and (9) is false — regardless of our initial hypothesis. In

Section 3, we will reconsider this example in a more formal context.

One thing to note is that, in Example 1.1, the revision process yields stable truth values for all

three sentences. The notion of a sentence stably true in all revision sequences will be a central

notion for the RTT. The revision-theoretic treatment contrasts, in this case, with the three-valued

approach: on most ways of implementing the three-valued idea, all three sentences, (7), (8) and

(9), turn out to be neither true nor false.[3] In this case, the RTT arguably better captures the

correct informal reasoning than does the three-valued approach: the RTT assigns to the sentences

(7), (8) and (9) the truth-values that were assigned to them by the informal reasoning given at the

beginning of the example.

2. Framing the problem

2.1 Truth languages

The goal of the RTT is to give an account of our often unstable and often paradoxical reasoning

about truth — a two-valued account that assigns to sentences stable classical truth values when

intuitive reasoning would assign stable classical truth values. We will present a formal semantics

for a formal language: we want that language to have both a truth predicate and the resources to

refer to its own sentences.

Let us consider a first-order language L, with connective& , ∨, and ¬, quantifiers ∀ and ∃, theequals sign =, variables, and some stock of names, function symbols and relation symbols. We

will say that L is a truth language, if it has a distinguished predicate T and quotation marks ‘ and ’,

which will be used to form quote names: if A is a sentence of L, then ‘ A’ is a name. Let Sent L =

{ A : A is a sentence of L}.

2.2 Ground models

Other than the truth predicate, we will assume that our language is interpreted completely

classically. So we will represent the T -free fragment of a truth language L by a ground model , i.e.,

a classical interpretation of the T -free fragment of L. By the T -free fragment of L, we mean the

first-order language L− that has the same names, function symbols and relation symbols as L,

except the unary predicate T . Since L− has the same names as L, including the same quote names,

L− will have a quote name ‘ A’ for every sentence A of L. Thus∀ xT x is not a sentence of L−, but

‘∀ xT x’ is a name of L−

and∀ x( x = ‘∀ xT x’) is a sentence of L−

. Given a ground model, we willconsider the prospects of providing a satisfying interpretation of T . The most obvious desideratum

is that the ground model, expanded to include an interpretation of T , satisfy Tarski's

T-biconditionals, i.e., the biconditionals of the form

T ‘ A’ iff A

for each A ∈ Sent L. To make things precise, let a ground model for L be a classical model M = <

D, I > for the T -free fragment of L, satisfying the following:

1. D is a nonempty domain of discourse;

2. I is a function assigninga. to each name of L a member of D;

b. to each n-ary function symbol of L a function from Dn to D; and





c. to each n-ary relation symbol, other than T , of L a function from Dn to one of the two

truth-values in the set {t, f };[4]

3. Sent L ∈ D; and

4. I (‘ A’) = A for every A ∈ Sent L.

Clauses (1) and (2) simply specify what it is for M to be a classical model of the T -free fragment

of L. Clauses (3) and (4) ensure that L, when interpreted, can talk about its own sentences. Given aground model M for L and a name, function symbol or relation symbol X , we can think of I ( X ) as

the interpretation or, to borrow a term from Gupta and Belnap, the signification of X . Gupta and

Belnap characterize an expression's or concept's signification in a world w as “an abstract

something that carries all the information about all the expression's [or concept's] extensional

relations in w.” If we want to interpret T x as ‘ x is true’, then, given a ground model M , we would

like to find an appropriate signification, or an appropriate range of significations, for T .

2.3 The liar's paradox (again)

We might try to assign to T a classical signification, by expanding M to a classical model M ′ = <

D′, I ′ > for all of L, including T . Recall that we want M′ to satisfy the T-biconditionals: the most

obvious thought here is to understand the ‘iff’ as the standard truth-conditional biconditional.

Unfortunately, not every ground model M = < D, I > can be expanded to such an M ′. Consider a

truth language L with a name λ , and a ground model M =< D, I > such that I (λ ) = ¬T λ . And

suppose that M ′ is a classical expansion of M to all of L. Since M ′ is an expansion of M , I and I ′

agree on all the names of L. So

I ′(λ ) = I (λ ) = ¬T λ = I (‘¬T λ ’) = I ′(‘¬T λ ’).

So the sentences T λ and T ‘¬T λ ’ have the same truth value in M ′. So the T-biconditional

T ‘¬T λ ’ ≡ ¬T λ

is false in M ′. This is a formalization of the liar's paradox, with the sentence ¬T λ as the offending

liar's sentence.

In a semantics for languages capable of expressing their own truth concepts, T will not, in general,

have a classical signification; and the ‘iff’ in the T-biconditionals will not be read as the classical

biconditional. We take these suggestions up in Section 4, below.

3. Basic notions of the RTT

3.1 Revision rules

In Section 1, we informally sketched the central thought of the RTT, namely, that we can use the

T-biconditionals to generate a revision rule — a rule for revising a hypothesis about the extension

of the truth predicate. Here we will formalize this notion, and work through an example from

Section 1.

In general, let L be a truth language and M be a ground model for L. An hypothesis is a function

h : D → {t, f }. A hypothesis will in effect be a hypothesized classical interpretation for T . Let's

work with an example that captures both the liar's paradox and Example 1.1 from Section 1. We

will state the example formally, but reason in a semiformal way, to transition from onehypothesized extension of T to another.





Example 3.1

Suppose that L contains four non-quote names, α, β, γ and λ and no predicates other

than T . Also suppose that M = < D, I > is as follows:

D = Sent L

I (α) = T

β ∨ T

γ I (β) = T α

I (γ) = ¬T α

I (λ ) = ¬T λ

It will be convenient to let

A be the sentence T β ∨ T γ

B be the sentence T αC be the sentence ¬T α

X be the sentence ¬T λ

Thus:

D = Sent L

I (α) = A

I (β) = B

I (γ) = C

I (λ ) = X

Suppose that the hypothesis h0 hypothesizes that A is false, B is true, C is false and X

is true. Thus

h0( A) = f

h0( B) = t

h0(C ) = f

h0( X ) = f

Now we will engage in some semiformal reasoning, on the basis of hypothesis h0.

Among the four sentences, A, B, C and X , h0 puts only B in the extension of T . Thus,

reasoning from h0, we conclude that

¬T α since the referent of α is not in the extension of T

T β since the referent of β is in the extension of T

¬T γ since the referent of γ is not in the extension of T

¬T λ since the referent of λ is not in the extension of T .





The T-biconditional for the four sentence A, B, C and X are as follows:

(T A) A is true iff T β ∨ T γ

(T B) B is true iff T α

(TC ) C is true iff ¬T α

(T X ) X is true iff ¬T λ

Thus, reasoning from h0, we conclude that

A is true

B is not true

C is true

X is true

This produces our new hypothesis h1:

h1( A) = t

h1( B) = f

h1(C ) = t

h1( X ) = t

Let's revise our hypothesis once again. So now we will engage in some semiformal

reasoning, on the basis of hypothesis h1. Hypothesis h1 puts A, C and X , but not B, inthe extension of the T . Thus, reasoning from h1, we conclude that

T α since the referent of a is in the extension of T

¬T β since the referent of β is in the extension of T

T γ since the referent of γ is not in the extension of T

T λ since the referent of λ is not in the extension of T

Recall the T-biconditional for the four sentence A, B, C and X , given above.

Reasoning from h1 and these T-biconditionals, we conclude that

A is true

B is true

C is not true

X is not true

This produces our new new hypothesis h2:

h2( A) = t

h2( B) = t





h2(C ) = f

h2( X ) = f

□

Let's formalize the semiformal reasoning carried out in Example 3.1. First we hypothesized that

certain sentences were, or were not, in the extension of T . Consider ordinary classical modeltheory. Suppose that our language has a predicate G and a name a, and that we have a model M =

< D, I > which places the referent of a inside the extension of G:

I (G)( I (α)) = t

Then we conclude, classically, that the sentence Ga is true in M . It will be useful to have some

notation for the classical truth value of a sentence S in a classical model M . We will write Val M (S ).

In this case, Val M (Ga) = t. In Example 3.1, we did not start with a classical model of the whole

language L, but only a classical model of the T -free fragment of L. But then we added a

hypothesis, in order to get a classical model of all of L. Let's use the notation M + h for the

classical model of all of L that you get when you extend M by assigning T an extension via the

hypothesis h. Once you have assigned an extension to the predicate T , you can calculate the truth

values of the various sentences of L. That is, for each sentence S of L, we can calculate

Val M + h(S )

In Example 3.1, we started with hypothesis h0 as follows:

h0( A) = f

h0( B) = t

h0(C ) = f

h0( X ) = f

Then we calculated as follows:

Val M +h0(T α) = f

Val M +h0(T β) = t

Val M +h0(T γ) = f

Val M +h0(T λ ) = f

And then we concluded as follows:

Val M +h0( A) = Val M +h0

(T β ∨ T γ) = t

Val M +h0( B) = Val M +h0

(¬T α) = f

Val M +h0(C ) = Val M +h0

(T α) = t

Val M +h0( X ) = Val M +h0

(¬T λ ) = t

These conclusions generated our new hypothesis, h1:





h1( A) = t

h1( B) = f

h1(C ) = t

h1( X ) = t

Note that, in general,

h1(S ) = Val M +h0(S ).

We are now prepared to define the revision rule given by a ground model M = < D, I >. In general,

given an hypothesis h, let M + h = < D, I ′ > be the model of L which agrees with M on the T -free

fragment of L, and which is such that I ′(T ) = h. So M + h is just a classical model for all of L. For

any model M + h of all of L and any sentence A if L, let Val M +h( A) be the ordinary classical truth

value of A in M + h.

Definition 3.2

Suppose that L is a truth language and that M = < D, I > is a ground model for L. The

revision rule, τ M , is the function mapping hypotheses to hypotheses, as follows:

τ M (h)(d ) ={ t, if d ∈ D is a sentence of L and Val M +h(d ) = t

f , otherwise

The ‘otherwise’ clause tells us that if d is not a sentence of L, then, after one application of

revision, we stick with the hypothesis that d is not true.[5] Note that, in Example 3.1, h1 = τ M (h0)

and h2 = τ M (h1). We will often drop the subscripted ‘ M ’ when the context make it clear which

ground model is at issue.

3.2 Revision sequences

Let's pick up Example 3.1 and see what happens when we iterate the application of the revision

rule.

Example 3.3 (Example 3.2 continued)

Recall that L contains four non-quote names, α, β, γ and λ and no predicates other

than T . Also recall that M = < D, I > is as follows:

D = Sent L

I (α) = A = T β ∨ T γ

I (β) = B = T α

I (γ) = C = ¬T α

I (λ ) = X = ¬T λ

The following table indicates what happens with repeated applications of the revision rule τ M to

the hypothesis h0 from Example 3.1. In this table, we will write τ instead of τ M :

S h0(S ) τ(h0)(S ) τ2(h0)(S ) τ3(h0)(S ) τ4(h0)(S ) …

A f t t t t …





B t f t t t …

C f t f f f …

X f t f t f …

So h0 generates a revision sequence (see Definition 3.7, below). And A and B are stably true in

that revision sequence (see Definition 3.6, below), while C is stably false. The liar sentence X is,unsurprisingly, neither stably true nor stably false: the liar sentence is unstable. A similar

calculation would show that A is stably true, regardless of the initial hypothesis: thus A is

categorically true (see Definition 3.8).

Before giving a precise definition of a revision sequence, we give an example where we would

want to carry the revision process beyond the finite stages, h, τ1(h), τ2(h), τ3(h), and so on.

Example 3.4

Suppose that L contains nonquote names α0, α1, α2, α3, …, and unary predicates G and

T . Now we will specify a ground model M = < D, I > where the name α0 refers to

some tautology, and where

the name α1 refers to the sentence T α0

the name α2 refers to the sentence T α1

the name a3 refers to the sentence T a2

…

More formally, let A0 be the sentence T α0 ∨ ¬T α0, and for each n ≥ 0, let An+1 be the

sentence T αn. Thus A1 is the sentence T α0, and A2 is the sentence T α1, and A3 is the

sentence T α2, and so on. Our ground model M = < D, I > is as follows:

D = Sent L

I (αn) = An

I (G)( A) = t iff A = An for some n

Thus, the extension of G is the following set of sentences: { A0, A1, A2, A3, … } =

{(T α0 ∨ ¬T α0), T α0, T a1, T a2, T a3, … }. Finally let B be the sentence∀ x(Gx ⊃ T x).

Let h be any hypothesis for which we have, for each natural number n,

h( An) = h( B) = f .

The following table indicates what happens with repeated applications of the revision

rule τ M to the hypothesis h. In this table, we will write τ instead of τ M :

S h(S ) t (h)(S ) τ2(h)(S ) τ3(h)(S ) τ4(h)(S ) …

A0 f t t t t …

A1 f f t t t …

A2 f f f t t …

A3 f f f f t …

A4 f f f f f …





B f f f f f …

At the 0th stage, each An is outside the hypothesized extension of T . But from the nth

stage onwards, An is in the hypothesized extension of T . So, for each n, the sentence

An is eventually stably hypothesized to be true. Despite this, there is no finite stage at

which all the An's are hypothesized to be true: as a result the sentence B =∀ x(Gx ⊃

T x) remains false at each finite stage. This suggests extending the process as follows:

S h(S ) τ(h)(S ) τ2(h)(S ) τ3(h)(S ) … ω ω+1 ω+2 …

A0 f t t t … t t t …

A1 f f t t … t t t …

A2 f f f t … t t t …

A3 f f f f … t t t …

A4 f f f f … t t t …

B f f f f … f t t …

Thus, if we allow the revision process to proceed beyond the finite stages, then the

sentence B =∀ x Gx ⊃ T x) is stably true from the ω+1st stage onwards. □

In Example 3.4, the intuitive verdict is that not only should each An receive a stable truth value of

t, but so should the sentence B =∀ x(Gx ⊃ T x). The only way to ensure this is to carry the revision

process beyond the finite stages. So we will consider revision sequences that are very long: not

only will a revision sequence have a nth stage for each finite number n, but a ηth stage for every

ordinal number η. (The next paragraph is to help the reader unfamiliar with ordinal numbers.)

One way to think of the ordinal numbers is as follows. Start with the finite natural numbers:

0, 1, 2, 3,…

Add a number, ω, greater than all of these but not the immediate successor of any of them:

0, 1, 2, 3, …, ω

And then take the successor of ω, its successor, and so on:

0, 1, 2, 3, …, ω, ω+1, ω+2, ω+3 …

Then add a number ω+ω, or ω×2, greater than all of these (and again, not the immediate successor

of any), and start over, reiterating this process over and over:

0, 1, 2, 3, …,

ω, ω+1, ω+2, ω+3, …,

ω×2, (ω×2)+1, (ω×2)+2, (ω×2)+3, …,

ω×3, (ω×3)+1, (ω×3)+2, (ω×3)+3, …

At the end of this, we add an ordinal number ω×ω or ω2:





0, 1, 2, …, ω, ω+1, ω+2, …, ω×2, (ω×2)+1, …,

ω×3, …, ω×4, …, ω×5, …, ω2, ω2+1, …

The ordinal numbers have the following structure: every ordinal number has an immediate

successor known as a successor ordinal ; and for any infinitely ascending sequence of ordinal

numbers, there is a limit ordinal which is greater than all the members of the sequence and which

is not the immediate successor of any member of the sequence. Thus the following are successorordinals: 5, 178, ω+12, (ω×5)+56, ω2+8; and the following are limit ordinals: ω, ω×2, ω2, (ω2+ω),

etc. Given a limit ordinal η, a sequence S of objects is an η-long sequence if there is an object S δfor every ordinal δ < η. We will denote the class of ordinals as On. Any sequence S of objects is

an On-long sequence if there is an object S δ for every ordinal δ.

When assessing whether a sentence receives a stable truth value, the RTT considers sequences of

hypotheses of length On. So suppose that S is an On-long sequence of hypotheses, and let ζ and η

range over ordinals. Clearly, in order for S to represent the revision process, we need the ζ+1st

hypothesis to be generated from the ζth hypothesis by the revision rule. So we insist that S ζ+1 = τ M

(S ζ). But what should we do at a limit stage? That is, how should we set S η(δ) when η is a limit

ordinal? Clearly any object that is stably true [false] up to that stage should be true [false] at that

stage. Thus consider Example 3.2. The sentence A2, for example, is true up to the ωth stage; so we

set A2 to be true at the ωth stage. For objects that do not stabilize up to that stage, Gupta and

Belnap 1993 adopt a liberal policy: when constructing a revision sequence S , if the value of the

object d ∈ D has not stabilized by the time you get to the limit stage η, then you can set S η(δ) to be

whichever of t or f you like. Before we give the precise definition of a revision sequence, we

continue with Example 3.3 to see an application of this idea.

Example 3.5 (Example 3.3 continued)

Recall that L contains four non-quote names, α, β, γ and λ and no predicates other

than T . Also recall that M = < D, I > is as follows:

D = Sent L

I (α) = A = T β ∨ T γ

I (β) = B = T α

I (γ) = C = ¬T α

I (λ ) = X = ¬T λ

The following table indicates what happens with repeated applications of the revisionrule τ M to the hypothesis h0 from Example 3.1. For each ordinal η, we will indicate

the ηth hypothesis by S η (suppressing the index M on τ). Thus S 0 = h0, S 1 = τ(h0), S 2 =

τ2(h0), S 3 = τ3(h0), and S ω, the ωth hypothesis, is determined in some way from the

hypotheses leading up to it. So, starting with h0 from Example 3.3, our revision

sequence begins as follows:

S S 0(S ) S 1(S ) S 2(S ) S 3(S ) S 4(S ) …

A f t t t t …

B t f t t t …

C f t f f f …

X f t f t f …





What happens at the ωth stage? A and B are stably true up to the ωth stage, and C is

stably false up to the ωth stage. So at the ωth stage, we must have the following:

S S 0(S ) S 1(S ) S 2(S ) S 3(S ) S 4(S ) … S ω(S )

A f t t t t … t

B t f t t t … tC f t f f f … f

X f t f t f … ?

But the entry for S ω( X ) can be either t or f . In other words, the initial hypothesis h0

generates at least two revision sequences. Every revision sequence S that has h0 as its

initial hypothesis must have S ω( A) = t, S ω( B) = t, and S ω(C ) = f . But there is some

revision sequence S , with h0 as its initial hypothesis, and with S ω( X ) = t; and there is

some revision sequence S ′, with h0 as its initial hypothesis, and with S ω′( X ) = f . □

We are now ready to define the notion of a revision sequence:

Definition 3.6

Suppose that L is a truth language, and that M = < D, I > is a ground model. Suppose

that S is an On-long sequence of hypotheses. Then we say that d ∈ D is stably t [f ] inS iff for some ordinal θ we have

S ζ(d ) = t [f ], for every ordinal ζ ≥ θ.

Suppose that S is a η-long sequence of hypothesis for some limit ordinal η. Then we

say that d ∈ D is stably t [f ] in S iff for some ordinal θ < η we have

S ζ(d ) = t [f ], for every ordinal ζ such that ζ ≥ θ and ζ < η.

If S is an On-long sequence of hypotheses and η is a limit ordinal, then S |η is the

initial segment of S up to but not including η. Note that S |η is a η-long sequence of

hypotheses.

Definition 3.7

Suppose that L is a truth language, and that M = < D, I > is a ground model. Suppose

that S is an On-long sequence of hypotheses. S is a revision sequence for M iff

• S ζ+1 = τ M (S ζ), for each ζ ∈ On, and• for each limit ordinal η and each d ∈ D, if d is stably t [f ] in S |η, then S η(d ) = t

[f ].

Definition 3.8

Suppose that L is a truth language, and that M =< D, I > is a ground model. We say

that the sentence A is categorically true [ false] in M iff A is stably t [f ] in every

revision sequence for M . We say that A is categorical in M iff A is either categorically

true or categorically false in M .

We now illustrate these concepts with an example. The example will also illustrate a new concept

to be defined afterwards.





Example 3.9

Suppose that L is a truth language containing nonquote names β, α0, α1, α2, α3, …, and

unary predicates G and T . Let B be the sentence

T β ∨∀ x∀ y(Gx & ¬T x & Gy & ¬T y ⊃ x=y).

Let A0 be the sentence ∃ x(Gx & ¬T x). And for each n ≥ 0, let An+1 be the sentence

T αn. Consider the following ground model M =< D, I >

D = Sent L

I (β) = B

I (αn) = An

I (G)( A) = t iff A = An for some n

Thus, the extension of G is the following set of sentences: { A0, A1, A2, A3, … } =

{T α0, T α1, T α2, T α3, … }. Let h be any hypothesis for which we have, h( B) = f andfor each natural number n,

h( An) = f .

And let S be a revision sequence whose initial hypothesis is h, i.e., S 0 = h. The

following table indicates some of the values of S γ(C ), for sentences C ∈ { B, A0, A1,

A2, A3, … }. In the top row, we indicate only the ordinal number representing the

stage in the revision process.

0 1 2 3 … ω ω+1 ω+2 ω+3 … ω×2 (ω×2)+1 (ω×2)+2 …

B f f f f … f t t t … t t t …

A0 f t t t … t f t t … t f t …

A1 f f t t … t t f t … t t f …

A2 f f f t … t t t f … t t t …

A3 f f f f … t t t t … t t t …

A4 f f f f … t t t t … t t t …

It is worth contrasting the behaviour of the sentence B and the sentence A0. From the

ω+1st stage on, B is stabilizes as true. In fact, B is stably true in every revision

sequence for M . Thus, B is categorically true in M . The sentence A0, however, never

quite stabilizes: it is usually true, but within a few finite stages of a limit ordinal, the

sentence A0 can be false. In these circumstances, we say that A0 is nearly stably true

(See Definition 3.10, below.) In fact, A0 is nearly stably true in every revision

sequence for M . □

Example 3.9 illustrates not only the notion of stability in a revision sequence, but also of near

stability, which we define now:

Definition 3.10.

Suppose that L is a truth language, and that M =< D, I > is a ground model. Suppose





that S is an On-long sequence of hypotheses. Then we say that d ∈ D is nearly stably

t [f ] in S iff for some ordinal θ we have

for every ζ ≥ θ, there is a natural number n such that, for every m ≥ n,

S ζ+m(d ) = t [f ].

Gupta and Belnap 1993 characterize the difference between stability and near stability as follows:

“Stability simpliciter requires an element [in our case a sentence] to settle down to a value x [in

our case a truth value] after some initial fluctuations say up to [an ordinal η]… In contrast, near

stability allows fluctuations after η also, but these fluctuations must be confined to finite regions

just after limit ordinals” (p. 169). Gupta and Belnap 1993 introduce two theories of truth, T * and

T #, based on stability and near stability. Theorems 3.12 and 3.13, below, illustrate an advantage of

the system T #, i.e., the system based on near stability.

Definition 3.11

Suppose that L is a truth language, and that M = < D, I > is a ground model. We say

that a sentence A is valid in M by T * iff A is stably true in every revision sequence.

And we say that a sentence A is valid in M by T # iff A is nearly stably true in everyrevision sequence.

Theorem 3.12

Suppose that L is a truth language, and that M =< D, I > is a ground model. Then, for

every sentence A of L, the following is valid in M by T #:

T ‘¬ A’ ≡ ¬T ‘ A’.

Theorem 3.13

There is a truth language L and a ground model M =< D, I > and a sentence A of L

such that the following is not valid in M by T *:

T ‘¬ A’ ≡ ¬T ‘ A’.

Gupta and Belnap 1993, Section 6C, note similar advantages of T # over T

*. For example, T # does,

but T * does not, validate the following semantic principles:

T ‘ A & B’ ≡ T ‘ A’ & T ‘ B’

T ‘ A ∨ B’ ≡ T ‘ A’ ∨ T ‘ B’

Gupta and Belnap remain noncommittal about which of T # and T * (and a further alternative that

they define, T c) is preferable.

4. Interpreting the formalism

The main formal notions of the RTT are the notion of a revision rule (Definition 3.2), i.e., a rule

for revising hypotheses; and a revision sequence (Definition 3.7), a sequence of hypotheses

generated in accordance with the appropriate revision rule. Using these notions, we can, given a

ground model, specify when a sentence is stably, or nearly stably, true or false in a particular

revision sequence. Thus we could define two theories of truth,T

*

andT

#

, based on stability andnear stability. The final idea is that each of these theories delivers a verdict on which sentences of

the language are categorically assertible, given a ground model.





Note that we could use revision-theoretic notions to make rather fine-grained distinctions among

sentences: Some sentences are unstable in every revision sequence; others are stable in every

revision sequence, though stably true in some and stably false in others; and so on. Thus, we can

use revision-theoretic ideas to give a fine-grained analysis of the status of various sentences, and

of the relationships of various sentences to one another.

Recall the suggestion made at the end of Section 2:

In a semantics for languages capable of expressing their own truth concepts, T will

not, in general, have a classical signification; and the ‘iff’ in the T-biconditionals will

not be read as the classical biconditional.

Gupta and Belnap fill out these suggestions in the following way.

4.1 The signification of T

First, they suggest that the signification of T , given a ground model M , is the revision rule τ M

itself. As noted in the preceding paragraph, we can give a fine-grained analysis of sentences'statuses and interrelations on the basis of notions generated directly and naturally from the

revision rule τ M . Thus, τ M is a good candidate for the signification of T , since it does seem to be

“an abstract something that carries all the information about all [of T 's] extensional relations” in

M . (See Gupta and Belnap's characterization of an expression's signification, given in Section 2,

above.)

4.2 The ‘iff’ in the T-biconditionals

Gupta and Belnap's related suggestion concerning the ‘iff’ in the T-biconditionals is that, rather

than being the classical biconditional, this ‘iff’ is the distinctive biconditional used to define a

previously undefined concept. In 1993, Gupta and Belnap present the revision theory of truth as aspecial case of a revision theory of circularly defined concepts. Suppose that L is a language with

a unary predicate F and a binary predicate R. Consider a new concept expressed by a predicate G,

introduced through a definition like this:

Gx =df ∀ y( Ryx ⊃ Fx) ∨ ∃ y( Ryx & Gx).

Suppose that we start with a domain of discourse, D, and an interpretation of the predicate F and

the relation symbol R. Gupta and Belnap's revision-theoretic treatment of concepts thus circularly

introduced allows one to give categorical verdicts, for certain d ∈ D about whether or not d satisfies G. Other objects will be unstable relative to G: we will be able categorically to assert

neither that d satisfies G nor that d does not satisfy G. In the case of truth, Gupta and Belnap takethe set of T-biconditionals of the form

T ‘ A’ =df A (10)

together to give the definition of the concept of truth. It is their treatment of ‘= df ’ (the ‘iff’ of

definitional concept introduction), together with the T-biconditionals of the form (10), that

determine the revision rule τ M .

4.3 The paradoxical reasoning

Recall the liar sentence, (1), from the beginning of this article:





(1) is not true (1)

In Section 1, we claimed that the RTT is designed to model, rather than block, the kind of

paradoxical reasoning regarding (1). But we noted in footnote 2 that the RTT does avoid

contradictions in these situations. There are two ways to see this. First, while the RTT does

endorse the biconditional

(1) is true iff (1) is not true,

the relevant ‘iff’ is not the material biconditional, as explained above. Thus, it does not follow that

both (1) is true and (1) is not true. Second, note that on no hypothesis can we conclude that both

(1) is true and (1) is not true. If we keep it firmly in mind that revision-theoretical reasoning is

hypothetical rather than categorical, then we will not infer any contradictions from the existence of

a sentence such as (1), above.

4.4 The signification thesis

Gupta and Belnap's suggestions, concerning the signification of T and the interpretation of the ‘iff’in the T-biconditionals, dovetail nicely with two closely related intuitions articulated in Gupta &

Belnap 1993. The first intuition, loosely expressed, is “that the T-biconditionals are analytic and

fix the meaning of ‘true’” (p. 6). More tightly expressed, it becomes the “Signification Thesis” (p.

31): “The T-biconditionals fix the signification of truth in every world [where a world is

represented by a ground model].”[6] Given the revision-theoretic treatment of the definition ‘iff’,

and given a ground model M , the T-biconditionals (10) do, as noted, fix the suggested

signification of T , i.e., the revision rule τ M .

4.5 The supervenience of semantics

The second intuition is the supervenience of the signification of truth. This is a descendant of M.

Kremer's 1988 proposed supervenience of semantics. The idea is simple: which sentences fall

under the concept truth should be fixed by (1) the interpretation of the nonsemantic vocabulary,

and (2) the empirical facts. In non-circular cases, this intuition is particularly strong: the standard

interpretation of “snow” and “white” and the empirical fact that snow is white, are enough to

determine that the sentence “snow is white” falls under the concept truth. The supervenience of

the signification of truth is the thesis that the signification of truth, whatever it is, is fixed by the

ground model M . Clearly, the RTT satisfies this principle.

It is worth seeing how a theory of truth might violate this principle. Consider the truth-teller

sentence, i.e., the sentence that says of itself that it is true:

(11) is true (11)

As noted above, Kripke's three-valued semantics allows three truth values, true (t), false (f ), and

neither (n). Given a ground model M = < D, I > for a truth language L, the candidate

interpretations of T are three-valued interpretations, i.e., functions h : D → { t, f , n }. Given a

three-valued interpretation of T , and a scheme for evaluating the truth value of composite

sentences in terms of their parts, we can specify a truth value Val M +h( A) = t, f or n, for every

sentence A of L. The central theorem of the three-valued semantics is that, given any ground

model M , there is a three-valued interpretation h of T so that, for every sentence A, we have

Val M +h(T ‘ A’) = Val M +h( A).[7] We will call such an interpretation of T an acceptable interpretation.Our point here is this: if there's a truth-teller, as in (11), then there is not only one acceptable

interpretation of T ; there are three: one according to which (11) is true, one according to which





(11) is false, and one according to which (11) is neither. Thus, there is no single “correct”

interpretation of T given a ground model M. Thus the three-valued semantics seems to violate the

supervenience of semantics.[8]

The RTT does not assign a truth value to the truth-teller, (11). Rather, it gives an analysis of the

kind of reasoning that one might engage in with respect to the truth-teller: If we start with a

hypothesis h according to which (11) is true, then upon revision (11) remains true. And if we startwith a hypothesis h according to which (11) is not true, then upon revision (11) remains not true.

And that is all that the concept of truth leaves us with. Given this behaviour of (11), the RTT tells

us that (11) is neither categorically true nor categorically false, but this is quite different from a

verdict that (11) is neither true nor false.

4.6 Yaqūb's interpretation of the formalism

We note an alternative interpretation of the revision-theoretic formalism. Yaqū b 1993 agrees with

Gupta and Belnap that the T-biconditionals are definitional rather than material biconditionals,

and that the concept of truth is therefore circular. But Yaqū b interprets this circularity in a

distinctive way. He argues that,

since the truth conditions of some sentences involve reference to truth in an essential,

irreducible manner, these conditions can only obtain or fail in a world that already

includes an extension of the truth predicate. Hence, in order for the revision process to

determine an extension of the truth predicate, an initial extension of the predicate

must be posited. This much follows from circularity and bivalence. (1993, 40)

Like Gupta and Belnap, Yaqū b posits no privileged extension for T . And like Gupta and Belnap,

he sees the revision sequences of extensions of T , each sequence generated by an initial

hypothesized extension, as “capable of accommodating (and diagnosing) the various kinds of

problematic and unproblematic sentences of the languages under consideration” (1993, 41). But,unlike Gupta and Belnap, he concludes from these considerations that “truth in a bivalent

language is not supervenient ” (1993, 39). He explains in a footnote: for truth to be supervenient,

the truth status of each sentence must be “fully determined by nonsemantical facts”. Yaqū b does

not explicitly use the notion of a concept's signification. But Yaqū b seems committed to the claim

that the signification of T — i.e., that which determines the truth status of each sentence — is

given by a particular revision sequence itself. And no revision sequence is determined by the

nonsemantical facts, i.e., by the ground model, alone: a revision sequence is determined, at best,

by a ground model and an initial hypothesis.[9]

5. Further issues

5.1 Three-valued semantics

We have given only the barest exposition of the three-valued semantics, in our discussion of the

supervenience of the signification of truth, above. Given a truth language L and a ground model

M , we defined an acceptable three-valued interpretation of T as an interpretation h : D → { t, f , n }

such that Val M +h(T ‘ A’) = Val M +h( A) for each sentence A of L. In general, given a ground model M ,

there are many acceptable interpretations of T . Suppose that each of these is indeed a truly

acceptable interpretation. Then the three-valued semantics violates the supervenience of the

signification ofT

.

Suppose, on the other hand, that, for each ground model M , we can isolate a privileged acceptable

interpretation as the correct interpretation of T . Gupta and Belnap present a number of





considerations against the three-valued semantics, so conceived. (See Gupta & Belnap 1993,

Chapter 3.) One principal argument is that the central theorem, i.e., that for each ground model

there is an acceptable interpretation, only holds when the underlying language is expressively

impoverished in certain ways: for example, the three-valued approach fails if the language has a

connective ~ with the following truth table:

A ~ At f

f t

n t

The only negation operator that the three-valued approach can handle has the following truth

table:

A ¬ A

t f f t

n t

But consider the liar that says of itself that it is ‘not’ true, in this latter sense of ‘not’. Gupta and

Belnap urge the claim that this sentence “ceases to be intuitively paradoxical” (1993, 100). The

claimed advantage of the RTT is its ability to describe the behaviour of genuinely paradoxical

sentences: the genuine liar is unstable under semantic evaluation: “No matter what we hypothesize

its value to be, semantic evaluation refutes our hypothesis.” The three-valued semantics can only

handle the “weak liar”, i.e., a sentence that only weakly negates itself, but that is not guaranteed to

be paradoxical: “There are appearances of the liar here, but they deceive.”

5.2 Amendments to the RTT

We note three ways to amend the RTT. First, we might put constraints on which hypotheses are

acceptable. For example, Gupta and Belnap 1993 introduce a theory, Tc, of truth based on

consistent hypotheses: an hypothesis h is consistent iff the set { A : h( A) = t} is a complete

consistent set of sentences. The relative merits of T*, T# and Tc are discussed in Gupta & Belnap

1993, Chapter 6.

Second, we might adopt a more restrictive limit policy than Gupta and Belnap adopt. Recall the

question asked in Section 3: How should we set S η(d ) when η is a limit ordinal? We gave a partial

answer: any object that is stably true [false] up to that stage should be true [false] at that stage. We

also noted that for an object d ∈ D that does not stabilize up to the stage η, Gupta and Belnap1993 allow us to set S η(d ) as either t or f . In a similar context, Herzberger 1982a and 1982b

assigns the value f to the unstable objects. And Gupta originally suggested, in Gupta 1982, that

unstable elements receive whatever value they received at the initial hypothesis S 0.

These first two ways of amending the RTT both, in effect, restrict the notion of a revision

sequence, by putting constraints on which of our revision sequences really count as acceptable

revision sequences. The constraints are, in some sense local: the first constraint is achieved by

putting restrictions on which hypotheses can be used, and the second constraint is achieved by putting restrictions on what happens at limit ordinals. A third option would be to put more global

constraints on which putative revision sequences count as acceptable. Yaqū b 1993 suggests, in





• A sentence B is valid on the definition D in the system S# (notation ⊨#, D B) iff

for all classical ground models M , we have M ⊨#, D B.

One of Gupta and Belnap's principle open questions is whether there is a complete calculus for

these systems: that is, whether, for each definition D, either of the following two sets of sentences

is recursively axiomatizable: { B : ⊨*, D B} and { B : ⊨#, D B}. Kremer 1993 proves that the answer is

no: he shows that there is a definition D such that each of these sets of sentences is of complexityat least Π1

2, thereby putting a lower limit on the complexity of S* and S

#. (Antonelli 1994b and

2002 shows that this is also an upper limit.)

Kremer's proof exploits an intimate relationship between circular definitions understood

revision-theoretically and circular definitions understood as inductive definitions: the theory of

inductive definitions has been quite well understood for some time. In particular, Kremer proves

that every inductively defined concept can be revision-theoretically defined. The expressive power

and other aspects of the revision-theoretic treatment of circular definitions is the topic of much

interesting work: see Welch 2001, Löwe 2001, Löwe and Welch 2001, and Kühnberger et al .

2005.

5.5 Applications

Given Gupta and Belnap's general revision-theoretic treatment of circular definitions-of which

their treatment of truth is a special case-one would expect revision-theoretic ideas to be applied to

other concepts. Antonelli 1994a applies these ideas to non-well-founded sets: a non-well-founded

set X can be thought of as circular, since, for some X 0, …, X n we have X ∈ X 0 ∈ … ∈ X n ∈ X .

And Chapuis 2003 applies revision-theoretic ideas to rational decision making.

5.5 An open question

We close with an open question about T* and T#. Recall Definition 3.11, above, which defines

when a sentence A of a truth language L is valid in the ground model M by T* or by T#. We will

say that A is valid by T* [alternatively, by T#] iff A is valid in the ground model M by T*

[alternatively, by T#] for every ground model M . Our open question is this: What is the complexity

of the set of sentences valid by T* [T#]?

Bibliography

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35: 204–218.

• Antonelli, G.A., 1994, “Non-well-founded sets via revision rules”, Journal of Philosophical

Logic, 23: 633–679.

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479–497.

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Journal of Logic and Computation, 11: 25–40.• Löwe, B., and Welch, P., 2001, “Set-theoretic absoluteness and the revision theory of truth”,

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