## 1. The Paradox and the Broader Phenomenon

### 1.1 Simple-falsity Liar

Consider a sentence named ‘FLiar’, which says of itself

(i.e., says of FLiar) that it is false.

This seems to lead to contradiction as follows. If the sentence

‘FLiar is false’ is true, then given what it says, FLiar

is false. But FLiar just is the sentence ‘FLiar is false’,

so we can conclude that if FLiar is true, then FLiar is false.

Conversely, if FLiar is false, then the sentence ‘FLiar is

false’ is true. Again, FLiar just is the sentence ‘FLiar

is false’, so we can conclude that if FLiar is false, then FLiar

is true. We have thus shown that FLiar is false if and only if FLiar

is true. But, now, if every sentence is true or false, FLiar itself is

either true or false, in which case—given our reasoning

above—it is both true and false. This is a contradiction.

Contradictions, according to many logical theories (e.g., classical

logic, intuitionistic logic, and much more) imply

absurdity—triviality, that is, that every sentence is true.

An obvious response is to deny that every sentence is true or false,

i.e., to deny the principle of bivalence. As we will discuss in

§4,

some descendants of this idea remain important in current work on the

Liar. Even so, a simple variant Liar sentence shows that this

immediate answer is not all there is to the story.

### 1.2 Simple-untruth Liar

Rather than work with falsehood, we can construct a Liar sentence with

the complex predicate ‘not

true’.^{[2]}

Consider a sentence named ‘ULiar’ (for

‘un-true’), which says of itself that it is not true.

**ULiar**: ULiar

is not true.

The argument towards contradiction is similar to the FLiar case. In

short: if ULiar is true, then it is not true; and if it is not true,

then it is true. But, now, if every sentence is true or not true,

ULiar itself is true or not true, in which case it is both true and

not true. This is a contradiction. According to many logical theories,

a contradiction implies absurdity—triviality.

The two forms of the Liar paradox we have so far reviewed rely on some

explicit self-reference—sentences talking directly about

themselves. Such explicit self-reference can be avoided, as is shown

by our next family of Liar paradoxes.

### 1.3 Liar cycles

Consider a very concise (viz., one-sentence-each) dialog between

siblings Max and Agnes.

**Max**:

Agnes’ claim is true.**Agnes**:

Max’s claim is not true.

What Max said is true if and only if what Agnes said is true. But what

Agnes said (viz., ‘Max’s claim is not true’) is true

if and only if what Max said is not true. Hence, what Max said is true

if and only if what Max said is not true. But, now, if what Max said

is true or not true, then it is both true and not true. And this, as

in the FLiar and ULiar cases, is a contradiction, implying, according

to many logical theories, absurdity.

Liar paradoxes can also be formed using more complex sentence

structure, rather than complex modes of reference. One that has been

important involves Boolean compounds.

### 1.4 Boolean compounds

Boolean compounds can enter into Liar sentences in many ways. One

relatively simple one is as follows. Consider the following sentence

named ‘DLiar’ (for ‘Disjunctive’).

**DLiar**: Either

DLiar is not true or (1 = 0).

First, observe that if DLiar is not true, then it must be true. If

DLiar is not true, then by similar reasoning to what we saw above, we

have that the left disjunct of DLiar is true. But a disjunction is

true if one of its disjuncts is, so DLiar is true. Thus, if DLiar is

not true, it is true and not true, and we have a contradiction. By

reductio, then, it must be true; so one of its disjuncts must be true.

If it’s the first one, we have a contradiction, so it must be

the second one; we can conclude that (1 = 0). We have thus proved

that (1 = 0). Moreover, the sentence ‘(1 = 0)’ played

no real role in the above reasoning. We could replace it with any

other sentence to get a proof of that sentence.

We pause to mention DLiar as it is connected with another important

paradox: Curry’s paradox, which involves conditionals that say

of themselves only that if they (the conditional itself) are true, so

too is some absurdity (e.g., ‘if this sentence is true, then (1

= 0)’ or ‘if this sentence is true, everything is

true’ or so on). At least in languages where the conditional is

the material conditional, and so (A supset B) is equivalent to

(neg A vee B), DLiar is equivalent to the Curry sentence

‘DLiar is true (supset 1 = 0)’. Though this may set up

some relations between the Liar and Curry’s paradox, we pause to

note an important difference. For the Curry paradox is most important

where the conditional is more than the material conditional (or some

modalized variant of it). In such settings, the Curry paradox does not

wear negation on its sleeve, as DLiar does. For more information,

consult the entry on

Curry’s paradox.

### 1.5 Infinite sequences

The question of whether the Liar paradox really requires some sort of

circularity has been the subject of extensive debate. Liar cycles

(e.g., the Max–Agnes dialog) show that explicit self-reference

is not necessary, but it is clear that such cycles themselves involve

circular reference. Yablo (1993b) has argued that a more complicated

kind of multi-sentence paradox produces a Liar without

circularity.

Yablo’s paradox relies on an infinite sequence of claims

(A_0), (A_1), (A_2), …, where each (A_i) says that all

of the ‘greater’ (A_k) (i.e., the (A_k) such that (k

gt i)) are untrue. (In other words, each claim says of the rest that

they’re all untrue.) Since we have an infinite sequence, this

version of the Liar paradox appears to avoid the sort of circularity

apparent in the previous examples; however, contradiction still seems

to emerge. If (A_0) is true, then all of the ‘greater’

(A_k) are untrue, and *a fortiori* (A_1) is untrue. But,

then, there is at least one true (A_k) greater than (A_1) (i.e.,

some (A_k) such that (k gt 1)), which contradicts (A_0).

Conversely, if (A_0) is untrue, then there’s at least one true

(A_k) greater than (A_0). Letting (A_m) be such a one (i.e., a

truth greater than (A_0)), we have it that (A_{m+1}) is untrue, in

which case there’s some truth greater than (A_{m+1}). But this

contradicts (A_m). What we have, then, is that if (A_0) (the first

claim in the infinite sequence) is true or untrue, then it is both.

And this, as in the other cases, is a contradiction.

Whether Yablo’s paradox really avoids self-reference is

much-debated. See, for instance, Barrio (2012), Beall (2001), Cook

(2006, 2014), Ojea (2012), Picollo (2012), Priest (1997), Sorensen

(1998), and Teijeiro (2012).

## 2. Basic Ingredients

We have already seen a kind of characteristic reasoning that goes with

the Liar. We have also seen some common structure across all our

example Liar paradoxes, such as the presence of truth predicates, and

something like negation. We pause here to discuss these ingredients of

the paradox, focusing on the *basic* Liars. Just what creates

the Liar paradox, and just which of the puzzles we just surveyed is

‘basic’, is a contentious matter; different approaches to

solving the Liar view these matters differently. Hence, our goal is

merely to illuminate some common themes across different Liars, not to

offer a full diagnosis of the source of the paradox.

We highlight three aspects of the Liar: the role of truth predicates,

the kinds of principles for reasoning about truth that are needed, and

the way that a paradox can be derived given these resources.

### 2.1 Truth predicate

The first ingredient in building a Liar is a truth predicate, which we

write here as (Tr). We follow the usual custom in logic of treating

this as a predicate of sentences. However, especially as we come to

consider some ways of resolving the Liar, it should be remembered that

this treatment can be seen as more for convenience in exposition than

a serious commitment to what truth bearers are.

We assume that we have, along with the truth predicate, appropriate

names of sentences. For a given sentence (A), suppose that

(leftulcorner Arighturcorner) is a name for it. A predication of

truth to (A) then looks like (Tr(leftulcorner

Arighturcorner)).

We shall say that a predicate (Tr(x)) is a *truth predicate for
language* (mathcal{L}) only if (Tr(leftulcorner

Arighturcorner)) is well-formed for every sentence (A) of

(mathcal{L}). We typically expect (Tr) to obey some principles

governing its behavior on sentences of a given language. It is to

those we now

turn.

^{[3]}

### 2.2 Principles of truth

The tradition, going back to Tarski (1935), is that the behavior of

the truth predicate (Tr) is described by the following

biconditional.

Indeed, Tarski took the biconditional here to be the material

biconditional of classical logic. This is usually called the

*T-schema*. For more on the T-schema, and Tarski’s views

of truth, see the entries on

Alfred Tarski

and

Tarski’s truth definitions.

The Liar paradox has been a locus of thinking about non-classical

logics (as we already saw a taste of, for instance, in the idea that

bivalence might be rejected as part of a solution to the Liar). Thus,

we should stop to consider what principles should govern the truth

predicate (Tr) if classical logic is not to hold.

The leading idea for what might replace the T-schema points to two

sorts of ‘rules’ (e.g., two sorts of ‘inference

rules’ in some sense) or principles that are characteristic of

the truth predicate. If you have a sentence (A), you can infer

(Tr(leftulcorner Arighturcorner)), that is, you can

‘capture’ (A) with the truth predicate. Conversely, if

you have (Tr(leftulcorner Arighturcorner)), you can infer

(A), that is, you can ‘release’ (A) from the truth

predicate. In some logics, capture and release wind up being

equivalent to the T-schema, but it is often helpful to break these

up:

**Capture**(A)

implies (Tr(leftulcorner Arighturcorner)).

(We also write this as (A vdash Tr(leftulcorner

Arighturcorner)).)**Release**(Tr(leftulcorner

Arighturcorner)) implies (A).

(We also write this as (Tr(leftulcorner Arighturcorner) vdash

A).)

*Implies* here is a logical notion, though just which one, and

what the options are, depends on what background logic is assumed. For

our discussion, we think of it in so-called rule form: that the

argument from (A) to (B) is valid, which we record (as above) via

the turnstile. In some logical settings (e.g., classical logic, in

which a certain so-called deduction theorem holds), this is equivalent

to the provability of a conditional, but in some settings, it is not.

Either way, capture and release jointly make (A) and

(Tr(leftulcorner Arighturcorner)) logically equivalent in the

sense of being inter-derivable. In strong forms, capture and release

can lead to the full intersubstitutability of (A) and

(Tr(leftulcorner Arighturcorner)) in extensional contexts. As

we discuss more in

section 4.1,

this is important to some views of the nature of truth. Thus,

(vdash) is being used here as a schematic placeholder for a range

of different logical notions, each of which will provide some notion

of valid inference in some logical theory.

(There are a number of logical subtleties here that we will not

pursue, especially about how to formulate rules, and which rules are

consistent. Different formulations of rules vary significantly in

logical strength as

well.^{[4]}

See the entry on

axiomatic theories of truth

for more on how consistent forms of capture and release can be

formulated in classical logic. In the terminology of Friedman and

Sheard (1987), the rule forms of capture and release are called

‘T-Intro’ and ‘T-Elim’, and the conditional

forms ‘T-In’ and ‘T-Out’. We prefer the

broader terminology, since it highlights a general form of behavior

common to a great variety of predicates and operators, e.g.,

*knowledge* releases but doesn’t capture;

*possibility* captures but doesn’t release; and so on;

and truth is special in doing both.)

### 2.3 The Liar in short

The Liar paradox begins with a language containing a truth predicate,

which obeys some form of capture and release. We now explore more

carefully how a paradox results from these assumptions.

#### 2.3.1 Existence of Liar-like sentences

Putting aside Yablo-type paradoxes, the Liar relies on some form of

self-reference, either direct, as in in the simple Liars above, or

indirect, as in Liar cycles. Most natural languages have little

trouble generating self-reference. The first sentence of this essay is

one example. Self-reference can be accidental, as in the case where

someone writes ‘The only sentence on the blackboard in room 101

is not true’, by chance writing this in room 101 itself (as C.

Parsons (1974) noted).

In formal languages, self-reference is also very easy to come by. Any

language capable of expressing some basic syntax can generate

self-referential sentences via so-called diagonalization (or more

properly, any language together with an appropriate theory of syntax

or

arithmetic).^{[5]}

A language containing a truth predicate and this basic syntax will

thus have a sentence (L) such that (L) implies (neg

Tr(leftulcorner Lrighturcorner)) and *vice versa*:

This is a ‘fixed point’ of (the compound predicate)

(negTr), and is, in effect, our simple-untruth Liar.

(Technically, it is simplest to put the fixed point property in terms

of implications, as we have done here. But intuitively, the idea is

that somehow (L) ‘just is’ (neg Tr(leftulcorner

Lrighturcorner)). This can be made more precise, if we think of

such the Liar sentence (L) arising from a *name* (c) that

denotes the sentence (neg Tr(c)). In this way, we can think of the

existence of the Liar as being reflected in the identity (c =

leftulcorner neg Tr(c)righturcorner). For more on the details

of this approach, see Heck 2012.)

#### 2.3.2 Other logical ‘laws’

Other conspicuous ingredients in common Liar paradoxes concern logical

behavior of basic connectives or features of implication. A few of the

relevant principles are:

- Excluded middle (LEM): (vdash A vee neg A).
- Explosion

(EFQ):^{[6]}

(A,neg A vdash B). - Disjunction principle

(DP):^{[7]}

If (A vdash C) and (B vdash C) then (A vee B vdash C). - Adjunction: If (A vdash B) and (A vdash C) then (A vdash B

wedge C).

(This is not to suggest that these are the *only* logical

features involved in common Liar paradoxes, but they’re arguably

the most important of the salient ones.)

#### 2.3.3 The Liar in abstract

Given the foregoing ingredients, we can now give a slightly more

abstract form of the paradox. (Our hope is to use this abstract form

to highlight different responses to the paradox.) We suppose that we

have a language (mathcal{L}) with a truth predicate (Tr), and

that (mathcal{L}) allows enough syntax to construct a sentence

(L) such that (L dashv vdash neg Tr(leftulcorner

Lrighturcorner)). We also suppose that the logic of (mathcal{L})

enjoys LEM and EFQ and satisfies DP and adjunction.

An argument that our Liar sentence (L) implies a contradiction runs

as follows.

- (Tr(leftulcorner Lrighturcorner) vee neg

Tr(leftulcorner Lrighturcorner)) [LEM] - Case One:
- (Tr(leftulcorner Lrighturcorner))
- (L) [2a: release]
- (neg Tr(leftulcorner Lrighturcorner)) [2b: definition of

(L)] - (neg Tr(leftulcorner Lrighturcorner) wedge

Tr(leftulcorner Lrighturcorner)) [2a, 2c: adjunction]

- Case Two:
- (neg Tr(leftulcorner Lrighturcorner))
- (L) [3a: definition of (L)]
- (Tr(leftulcorner L)righturcorner)) [3b: capture]
- (neg Tr(leftulcorner Lrighturcorner) wedge

Tr(leftulcorner Lrighturcorner)) [3a, 3c: adjunction]

- (neg Tr(leftulcorner Lrighturcorner) wedge

Tr(leftulcorner Lrighturcorner)) [1–3: DP]

This version of the Liar is one of many. With a little more

complexity, for instance, either capture or release can be avoided in

favor of some other background assumptions. Intuitionistic variants of

the Liar are also available, though we shall not explore

intuitionistic logic

here.^{[8]}

We have so far shown that with the given ingredients our Liar sentence

(L) implies a contradiction (thus formalizing the reasoning in

ULiar). From here, it is one short step to all-out absurdity—if

the lone contradiction weren’t already absurd enough. We invoke

EFQ to finish the proof. (Well, we also assume that (A wedge B)

implies (A) and (B), i.e., that simplification is valid in

(mathcal{L}); but in fact this assumption is not really

necessary.)

- (B) [4: EFQ]

(B), here, may be any—every—sentence that you like (or

don’t like, as the case may be)! EFQ is the principle that every

sentence follows from a contradiction; it sanctions the step from a

single contradiction to outright triviality of logic.

In the face of such absurdity (triviality), we conclude that something

is wrong in the foregoing Liar reasoning. The question is: what? This,

in the end, is the question that the Liar paradox raises.

## 3. Significance

We have now seen that with some elementary assumptions about truth and

logic, a logical disaster ensues. What is the wider significance of

such a result?

From time to time, the Liar has been argued to show us something

far-reaching about philosophy. For instance, Grim (1991) has argued

that it shows the world to be essentially ‘incomplete’ in

some sense, and that there can be no omniscient being. McGee (1991)

and others suggest that the Liar shows the notion of truth to be a

vague notion. Glanzberg (2001) holds that the Liar shows us something

important about the nature of context dependence in language, while

Eklund (2002) holds that it shows us something important about the

nature of semantic competence and the languages we speak. Gupta and

Belnap (1993) claim that it reveals important properties of the

general notion of definition. And there are other lessons, and

variations on such lessons, that have been drawn.

Of more immediate concern, at least for our purposes here, is what the

Liar shows us about the basic principles governing truth, and about

logic. In a skeptical vein, Tarski himself (1935, 1944) seems to have

thought the Liar shows the ordinary notion of truth to be incoherent,

and in need of replacement with a more scientifically respectable one.

(For more on Tarski, see the entries on

Tarski

and

Tarski’s truth definitions.

For more on Tarski’s aims and purposes, see Heck 1997.) More

common, and perhaps the dominant thread in the solutions to the Liar,

is the idea that the basic principles governing truth are more subtle

than the T-schema reflects.

The Liar has also formed the core of arguments against classical

logic, as it is some key features of classical logic that allow

capture and release to result in absurdity. Notable among these are

the arguments for logics that are paracomplete (e.g., Kripke 1975;

Field 2008) and paraconsistent (e.g., Asenjo 1966; Priest 1984, 2006).

However, Ripley (2013b) argues that classical logic can be maintained

while shedding the features in question.

In many cases inspired by wider views of the significance of the

paradox, there have been a number of attempts to one way or another

resolve the paradox. It is to these proposed solutions that we now

turn.

## 4. Some Families of Solutions

In this section, we briefly survey some approaches to resolving the

Liar paradox. We group proposed solutions into families, and try to

explain the basic ideas behind them. In many cases, a full exposition

would involve a great deal of technical material, that we will not go

into here. Interested readers are encouraged to follow the references

we provide for each basic idea.

### 4.1 Paracomplete and paraconsistent logics

One of the leading ideas for how to resolve the Liar paradox is that

it shows us something about logic, in fact, something far-reaching

about logic. The main idea is that the principles of capture and

release are the fundamental conceptual principles governing truth, and

cannot be modified. Instead, basic logic must be non-classical, to

avoid a logical disaster of the kind we reviewed in

§2.

One important way to motivate non-classical solutions is to appeal to

a form of *deflationism* about truth. Such views take something

akin to the T-schema to be the defining characteristic of truth, and

as such, not open to modification (see, e.g., Horwich 1990). Most

strictly, so-called transparency or ‘see-through’ or

‘pure disquotational’ conceptions of truth (e.g., Field

1994, 2008; Beall 2005) take the defining property of truth to

be *intersubstitutability* of (A) and (Tr(leftulcorner

Arighturcorner)) in all non-opaque contexts. This makes capture and

release, in unrestricted form applying to all sentences of a language,

a requirement for truth (at least where we have (A vdash A) or,

more strongly, (vdash A rightarrow

A)).^{[9]}

See the entry on

truth

for further discussion.

Holding capture and release fixed, and applying it to all sentences

without restriction, yields triviality unless the logic is

non-classical. There are two main sub-families of non-classical

(transparency) truth theories: *paracomplete* and

*paraconsistent*. We sketch the main ideas of each.

#### 4.1.1 Paracomplete

According to paracomplete approaches to the Liar, the main lesson of

the Liar is that LEM ‘fails’ in some sense. In other

words: the Liar teaches us that some sentences (notably, Liars!)

‘neither hold nor do not hold’ (in some sense), and so are

neither true nor false. As a result, the logic of truth is

non-classical.

This idea is perhaps most natural in response to the simple-falsity

Liar. There, it is tempting to say that there is some status other

than truth and falsity, and the Liar sentence (L) has it. But this

will not suffice, for instance, for the simple-untruth Liar. This says

nothing about falsity. Rather, in some way the basic reasoning

reviewed in

§2.3

must fail, and the culprit, in the paracomplete view, is LEM.

Liar-instances of LEM ‘fail’ (in some sense) according to

the paracomplete approach; such sentences fall into the

‘gap’ between truth and falsity (to use a common

metaphor).

There have been many proposals for using such non-classical logics to

address the Liar. An early example is van Fraassen (1968, 1970). But

Kripke’s work has been the most influential in recent times, not

only to approaches to the Liar based on non-classical logic, but a

range of other approaches we will survey in

§4.2

as well. Thus, we pause to describe at least a little of

Kripke’s framework.

##### Kripke’s theory

Logics where LEM fails are not themselves hard to come by. Among many

such logics are a number of three-valued logics that allow sentences

to take a third value over and above true and false. Sentences like

Liar sentences take the third value. One of the most commonly applied

logics is the Strong Kleene logic (K_3). We do not go into the

details of (K_3) here, but only note the properties of (K_3) we

need. (For more details, see the entry on

many-valued logic,

or Priest 2008.) First and foremost,

we have:

LEM fails. In fact, there are no logical truths

(or valid sentences) according to (K_3). (We return to this on the

topic of a ‘suitable conditional’ below.)

The challenge to using (K_3) to flesh out a paracomplete theory is

to explain how anything like (even rule-form) capture and release

hold, and if you follow the deflationist line, how full unrestricted

capture and release hold. One way of understanding the important work

of Kripke (1975) (and related work of Martin and Woodruff 1975) is as

a way of achieving just that.

Kripke begins with a fully classical language (mathcal{L}_0)

containing no truth predicate (or more generally, no semantic terms).

(Recall, we are assuming a language comes equipped with a valuation

scheme. For (mathcal{L}_0) it is classical.) He then considers

extending it to a language (mathcal{L}^{+}_0) which contains a

truth predicate (Tr). The predicate (Tr) is taken to apply to

every sentence of the expanded language (mathcal{L}^{+}_0),

including those of the original (mathcal{L}_0). Thus, it is a

self-applicative truth predicate (as the deflationist-inspired picture

we mentioned must require), even though we begin with a language

without a truth predicate.

We can think of (mathcal{L}_0) as interpreted by a classical model

(mathcal{M}_0). Kripke shows us how to build an interpretation

(mathcal{M}^{+}_0) for the expanded language. The main innovation

is to see the truth predicate as *partial*. Rather than simply

having an extension, it has an extension (set of things of which it is

true), and an anti-extension (set of things of which it is false). The

extension and anti-extension are mutually exclusive, but they need not

jointly exhaust the domain of (mathcal{M}_0). Pathological

sentences like (L) fall in neither in the extension or the

anti-extension of (Tr). (Actually, we could have interpreted the

base language (mathcal{L}_0) by a partial model as well, but the

intended application sees partiality as only arising with semantic

predicates like (Tr).)

Falling into neither the extension or the anti-extension of (Tr)

acts like having a third value, and we can interpret

(mathcal{L}^{+}_0) as acting like a language with a (K_3)

valuation scheme. Treating the language this way, Kripke shows up to

build up a very plausible extension and anti-extension for (Tr),

typically written (mathcal{E}) and (mathcal{A}). The important

property of the new extended model (langle mathcal{M}_0,langle

mathcal{E},mathcal{A}rangle rangle) is that the truth value of

any sentence (A) and (Tr(leftulcorner Arighturcorner)) are

exactly the same. (A) is true, false, or neither, just in case

(Tr(leftulcorner Arighturcorner)) is. Furthermore, interpreting

the expanded language (mathcal{L}^{+}_0) as a (K_3) language, we

have for (K_3) consequence (A dashv vdash Tr(leftulcorner

Arighturcorner)), just as we desired.

Kripke shows how to build up (mathcal{E}) and (mathcal{A}) by an

inductive process. One starts with an ‘approximation’ of

the extension and anti-extension of (Tr), and successively improves

it until the improvement process ceases to be productive (it reaches a

‘fixed point’). In fact, for the (K_3)-based solution,

the natural thing to do is start with an empty extension and

anti-extension, and throw in sentences that are true at successive

stages of the process.

Kripke’s construction can be applied to a number of different

logics, including other many-valued logics such as the ‘Weak

Kleene’ logic, and supervaluation logics. See, for instance,

Burgess 1986 and McGee 1991 for discussion. Kripke-style constructions

engage a fair bit of mathematical subtlety. For an accessible overview

of more of the details, see Soames 1999. For a more mathematically

rich exposition, see McGee 1991.

##### Suitable conditionals

Logics like (K_3) suffer from the lack of a natural or

‘suitable’ conditional (in particular, one that satisfies

(A,A rightarrow B vdash B) and (vdash A rightarrow A)). This

reveals a limitation of the Kripkean approach to the Liar. The

language (mathcal{L}^{+}_0) cannot report the capture and release

properties of truth itself in conditional form (i.e.,

T-biconditionals): (Tr) is transparent on this picture, and so

(Tr(leftulcorner Arighturcorner)) and (A) are fully

intersubstitutable. We don’t have (neg A vee A) true for all

sentences (A) in this theory, and hence don’t have (neg

Tr(leftulcorner Arighturcorner) vee) A for all (A). But

(neg Tr(leftulcorner Arighturcorner) vee A) is equivalent to

(Tr(leftulcorner Arighturcorner) rightarrow A) in the theory,

since (in the theory) (rightarrow) is just the material

conditional. The Kripke construction at hand, then, thus fails to

enjoy all T-biconditionals—the natural candidates for expressing

in the theory the basic capture and release features of truth.

A recent, major step towards supplementing Kripke’s framework

with a suitable conditional is that of Field (2008). Field’s

theory is a major advance, but complex enough to be beyond the scope

of this (very basic) introduction. Readers should consult

Field’s own discussion for a taste of how such a modification

might proceed. See Field (2008), and further discussion in Beall

(2009).

One important use for conditionals in logic is in formalizing

*restricted universal quantification*, expressing the

connection between (A) and (B) in ‘All (A)s are

(B)s’. This has recently played a key role in a number of

discussions of conditionals and paradoxes; see for example Beall et

al. (2006); Beall (2011); Field (2014); and Ripley (2015).

#### 4.1.2 Paraconsistent

As we mentioned, two important approaches to the Liar paradox that

focus on non-classical logics are paracomplete and paraconsistent

approaches. We sketched a paracomplete option above. We now turn to a

paraconsistent option. Here, the basic idea is to allow the

contradiction (e.g., up to and including step 4 of the derivation in

§2.3.3),

but alter the logic by rejecting EFQ—and, hence, avoid the

absurdity involved in step 5.

Like the paracomplete approach we just surveyed, paraconsistent

approaches to the Liar find easy, natural motivation in transparency

or otherwise suitably ‘minimalist’ views of truth that

require full intersubstitutability of (A) and (Tr(leftulcorner

Arighturcorner)), and thus cannot restrict capture and release. But

paraconsistent approaches have also found motivation in a

Dummett-inspired anti-deflationist view, which takes the role of truth

as the aim of assertion seriously (cf. Dummett 1959). Indeed, Priest

(2006) argues that this (non-transparency) view of truth motivates

both the T-schema and LEM, and that this implies that the Liar

sentence (L) is both true and not true. Hence, according to any such

dialetheic line (according to which at least one sentence is both true

and not true), the only option is to reject EFQ.

##### Dialetheism

Priest (1984, 2006) has been one of the leading voices in advocating a

paraconsistent approach to solving the Liar paradox. He has proposed a

paraconsistent (and non-paracomplete) logic now known as *LP*

(for *L*ogic of *P*aradox), which retains LEM, but not

EFQ.^{[10]}

It has the distinctive feature of allowing true contradictions. This

is what Priest calls the dialetheic approach to truth. (See the entry

on

dialetheism

for a more extensive discussion.)

Formally, *LP* can be seen as a three-valued logic; but where

(K_3) has truth-value gaps, *LP* has truth-value

*gluts*. Thus, sentences in *LP* can be both true and

false. However, as we discuss further in

section 4.1.3,

just how to describe both gaps and gluts is a delicate matter. For

now, we only make the rough observation that in the same sense that

(K_3), in virtue of having a third truth value, can be said to have

gaps, *LP* correspondingly has gluts.

Likewise, Kripke-style techniques can be applied to produce an

interpretation for a truth predicate, starting with a classical

language (mathcal{L}_0) not containing a truth predicate. Again, an

extension and anti-extension are assigned to (Tr). Whereas

Kripke’s original construction had the extension and

anti-extension disjoint but not exhausting the domain, in this case we

allow the extension and anti-extension to overlap, but suppose that

the two together exhaust the domain of the model. This implements the

idea of gluts, as the earlier version implemented the idea of gaps.

Related techniques to Kripke’s can then be used to build an

extension and anti-extension for (Tr). The result is again an

interpretation where (A) and (Tr(leftulcorner

Arighturcorner)) get the same truth value in the model.

This construction was not given by Kripke himself, but variants have

been pursued by a number of authors, including Dowden (1984), Leitgeb

(1999), Priest (1984, 2006), Visser (1984), and Woodruff (1984).

##### Combining paracompleteness and paraconsistency

Though we have identified paracomplete and paraconsistent approaches

to the Liar as two distinct options, they are not incompatible.

Indeed, seen as theories of negation (if one wants), one might think

that negation is neither exhaustive nor ‘explosive’

– i.e., satisfies neither LEM nor EFQ. An approach like this is

the *FDE*-based (transparent) truth theory discussed in Dunn

1969 (see

Other Internet Resources);

Gupta and Belnap 1993; Leitgeb 1999; Visser 1984; Woodruff 1984;

Yablo 1993a; and—in effect—Brady 1989.

(The *LP*-based theories and (K_3)-based theories

are—at least on one (standard-first-order) level—simply

strengthened logics of the broader *FDE* logic. For general

discussion of such frameworks, see, e.g. Priest 2008.)

#### 4.1.3 Expressive power and ‘revenge’

Working in classical logic, Tarski (1935) famously concluded from the

Liar paradox that a language cannot define its own truth predicate.

More generally, he took the lesson of the Liar to be that languages

cannot express the full range of semantic concepts that describe their

own workings. One of the main goals of the non-classical approaches to

the Liar we have surveyed here is to avoid this conclusion, which many

have seen as far too drastic. However, how successful these approaches

have been in this regard remains a highly contentious issue.

In one sense, both the paracomplete and paraconsistent approaches

achieve the desired result: they present languages which contain truth

predicates which apply to sentences of that very language, and have

the feature that (A) and (Tr(leftulcorner Arighturcorner))

have the same truth value. In this respect, they both present

languages which contain their own truth predicate.

In the paracomplete case, the issue of whether this suffices has been

much debated. The paracomplete view holds that the Liar sentence (L)

is neither true nor false, and this is key to retaining consistency.

But note, the paracomplete approach we discussed above cannot state

this fact, as it cannot come out true that (neg Tr(leftulcorner

Lrighturcorner)). If this were true, then (L) would be true, and

then (Tr(leftulcorner Lrighturcorner)) would be true, bringing

us back to contradiction.

One further point follows from this. As we alluded to above, this

shows that (K_3) with a truth predicate will not state the gappy

status of gaps, while *LP* will state both gap and glut

properties. Hence, as we mentioned, the status of gaps and gluts can

be complicated.

For the issue of revenge, the key problem is simply that the

paracomplete approach cannot accurately state its own solution to the

Liar. Just what to make of this has been debated. It is certainly the

case that the set of true sentences in the kind of model Kripke

constructs does not include (neg Tr(leftulcorner

Lrighturcorner)). Because of this, some authors, such as McGee

(1991), T. Parsons (1984), and Soames (1999) have in effect maintained

that the Liar sentence failing to be true is a further fact that is

goes beyond what the truth predicate needs to express, and so is

immaterial to the success of the solution to the Liar. (Actually,

McGee’s view has another aspect, which we discuss in

§4.2.3.)

$$

But nonetheless, it does appear that there is an important semantic

fact about truth in the paracomplete language, closely related to if

not identical to a fact about truth *per se*, which the

language cannot express. It thus has been argued to fail to achieve a

fully adequate theory of truth. Kripke himself notes that there are

some semantic concepts that cannot be expressed, and the argument has

been pressed by C. Parsons (1974).

One way of spelling out what is missing in the paracomplete language

is to introduce a new notion of *determinateness*, so that the

status of the Liar is that of not being determinately true. If so,

then the Kripke paracomplete language cannot express this concept of

determinateness. Some approaches taking paracomplete ideas on board

have sought to supplement the Kripke approach by adding notions of

determinate truth. McGee (1991) does so in a basically classical

setting. In a non-classical, paracomplete setting, Field (2008)

supplements the basic paracomplete approach with infinitely many

different ‘determinately’ operators, each defined in terms

of Field’s ‘suitable conditional’, and each giving a

different (stronger) notion of ‘truth’. (See also some of

the papers in Beall (ed.) 2008.)

It is often argued in favor of paraconsistent approaches that they

have no trouble ‘characterizing’ the status of Liars:

they’re true and false (i.e., true and have true negation).

*LP* theories can state this. On the other hand, some such as

Littmann and Simmons (2004) and S. Shapiro (2004), have thought that

there is a dual problem: namely, characterizing ‘normal’

sentences that are not both true and false. (Some put this alleged

problem as the problem of characterizing being *just true*.)

Whether this is a problem is something we leave open. (For some

discussion, see Field 2008 and Priest 2006.)

One other issue that arises here is that of so-called ‘revenge

paradoxes’. We can illustrate this with the simple-falsity Liar.

Suppose one starts with this as the bench-mark Liar paradox, and

proposes a simple solution that rejects bivalence. In response one is

shown the simple-untruth Liar, which undercuts the simple solution.

This is the pattern of ‘revenge’, where a solution to the

paradox is rejected on the basis of what might be taken to be a

slightly modified form of the paradox. Revenge paradoxes for

paracomplete solutions are often proposed: many points where the

paracomplete language fails to express some semantic concept offer

ways to construct a revenge problems. Failing to correctly state the

status of the simple-untruth Liar is one example. Another example

involves the notion of determinateness. If we take the determinateness

route, and assign the Liar sentence the status of not being

determinately true, then one can construct a revenge problem via a

sentence which says of itself that it is not determinately true.

In a similar vein, it is sometimes argued that paraconsistent

approaches face a kind of revenge problem, as they have to treat the

Curry paradox we discussed in

section 1.4

separately from the Liar. This is a somewhat difficult technical

issue, as it depends on the nature of the conditional used to

formulate the Curry sentence. If that conditional obeys the detachment

property, then it cannot be a glut, as the Liar is in paraconsistent

settings. But, whether that is the correct approach to the conditional

has been controversial. For more discussion, see Beall (2014, 2015).

We have seen at least some approaches (e.g., McGee 1991; in some

respects, T. Parsons 1984 and Soames 1999) reject the revenge problem,

while some seek to solve it by additional apparatus (e.g., Field

2008). As we discuss further in

§4.3,

contextualist views such as those of Burge (1979), Glanzberg (2004a), and C. Parsons (1974) tend to see revenge not as a separate

problem, but as the core Liar phenomenon. For more discussion about

revenge and its nature, see the papers in Beall (ed.) (2008) and L.

Shapiro (2006).

### 4.2 Substructural logics

There is another way to see the paradox as arising from mistaken

assumptions built into standard logics. This way doesn’t see the

trouble as attaching to any particular connective or piece of

vocabulary, but instead as attaching to some of the *structural
rules* that govern the consequence relation in question. These

approaches, based on so-called

*substructural logics*, fall

into three main camps: the noncontractive, the nontransitive, and the

nonreflexive. (There is a great deal more diversity among

substructural logics than this suggests; in particular, many do not

fall into any of these camps, or fall into more than one. But these

are the three that seem to be best-suited for addressing the paradoxes

we are concerned with here.)

#### 4.2.1 Noncontractive logics

The best-developed substructural approach to paradoxes works by

attacking the structural rule of *contraction*. Contraction is

the principle that tells us that whenever (Gamma , A, A vdash B),

then (Gamma , A vdash B); that is, it is the principle that tells

us that we can use premises repeatedly while only counting them once.

Returning to the argument given in

section 2,

we can see that in two cases, an assumption is used twice in reaching

a conclusion: assumption 2a is used twice on the way to 2d, and

assumption 3a is used twice on the way to 3d. As we presented the

argument, we did not call attention to this feature, but it is one

place a noncontractive approach will focus.

The details of the response will depend on how the connectives we have

written as ‘(vee)’ and ‘(wedge)’ are

interpreted; in the absence of contraction, each of conjunction and

disjunction comes in ‘additive’ and

‘multiplicative’ flavours, and different proponents of

noncontractive views differ in which of these they acknowledge. The

difference between additive and multiplicative conjunction is this: an

additive conjunction can do the work that *either* of its

conjuncts can do, while a multiplicative conjunction can do the work

that *both* of its conjuncts can do together. In the presence

of contraction, the additive conjunction suffices for the

multiplicative: it can be used once to fill the role of the first

conjunct and again to fill the role of the second conjunct.

Contraction allows these two uses to count as one. Without

contraction, though, the additive conjunction need not suffice for the

multiplicative. (The multiplicative conjunction suffices for the

additive in the presence of a structural rule called weakening, not

otherwise discussed in this article.) The situation for disjunction is

dual: in the presence of contraction, the additive disjunction

suffices for the multiplicative, but it need not otherwise.

The double use pointed to above will loom largest if these connectives

are read multiplicatively: if 2d really is to do the work of (neg

Tr(leftulcorner Lrighturcorner)) and (Tr(leftulcorner

Lrighturcorner)) *together*, then it really does use two

copies of 2a, one for each conjunct. On an additive reading of 2d and

3d, this seeming double use need not be troubling, since 2d itself

only needs to do the work of one of its conjuncts. Although this can

be either one, whichever conjunct it is, a single use of 2a will

suffice. On this additive reading, it is the principles LEM and EFQ

that come into question; for example, with (wedge) read additively

it takes *two* occurrences of the same contradiction to entail

an arbitrary sentence (since both conjuncts must be used), while the

derivation above only yields one. (The situation for 3d and 3a is

similar, in either case.) We do not here consider further details; for

more on these choices and noncontractive approaches in general, see

Beall and Murzi (2013), Grishin (1982), Petersen (2000), Restall

(1994), Ripley (2015), L. Shapiro (2011a, 2015), and Zardini (2011,

2013). (Some of these focus on set-theoretic paradoxes rather than

truth-theoretic paradoxes, but many of the issues are parallel. See

also the entry on

Russell’s paradox.)

#### 4.2.2 Nontransitive logics

Another kind of substructural approach works by attacking various

structural rules associated with *transitivity* of consequence.

The best-known of these rules is the rule of *cut*, which

allows us to move from (Gamma vdash B) and (Delta , B vdash C)

to (Delta , Gamma vdash C). But it can also be worth considering

other transitivity-related properties, such as the one called

*simple transitivity* in Weir 2015, proceeding from (A vdash

B) and (B vdash C) to (A vdash C). (That is, simple

transitivity is the special case of cut where (Delta) is empty and

(Gamma) is a singleton.)

Some nontransitive approaches can be understood through the same

three-valued models as are used for K(_3) and LP (again, we refer

you to the entry on

many-valued logic

for details). The difference is in how consequence is defined on

these models. In all cases, consequence amounts to the absence of a

countermodel, but there are different understandings available of what

a model has to be like to be a countermodel to an argument. Depending

on what understanding of countermodel is adopted, the very same

three-valued models can give rise to the paracomplete logic K(_3),

the paraconsistent logic LP, a paracomplete and paraconsistent logic

sometimes called S(_3) or FDRM, or—our present topic—two

different logics that include counterexamples to the rule of cut, and

have come to be known as *nontransitive*.

One kind of approach without cut is developed and defended in Weir

2005, 2015 (and for naive set theory in Weir 1998, 1999), and is there

dubbed ‘neoclassical’. On this approach, the third value

in the models is taken to be neither true nor false, and a

countermodel to an argument from (Gamma) to (B) must either: make

every sentence in (Gamma) true and (B) untrue, or else make (B)

false and every sentence *but one* in (Gamma) true, while

making that remaining sentence in (Gamma) unfalse. The motivating

idea is that valid arguments must preserve truth, and must also

preserve falsity backwards in a certain sense: if a valid argument has

all its premises but one true and its conclusion false, then the

remaining premise must be false. This allows for counterexamples to

cut, but not to simple transitivity, and allows for consistency to be

maintained. The resulting logic is weaker than classical logic. In our

version of the liar paradox, the trouble is at LEM: Weir’s

approach allows for counterexamples to excluded middle.

A different kind of approach without cut is developed and explored in

Barrio et al. 2015; Cobreros et al. 2013, 2015; Fjellstad 2016; and

Ripley 2013a, 2015. On this approach, a countermodel to an argument

cannot assign the third value to any sentence that occurs in the

argument. That is, a countermodel to an argument from (Gamma) to

(B) must do just what a classical countermodel does with regards to

the argument. If it assigns the third value to any sentence at all,

that sentence cannot be in (Gamma) and it cannot be (B). This

allows for counterexamples to cut, and unlike Weir’s approach,

it also allows for counterexamples to simple transitivity. It also has

the curious feature that every argument valid in classical logic

remains valid. That is, all the counterexamples to cut and simple

transitivity involve appeal to capture, release, or some other special

behaviour of the truth predicate. Despite this classical flavour,

these approaches are also dialetheist; the claim that the liar

sentence is both true and not true turns out to be a theorem. Such a

claim is forced to take the third value, and so there can be no

countermodel to any argument involving it.

Perhaps because of the importance of the rule of cut in proof theory,

nontransitive approaches are often studied via proof systems rather

than via models. The essential use of transitivity properties in

paradoxical derivations was noted in Tennant 1982; an approach to

paradoxes that rejects both cut and simple transitivity in a general

setting can be found in Hallnäs 1991; Hallnäs and

Schroeder-Heister 1991; and Schroeder-Heister 2004. There are helpful

philosophical remarks on cut in Schroeder-Heister 1992, which also

notes some relations between noncontractive and nontransitive

approaches.

#### 4.2.3 Nonreflexive logics

A third possibility for a substructural approach to

paradoxes comes from attacking *reflexivity*, the principle

that every sentence entails itself. There is a close analogy between

reflexivity and transitivity, as explained in Frankowski 2004; Girard

et al. 1989 (p.28); and Ripley 2012, so this kind of approach ends up

having commonalities with the nontransitive family. Nonreflexive

approaches to paradoxes have so far been less-explored, but seem to be

a promising direction for further work; see French (2016) and Meadows

(2014) for more. See also Malinowski (1990) for general work on

nonreflexive logics.

### 4.3 Classical logic

We have now seen a range of options for responding to the Liar paradox

by reconsidering basic logic. There are also a number of approaches

that leave classical logic fixed, and try to find other ways of

defusing the paradox.

One hallmark of most of these approaches is a willingness to somehow

*restrict* the range of application of capture and release, to

block the paradoxical reasoning. This is antithetical to the kind of

deflationist view of truth we discussed in

§4.1,

but it is consistent with another view of truth. This other view

takes the main feature of truth to be that it reports a non-trivial

semantic property of sentences (e.g., corresponding with a fact in the

world, or having a value in a model). Many approaches within classical

logic embody the idea that a proper understanding of this feature

allows for restricted forms of capture and release, and this in turn

allows the paradox to be blocked, without any departure from classical

logic.

We will consider a number of important approaches to the paradox

within classical logic, most of which embody this idea in some form or

another.

#### 4.3.1 Tarski’s hierarchy of languages

Traditionally, the main avenue for resolving the paradox within

classical logic is Tarski’s hierarchy of languages and

metalanguages. Tarski concluded from the paradox that no language

could contain its own truth predicate (in his terminology, no language

can be ‘semantically closed’).

Instead, Tarski proposed that the truth predicate for a language is to

be found only in an expanded metalanguage. For instance, one starts

with an interpreted language (mathcal{L}_0) that contains no truth

predicate. One then ‘steps up’ to an expanded language

(mathcal{L}_1), which contains a truth predicate, but one that only

applies to sentences of (mathcal{L}_0). With this restriction, it

is easy enough to define a truth predicate which completely accurately

states the truth values of every sentence in (mathcal{L}_0), obeys

capture and release, and yields no paradox. Of course, this process

does not stop. If we want to describe truth in (mathcal{L}_1), we

need to step up to (mathcal{L}_2) to get a truth predicate for

(mathcal{L}_1). And so on. The process goes on indefinitely. At

each stage, a new classical interpreted language is produced, which

expresses truth for languages below it. (For more on the mathematics

of this sort of hierarchy of languages, see Halbach (1997).)

Why is there no Liar paradox in this sort of hierarchy of languages?

Because the restriction that no truth predicate can apply to sentences

of its own language is enforced as a syntactic one. Any sentence (L)

equivalent to (neg Tr(leftulcorner Lrighturcorner)) is not

syntactically well-formed. There is no Liar paradox because there is

no Liar sentence. See the entries on

Tarski

and

Tarski’s truth definitions

for more on Tarski’s views of truth.

Tarski’s hierarchical approach has been subject to a number of

criticisms. One is that in light of naturally occurring cases of

self-reference, his ruling Liar sentences syntactically not

well-formed seems overly drastic. Though Tarski himself was more

concerned to resolve the Liar for formal languages, his solution seems

implausible as applied to many naturally occurring uses of

‘true’. Another important problem was highlighted by

Kripke (1975). As Kripke notes, any syntactically fixed set of levels

will make it extremely hard, if not impossible, to place various

non-paradoxical claims within the hierarchy. For instance, if Jc says

*that everything Michael says is true*, the claim has to be

made from a level of the hierarchy higher than everything Michael

says. But if among the things Michael says is *that everything Jc
says is true*, Michael’s claims must be at a higher level

than all of Jc’s claims. Thus, some of Michael’s claims

must be higher than some of Jc’s, and

*vice versa*. This

is impossible. It is also difficult to explain what level of the

hierarchy an utterance winds up at when it can be coherently assigned

a level. What fixes that it is to involve truth at one level rather

than another?

Another challenge Tarski’s hierarchy faces is explaining why we

cannot just define truth for the whole hierarchy, by quantifying over

levels. We would thus have a predicate like ‘true at some

level’. If such predicates are allowed, we are back in paradox,

so defenders of the Tarskian hierarchy must say they are not possible.

Explaining why is a problem for all hierarchical views. (See Glanzberg

(215) for further discussion.)

In light of these sorts of problems, many have concluded that

Tarski’s hierarchy of languages and metalanguages buys a

solution to the Liar paradox at the cost of implausible

restrictiveness.

#### 4.3.2 The closed-off Kripke construction

In light of these sorts of criticisms of Tarski’s theory, a

number of approaches to the Liar have sought to retain classical

logic, but have some degree of self-applicability for the truth

predicate. We know from the reasoning in

§2.3

that some restrictions on capture and release will then be required.

One goal has been to work out which ones are well-motivated, and how

to implement them.

One way to do this was suggested by Kripke himself. Rather than see

the Kripke apparatus we reviewed briefly in

§4.1.1

as part of a non-classical logical approach, one can see it as an

intermediate step towards building a classical interpretation of a

self-applicative (Tr).

Recall that the Kripke construction starts with a classical language

(mathcal{L}_0) with no truth predicate. It passes to an expanded

language (mathcal{L}^{+}_0), but unlike a Tarskian metalanguage,

this language contains a truth predicate (Tr) that applies to all

of (mathcal{L}^{+}_0). Kripke shows how to build a partial

interpretation of (Tr), providing an extension (mathcal{E}) and

an anti-extension (mathcal{A}). But one can then simply consider

the *classical* model (langle

mathcal{M}_0,mathcal{E}rangle), using only the extension. This is

the ‘closed-off’ construction, as the gap between

extension and anti-extension is closed off by throwing everything in

the gap into the false category of a classical model.

We know this interpretation cannot make true all of capture and

release (nor the full intersubstitutability of (A) and

(Tr(leftulcorner Arighturcorner))). But it does make a

restricted form true. The following holds in the closed-off model:

Arighturcorner)] rightarrow[Tr(leftulcorner Arighturcorner)

leftrightarrow A].]

This tells us that capture and release (in the form of the

T-schema) holds for sentences that are well-behaved, in the sense of

satisfying (Tr(leftulcorner Arighturcorner) vee

Tr(leftulcorner neg Arighturcorner)).

What happens to the Liar sentence on this approach? As in the

three-valued case, the Liar is interpreted as falling within the gap.

(L) is neither in (mathcal{E}) nor (mathcal{A}. L) thus falls

outside of the domain where (Tr) is interpreted as well-behaved.

Because the situation is classical, and (leftulcorner

Lrighturcornernotin mathcal{E}), we know that (neg

Tr(leftulcorner Lrighturcorner)) is true in the closed-off

model; likewise, so is (neg Tr(leftulcorner neg

Lrighturcorner)).

On well-behaved sentences, we have the fixed point property that (A)

and (Tr(leftulcorner Arighturcorner)) have the same truth

value, and so the semantics of (mathcal{L}^{+}_0) and the semantics

it assigns to (Tr) correspond exactly. On pathological sentences

like (L), they do not, and indeed, cannot, on pain of

triviality.

In a point related to the closed-off construction, it was observed by

Feferman (1984) that if we are careful about negation, we can dispense

with (mathcal{A}) altogether in the Kripke construction. Thus, the

construction can be done without any implicit appeal to many-valued

logic. Related ways of thinking about Kripke’s construction are

discussed by McGee (1991).

#### 4.3.3 Determinateness revisited

In

§4.1.3

we noted that paracomplete approaches to the paradox can be

vulnerable to ‘revenge paradoxes’ based on some idea of

indeterminate truth or lacking a truth value. Related issue bear in

the classical case. We will discuss a few in turn.

##### Grounding

The closed-off Kripke construction can help fill in the idea of a

*determinately* operator discussed in

§4.1.3.

Instead of an operator, it allows us to define a predicate

(D(leftulcorner Arighturcorner)) by (Tr(leftulcorner

Arighturcorner) vee Tr(leftulcorner neg Arighturcorner). D)

represents ‘determinately’ in the sense of applying to

sentences that have a truth value according to (Tr), as it were,

‘determined’ by the model produced by the Kripke

construction. It also, as we observed, applies to all the sentences

which are well-behaved in the sense of obeying the T-schema (or

capture and release).

Formally, the sentences to which (D) applies in the model generated

by the Kripke construction are those which fall in (mathcal{E}) or

have their negations fall in (mathcal{E}) (equivalently fall in

(mathcal{A})). Kripke labeled this being

*grounded*.^{[11]}

It has often been noted that there is also a more informal notion of

determinateness or grounding, to which the formal notion expressed by

(D) at least roughly corresponds (cf. Herzberger 1970). The idea is

that the determinate sentences are the ones with well-defined semantic

properties. Where we have no such well-defined semantic properties, we

should not expect the truth predicate to report anything well-behaved,

nor should we expect properties like capture and release to hold.

Kripke’s construction builds up (mathcal{E}) in stages,

starting with sentence with no semantic terms, and adding semantic

complexity at each stage. One reaches (mathcal{E}) at the limit of

this process, which allows us to think of (mathcal{E}) as

indicating the limit of where semantic values are assigned by a

well-defined process. Thus, the formal notion of grounding provided by

(D) is sometimes suggested to reflect the extent to which sentences

have well-defined semantic properties.

The notion of grounding has spawned its own literature, with Leitgeb

(2005) a key impetus. See also Bonnay and van Vugt (2015), Meadows

(2013), and Schindler (2014).

##### McGee on truth and definite truth

Another view which makes use of a form of determinateness is advocated

by McGee (1991). McGee’s theory, like many we have surveyed

here, is rich in complexity to which we cannot do justice. The theory

has many components, including a mathematically sophisticated

approaches to truth related to the Kripkean ideas we have been

discussing, in a setting which holds to classical logic.

McGee relies on two notions: truth and definite truth. Definite truth

is a form of the idea we glossed as determinateness. But, McGee

describes this idea using some very sophisticated logical techniques.

We will mention them briefly, for those familiar with the technical

background. Formally, for McGee, definite truth is identified with

provability in a partially interpreted language, using an extension of

classical logic which takes in facts about the partial interpretation

known as (mathcal{A})-logic. It is thus different from the

grounding notion we just discussed. McGee treats definitely as a

*predicate*, on par with the truth predicate, and not as an

operator on sentences as some developments do. With the right notion

of definite truth, McGee shows that a partially interpreted language

containing its own truth predicate can meet restricted forms of

capture and release put in terms of definite truth. Where (Def) is

the definiteness predicate, McGee show how to link truth and definite

truth, by showing how to validate:

begin{align}

Def (leftulcorner Arighturcorner)

& textrm{ iff }

Def (leftulcorner Tr(leftulcorner Arighturcorner)righturcorner)\

Def (leftulcorner neg Arighturcorner)

& textrm{ iff }

Def (leftulcorner neg Tr(leftulcorner Arighturcorner)righturcorner)

end{align}

]

Indeed, McGee shows that these conditions can be met within a theory

of both truth and definite truth, where truth meets appropriate forms

of capture and release, and also where a formal statement of bivalence

for truth comes out definitely true. McGee thus provides a theory

which has strongly self-applicative truth and definite truth, within a

classical setting.

Though truth may satisfy the formal property of bivalence, it is

crucial to McGee’s approach that definite truth is an open-ended

notion, which may be strengthened (formally, by strengthening a

partially interpreted language). Thus, definite truth meets weaker

forms of capture and release than truth itself. (Some instances of

(Def(leftulcorner Arighturcorner) rightarrow A) fail to be

definitely true, according to McGee.) Furthermore, McGee suggests that

this behavior of truth and definite truth makes truth a *vague*

predicate. It remains disputed whether McGee’s theory avoids the

kind of revenge problems that plague other Kripkean approaches.

#### 4.3.4 Other classical approaches

We have now surveyed some important representatives of approaches to

resolving the Liar within classical logic. There are a number of

others, may of them involving some complex mathematics. We will pause

to mention a few of the more important of these, though given the

mathematical complexity, we will only gesture towards them.

##### Axiomatic theories of truth

There is an important strand of work in proof theory, which has sought

to develop axiomatic theories of self-applicative truth in classical

logic, including work of Cantini (1996), Feferman (1984, 1991),

Friedman and Sheard (1987), Halbach (2011), and Horsten (2011). The

idea is to find ways of expressing rules like capture and release that

retain consistency. Options include more care about how

proof-theoretic rules of inference are formulated, and more care about

formulating restricted rules. The main ideas are discussed in the

entry on

axiomatic theories of truth,

to which we will leave the details.

##### Truth and inductive definitions

Kripke’s work on truth was developed in conjunction with some

important ideas about inductive definitions (as we see, for instance,

in the later parts of Kripke 1975). These connections are explored

further in work of Burgess (1986) and McGee (1991). We also pause to

mention work of Aczel (1980) combining ideas about inductive

definitions and the lambda calculus.

### 4.4 Contextualist approaches

Another family of proposed solutions to the Liar are *contextualist
solutions*. These also make use of classical logic, but base their

solutions primarily on some ideas from the philosophy of language.

They take the basic lesson of the Liar to be that truth predicates

show some form of

*context dependence*, even in otherwise

non-context-dependent fragments of a language. They seek to explain

how this can be so, and rely on it to resolve the problems faced by

the Liar.

Contextualist theories share with a number of approaches we have

already seen the idea that there is something indeterminate or

semantically not well-formed about our Liar sentence (L). But,

contextualist views give a special role to issues of

‘revenge’ and lack of expressive power.

#### 4.4.1 Instability and revenge

One way of thinking about why the truth predicate is not well-behaved

on the Liar sentence is that there is not really a well-defined truth

bearer provided by the Liar sentence. To make this vivid (as discussed

by C. Parsons (1974)), suppose that

truth bearers are propositions expressed by sentences in contexts, and

that the Liar sentence fails to express a proposition. This is the

beginnings of an account of how the Liar winds up ungrounded or in

some sense indeterminate. At least, we should not expect (Tr) to be

well-behaved where sentences fail to express propositions.

But, it is an unstable proposal. We can reason that if the Liar

sentence fails to express a proposition, it fails to express a true

proposition. In the manner of a revenge paradox, if our Liar sentence

had originally said ‘this sentence does not express a true

proposition’, then we would have our Liar sentence back. And, we

have shown that this sentence says something true, and so expresses a

true proposition. Thus, from the assumption that the Liar sentence is

indeterminate or lacks semantic status, we reason that it must have

proper semantic status, and indeed say something true. We are hence

back in paradox.

Contextualists do not see this as a new ‘revenge’ paradox,

but the basic problem posed by the Liar. First of all, in a setting

where sentences are context dependent, the natural formulation of a

truth claim is always in terms of expressing a true proposition, or

some related semantically careful application of the truth predicate.

But more importantly, to the contextualist, the main issue behind the

Liar is embodied in the reasoning on display here. It involves two key

steps. First, assigning the Liar semantically defective

status—failing to express a proposition or being somehow

indeterminate. Second, concluding from the first step that the Liar

must be true—and so not indeterminate or failing to express a

proposition—after all. Both steps appear to be the result of

sound reasoning, and so the conclusions reached at both must be true.

The main problem of the Liar, according to a contextualist, is to

explain how this can be, and how the second step can be

non-paradoxical. (Such reasoning is explored by Glanzberg (2004c) and C. Parsons (1974). For a critical discussion, see Gauker

(2006).)

Thus, contextualists seek to explain how the Liar sentence can have

unstable semantic status, switching from defective to non-defective in

the course of this sort of inference. They do so by appealing to the

role of *context* in fixing the semantic status of sentences.

Sentences can have different semantic status in different contexts.

Thus, to contextualists, there must be some non-trivial effect of

context involved in the Liar sentence, and more generally, in

predication of truth.

#### 4.4.2 Contextual parameters on truth predicates

One prominent contextualist approach, advocated by Burge (1979) and

developed by Koons (1992) and Simmons (1993), starts with the idea

that the Tarskian hierarchy itself offers a way to see the truth

predicate as context dependent. Tarski’s hierarchy postulates a

hierarchy of truth predicates (Tr_i). What if (i) is not merely a

marker of level in a hierarchy, but a genuine contextual parameter? If

so, then the Liar sentence is in fact context-dependent: it has the

form (neg Tr_i (leftulcorner Lrighturcorner)), where (i) is

set by context. Context then sets the level of the truth

predicate.

This idea can be seen as an improvement on the original Tarskian

approach in several respects. First, once we have a contextual

parameter, the need to insist that Liar sentences are never

well-formed disappears. Hence, we can think of each (Tr_i) as

including some limited range of applicability to sentences of its own

language. Using the Kripkean techniques likes the closed-off

construction we reviewed above, predicates like (Tr_i) can be

constructed which have as much self-applicability as Kripke’s

own. (Burge 1979 and the postscript to C. Parsons 1974 consider

briefly how Kripkean techniques could be applied in this setting.

Though he works in a very different setting, ideas of Gaifman (1988,

1992) can be construed as showing how even more subtle ways of

interpreting a context-dependent truth predicate can be

developed.)

With suitable care, other problems for the Tarskian hierarchy can be

avoided as well. Burge proposes that the parameter (i) in (Tr_i)

is set by a Gricean pragmatic process. In effect, speakers implicate

that (i) is to be set to a level for which the discourse they are in

can be coherently interpreted (with a maximal coherent extension for

(Tr_i)). Thus, truth does indeed find its own level, and so

Kripke’s objection about how to fix levels for non-paradoxical

sentences may be countered.

This approach gives substance to the idea that the Liar sentence is

context dependent. Any sentence containing (Tr_i) will be context

dependent, inheriting a contextual parameter along the way. This

offers a way to make sense of the arguments for the instability of the

semantic status of (L) that motivated contextualism. In an initial

context, we fix some level (i). This is the level at which (L) is

interpreted. Call this interpretation (L_i . L_i) says (neg Tr_i

(leftulcorner L_{i}righturcorner)). By the usual Liar reasoning,

we show that (L_i) must lack determinate semantic status—or

fail to express a proposition. As we discussed, we then reason that

(L) must come out true. According to the contextualist view at hand,

this is the claim that (L_i) is true according to some other

context, where a wider truth predicate is in play. This amounts to

being true at some higher level of the hierarchy. We can conclude, for

instance, that the Liar sentence as it was used at level (i) is true

according to a wider level (k gt i). Hence, (Tr_k (leftulcorner

L_{i}righturcorner)), where (k gt i).

This form of contextualism thus maintains that once we see the

context-dependent behavior of (Tr_i), we can make good sense of the

instability of (L). This can be seen as an improvement on both the

Tarskian view, and embodying some of the techniques of classical logic

we reviewed in

§4.2.

Depending how the Burge view is spelled out technically, it will

either have full capture and release at each level, or capture and

release with the same restrictions as the closed-off Kripke

construction.

The view that posits contextual parameters on the truth predicate does

face a number of questions. For instance, it is fair to ask why we

think the truth predicate really has a contextual parameter,

especially if we mean a truth predicate like the one we use in natural

language. Merely noting that such a parameter would avoid paradox does

not show that it is present in natural language. Furthermore, whether

it is acceptable to see truth as coming in levels at all,

context-based or not, remains disputed. (Not all those who advocate

contextual parameters on the truth predicate agree about the role of

hierarchy. In particular, Simmons (1993) advocates a view he labels

the ‘singularity theory’ which he proposes avoids outright

hierarchical structures.) Finally, the Burgean appeal to Gricean

mechanisms to set levels of truth has been challenged. (For instance,

Gaifman (1992) asks if the Gricean process does any substantial work

in Burge’s account.)

Contextualist approaches come in many varieties, each of which makes

use of slightly different apparatus. With contextualist theories the

choice often turns on issues in philosophy of language as well as

logic. We already noted a different way of developing contextualist

ideas from Gaifman (1988, 1992). We will now briefly review a few more

alternatives.

#### 4.4.3 Contextual effects on quantifier domains

Another contextualist approach, stemming from work of C. Parsons

(1974), seeks to build up the

context dependence of the Liar sentence, and ultimately the context

dependence of the truth predicate, from more basic components. The key

is to see the context dependence of the Liar sentence as derived from

the context dependence of quantifier domains.

Quantification enters the picture when we think about how to account

for predication of truth when sentences display context dependence. In

such an environment, it does not make good sense to predicate truth of

sentences directly. Not all sentences will have the right kind of

determinate semantic properties to be truth bearers; or, as we have

been putting it, not all sentences will express propositions. But

then, to say that a sentence (S) is true in context (c) is to say

that *there is* a proposition (p) expressed by (S) in

(c), and that proposition (p) is true.

The current contextualist proposal starts with the observation that

quantifiers in natural language typically have context-dependent

domains of quantification. When we say ‘Everyone is here’,

we do not mean everyone in the world, but everyone in some

contextually provided subdomain. Context dependence enters the Liar,

according to this contextualist view, in the contextual effects on the

domain of the propositional quantifier (exists p).

In particular, this domain must *expand* in the course of the

reasoning about the semantic status of the Liar. In the initial

context, (exists p) must range over a small enough domain that

there is no proposition for (L) to express. In the subsequent

context, the domain expands to allow (L) to express some true

proposition. Proposals for how this expansion happens, and how to

model the truth predicate and the relation of expressing a proposition

in the presence of the Liar, have been explored by Glanzberg (2001,

2004a), building on work of C. Parsons (1974). Defenders of this

approach argue that it does better in locating the locus of context

dependence than the parameters on truth predicates view.

#### 4.4.4 Situation theory

Another variant on the contextualist strategy for resolving the Liar,

developed by Barwise and Etchemendy (1987) and Groeneveld (1994),

relies on *situation theory* rather than quantifier domains to

provide the locus of context dependence. Situation theory is a highly

developed part of philosophy of language, so we shall again give only

the roughest sketch of how their view works.

A situation is a partial state the world might be in: something like

(a) being (F). Situations are classified by what are called

situation types. A proposition involves classifying a situation as

being of a situation type. Thus, a proposition ({s); [(sigma]})

tell us that situation (s) is of type (sigma). The situation

(s) here plays a number of roles, including that of providing a

context.

When it comes to the Liar, Barwise and Etchemendy construe Liar

propositions as having the form (f_s = {s); [(Tr,f_s); 0](}),

relative to an initial situation (s). This is a proposition (f_s)

which says of itself that its falsity is a fact that holds in (s).

(In Barwise and Etchemendy’s notation, the 0 indicates falsity,

so the situation type is that the state of affairs of the proposition

being false holds. The proposition says this is a fact that holds in

(s).) There is a sense in which this proposition cannot be

expressed. In particular, the state of affairs (langle Tr,f_s);

(0rangle) cannot be in (s). (Actually, Barwise and Etchemendy say

that the proposition is expressible, but give up on what they call the

(F)-closure of (s). But there is a core observation in common

between these two points, and the details do not matter for our

purposes here.) There is then a distinct situation (s’ = s cup

{langle Tr,f_s); (0rangle }), and the proposition ({s’);

[(Tr,f_s); 0](}) relative to this new situation—this new

‘context’—is true.

This idea clearly has a lot in common with the restriction on

quantifier domains view. In particular, both approaches seek to show

how the domain of contents expressible in contexts can expand, to

account for the instability of the Liar sentence. For discussion of

relations between the situation-theoretic and quantifier domain

approaches, see Glanzberg (2004a). Barwise and Etchemendy discuss

relations between their situation-based and a more traditional

approach in 1987 (Ch. 11). For a detailed match-up between the Barwise

and Etchemendy framework and a Burgean framework of indexed truth

predicates, see Koons (1992).

#### 4.4.5 Issues for contextualism

It is a key challenge to contextualists to provide a full and

well-motivated account of the source and nature of the shift in

context involved in the Liar, though of course, many contextualists

believe they have met this challenge. In favor of the contextualist

approach is that it takes the revenge phenomenon to be the basic

problem, and so is largely immune to the kinds of revenge issues that

affect other approaches we have considered. But, it may be that there

is another form of revenge which might be applied. To retain

consistency, contextualists must apply restrictions on quantifiers to

such quantifiers as ‘all contexts’. To achieve this, it

must presumably be denied that there are any absolutely unrestricted

quantifiers. Glanzberg (2004b, 2006) argues this is the correct

conclusion, but it is highly controversial. For a survey of thinking

about this, see the papers in Rayo and Uzquiano 2006.

### 4.5 The revision theory

Another approach to the Liar, advocated by Gupta (1982), Herzberger

(1982), Gupta and Belnap (1993), and a number of others, is the

*revision theory of truth*. This approach shares some features

with the views we surveyed in

§4.3,

in that it takes classical logic for granted. We also believe it has

an affinity with the views discussed in

§4.4,

as it rethinks some basic aspects of semantics. But it is a

distinctive approach. We will sketch some of the fundamentals of this

view. For a discussion of the foundations of the revision theory, and

its relations to contextualism, see L. Shapiro (2006). More details,

and more references, may be found in the entry on

the revision theory of truth.

The revision theory of truth starts with the idea that we may take the

T-schema at face value. Indeed, Gupta and Belnap (1993) take up a

suggestion from Tarski (1944), that the instances of the T-schema can

be seen as partial definitions of truth; presumably with all the

instances together, for the right language or family of languages

constituting a complete definition. At the same time, the revision

theory holds fast to classical logic. Thus, we already know, we have

the Liar paradox for any language with enough expressive resources to

produce Liar sentences.

In response, the revision theory proposes a different way of

approaching the semantic properties of the truth predicate. In keeping

with our practices here, we may begin with a classical model

(mathcal{M}_0) for a language (mathcal{L}_0) without a truth

predicate, and consider what happens when we add a truth predicate

(Tr) to form the extended language (mathcal{L}^{+}_0). This

language has a full self-applicative truth predicate, and so can

generate the Liar sentence (L).

To build a classical model for (mathcal{L}^{+}_0), we need an

extension for (Tr). Let us pick a set: call it (H) for a

*hypothesis* about what the extension of (Tr) might be.

(H) may be (varnothing), it may be the entire domain of

(mathcal{M}_0), or it may be anything else. It need not be a

particularly good approximation of the semantic properties of

(Tr).

Even if it is not, (langle mathcal{M}_0,Hrangle) still provides a

classical model, in which we can interpret (mathcal{L}^{+}_0). With

that, we can in effect apply the T-schema, relative to our hypothesis

(H), and see what we get. More precisely, we can let (tau(H) =

{leftulcorner Arighturcorner | A) is true in (langle

mathcal{M}_0,Hrangle }. tau(H)) is generally a better hypothesis

about what is true in our language than (H) might have been. At

least, clearly, if (H) made foolish guesses about the truth of

sentence of the truth-free fragment (mathcal{L}_0), they are

corrected in (tau(H)), which contains everything from

(mathcal{L}_0) true in (mathcal{M}_0). Thus, (langle

mathcal{M}_0,tau(H)rangle) is generally a better model of

(mathcal{L}^{+}_0) then (langle mathcal{M}_0,Hrangle).

Better in many respects. But when it comes to paradoxical sentences

like (L), we see something different. As a starting hypothesis, let

us consider (H = varnothing). Consider what happens to the truth of

(L) as we apply (tau):

(n) | truth value of (L) in (langle mathcal{M}_0,tau^n (varnothing)rangle) |

0 | true |

1 | false |

2 | true |

3 | false |

4 | true |

(vdots) | (vdots) |

The Liar sentence never stabilizes under this process. We reach an

alternation of truth values which will go on for ever. This shows,

according the revision theory, that truth is a circular concept. As

such, it does not have an extension in the ordinary sense. Rather, it

has a rule for revising extensions, which never stabilizes.

In the terminology of the revision theory, (tau) is a *revision
rule*. It takes us from one hypothesis about the interpretation of

(Tr) to another. Sequences of values we generate by such revision

rules, starting with a given initial hypothesis, are

*revision*

sequences. We leave to a more full presentation the important

sequences

issue of the right way to define

*transfinite*revision

sequences. (See the entry on

revision theories of truth.)

The characteristic property of paradoxical sentences like the Liar

sentence is that they are unstable in revision sequences: there is no

point in the sequence at which they reach a stable truth value. This

classifies sentences as stably true, stably false, and unstable. The

revision theory develops notions of consequence based on these, and

related notions. See the entry on

revision theories of truth

for further exposition of this rich theory.

### 4.6 Inconsistency views

In

§2.3.3

we saw that the Liar paradox, in the presence of unrestricted capture

and release and classical logic, leads to contradiction. So long as we

have EFQ (as classical logic does), this results in triviality. Most

of the proposed solutions we have considered (with the exception of

the revision theory) try to avoid this result somehow, either by

restricting capture and release or departing from classical logic. But

there is another idea that has occasionally been argued, that the Liar

paradox simply shows that the kinds of languages we speak, which

contain their own truth predicates, are *inconsistent*.

This is not an easy view to formulate. Though Tarski himself seemed to

suggest something along these lines (for natural languages,

specifically), it was argued by Herzberger (1967) that it is

impossible to have an inconsistent language.

In contrast, Eklund (2002) takes seriously the idea that our semantic

intuitions, expressed, for instance, by unrestricted capture and

release, really are inconsistent. Eklund grants that this does not

make sense if these intuitions have their source simply in our grasp

of the truth conditions of sentences. But he suggests an alternative

picture of semantic competence which does make sense of it (closely

related to conceptual role views of meaning). He suggests that we

think of semantic competence in terms of a range of principles

speakers are disposed to accept in virtue of knowing a language. Those

principles may be inconsistent. But even so, they determine semantic

values. Semantic values will be whatever comes closest to satisfying

the principles—whatever makes them maximally correct—even

if nothing can satisfy all of them due to an underlying

inconsistency.

Eklund thus supports an idea suggested by Chihara (1979).

Chihara’s main aim is to provide what he calls a

*diagnosis* of the paradox, which should explain why the

paradox arises and why it appears compelling. But along the way, he

suggests that the source of the paradox is our acceptance of the

T-schema (by convention, he suggests), in spite of its

inconsistency.

A related, though distinct, view is defended by Patterson (2007,

2009). Patterson argues that competence with a language puts one in a

cognitive state relating to an inconsistent theory—one including

the unrestricted T-schema and governed by classical logic. He goes on

to explore how such a cognitive state could allow us to successfully

communicate, in spite of relating us to a false theory.

A different sort of inconsistency theory is advocated by Scharp

(2013). Scharp argues that truth is an inconsistent concept, like the

pre-relativistic concept of mass. As such, it is unsuitable for

careful theorizing. What we need to do, according to Scharp, is

replace the inconsistent concept of truth with a family of consistent

concepts that work better. Scharp develops just such a family of

concepts, and offers a theory of them.

## 5. Concluding Remarks

There is much more to say about the Liar paradox than we have covered

here: there are more approaches to the Liar variants we have

mentioned, and more related paradoxes like those of denotation,

properties, etc. There are also more important technical results, and

more important philosophical implications and applications. Our goal

here has been to be more suggestive than exhaustive, and we hope to

have given the reader an indication of what the Liar paradox is, and

what its consequences might be.

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