Stanford University

Liar Paradox (Stanford Encyclopedia of Philosophy)

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.

[Tr(leftulcorner Arighturcorner) leftrightarrow A.]

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:

[L dashv vdash neg Tr(leftulcorner Lrighturcorner).]

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.

  1. (Tr(leftulcorner Lrighturcorner) vee neg
    Tr(leftulcorner Lrighturcorner)) [LEM]
  2. Case One:
    1. (Tr(leftulcorner Lrighturcorner))
    2. (L) [2a: release]
    3. (neg Tr(leftulcorner Lrighturcorner)) [2b: definition of
      (L)]
    4. (neg Tr(leftulcorner Lrighturcorner) wedge
      Tr(leftulcorner Lrighturcorner)) [2a, 2c: adjunction]
  3. Case Two:
    1. (neg Tr(leftulcorner Lrighturcorner))
    2. (L) [3a: definition of (L)]
    3. (Tr(leftulcorner L)righturcorner)) [3b: capture]
    4. (neg Tr(leftulcorner Lrighturcorner) wedge
      Tr(leftulcorner Lrighturcorner)) [3a, 3c: adjunction]
  4. (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.)

  1. (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:

[notvdash_{K3} A vee neg A.]

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 Logic of Paradox), 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:

[ [Tr(leftulcorner Arighturcorner) vee Tr(leftulcorner neg
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
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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