contradiction by reasoning about a Liar Sentence. The classical Liar
Sentence is the following self-referential sentence:
(L) This sentence is false.
Experts in the field of philosophical logic have never agreed on the
way out of the trouble despite 2,300 years of attention. Here is the
trouble–a sketch of the Liar Argument that reveals the contradiction:
If L is true, then L is false. But we can also establish the
converse, as follows. Assume L is false. Because the Liar Sentence is
saying precisely that (namely that it is false), the Liar Sentence is
true, so L is true.
We've now shown that L is true if and only if it is false. Since L
is one or the other, it is both.
The argument depends upon a few more assumptions and steps, but these
appear to be as uncontroversial as those above. The contradictory
result apparently throws us into the lion's den of semantic
incoherence. This article explores the details of the principal
attempts to resolve the paradox. Most logical paradoxes are based on
circular definitions or self-referential statements, and the liar
paradox is no exception.
Many people, when first encountering the Liar Paradox, will react by
saying that the Liar Sentence must be meaningless. This popular
solution does stop the argument of the paradox, but it isn't an
adequate solution if it answers the question, "Why is the Liar
Sentence meaningless?" simply with the ad hoc remark, "Otherwise we
get a paradox." An adequate solution would offer a more systematic
treatment. For example, the sentence, "This sentence is in English,"
is very similar to the Liar Sentence. Is it meaningless, too? What
ingredients of the Liar Sentence make it meaningless such that other
sentences with those same ingredients will also be meaningless? Are
disjunctions with the Liar Sentence also meaningless? The questions
continue, and an adequate solution should address them systematically.
1. History of the Paradox
The Liar Paradox has been discussed continually in philosophy since
the middle of the 4th century BCE. The most ancient attribution is to
Eubulides of Miletus who included it among a list of seven puzzles. He
said, "A man says that he is lying. Is what he says true or false?"
Eubulides' commentary on his puzzle has not been found. An ancient
gravestone on the Greek Island of Cos was reported by Athenaeus to
contain this poem about the difficulty of solving the paradox:
O Stranger: Philetas of Cos am I,
'Twas the Liar who made me die,
And the bad nights caused thereby.
Theophrastus, Aristotle's successor, wrote three papyrus rolls about
the Liar Paradox, and the Stoic philosopher Chrysippus wrote six, but
their contents are lost in the sands of time. In the New Testament of
the Bible, Saint Paul warned, "One of themselves, even a prophet of
their own, said that the Cretans are always liars." Paul, however,
gave no indication he recognized anything paradoxical about the
Cretan's remark, but it would be paradoxical if no other Cretan utters
a truth and if "liar" means utterer only of falsehoods.
In the late medieval period in Europe, Buridan put the Liar Paradox to
devious use with the following proof of the existence of God. It uses
the pair of sentences:
God exists.
None of the sentences in this pair is true.
The only consistent way to assign truth values, that is, to have these
two sentence be either true or false, requires making "God exists" be
true. So, Buridan has "proved" that God does exist.
There are many other versions of the Paradox. Some liar paradoxes
begin with a chain of sentences:
The following sentence is true.
The following sentence is true.
The following sentence is true.
The first sentence in this list is false.
The Strengthened Liar Paradox begins with the Strengthened Liar Sentence
This sentence is not true.
This version is called "Strengthened" because some promising solutions
to the classical liar paradox beginning with (L) fail completely when
faced with the Strengthened Liar. So, finding one's way out of the
Strengthened Liar is the acid test of a successful solution.
There are also Contingent Liars which depend upon what occurs in the
empirical world. Suppose that the last sentence in today's edition of
The New York Times newspaper is:
The last sentence in tomorrow's edition of The New York Times
newspaper is true.
Was that sentence grammatical? Was it meaningful? Was it true or
false, even if we don't know which at the moment? The common sense
answers are "yes" to all these questions. Perhaps we should not retain
those intuitive answers tomorrow when the Times's presses print a
newspaper whose last sentence is
The last sentence in yesterday's edition of The New York Times
newspaper is not true.
If we adopt the metaphor of a paradox as being an argument which
starts from the home of seemingly true assumptions and which travels
down the garden path of seemingly valid steps into the den of a
contradiction, then a solution to the paradox has to find something
wrong with the home, find something wrong with the garden path, or
find a way to live within the den. Less metaphorically, the main kinds
of ways out of the Paradox are the following: (a) The Liar Sentence
isn't grammatical. (b) The Liar Sentence isn't meaningful. (c) The
Liar Sentence is grammatical and meaningful but isn't true or false.
(d) There is some other error in one of the steps of the argument that
leads to the contradiction. (e) The Liar Sentence is both true and
false. Two philosophers might take one of these ways out but for very
different reasons, and they might offer different changes in our naive
system of beliefs and concepts in order to take this way out.
To put the Liar Paradox in perspective, it is essential to appreciate
why such an apparently trivial problem in fact is a deep problem.
Suppose we ask the larger question: What is truth? As a question about
what are the significant paths of life to be followed or the
significant things to know in order to have the best grasp on reality,
the question is just too difficult, and also too vague, to be a center
of attention for the analytical philosophers of the present age.
However, as a question asking simply for general characteristics of
all true sentences, the question is more amenable to solution.
Nevertheless, it is still a very difficult one. For instance, in the
attempt to generally characterize the grounds of validity of a true
sentence, that is, in the attempt to characterize why a true sentence
is true, philosophers have created several ingenious, and alluring
theories of truth: the correspondence theory of truth, the coherence
theory of truth, and the pragmatic theory of truth, among others. Yet
none of these has produced any detailed theory. At best, each is still
at the stage of being a suggestive, but uncompelling, metaphor.
[Tarski's Semantic Theory is a detailed theory, but it is not designed
to characterize why a true sentence is true.]
More progress on answering the question "What is truth?" will be had
by concentrating not on why a sentence is true, but on what other
sentences are true when a sentence is true. By concentrating this way
on truth's logical liaisons, Aristotle offered what many philosophers
consider to be a partially correct answer to our question about truth.
Stripped of its overtones suggesting a correspondence theory of truth,
Aristotle proposed what is essentially sentence (T):
(T) A declarative sentence is true if and only if what it says is so.
If pairs of quotation marks serve to name a sentence, then (T)
requires that "It is snowing" be true just in case it is snowing.
Similarly, if the sentence about snow were named with the numeral 88
inside a pair of parentheses, then (88) would be true just in case it
is snowing. What could be less controversial? Unfortunately, this
seemingly correct, but trivial response to our question "What is
truth?" is neither obviously correct nor trivial; and the resolution
of the difficulty is still an open problem in philosophical logic. Why
is that? The brief answer is that (T) can be used to produce the Liar
Paradox. The longer answer refers to Tarski's Undefinability Theorem
of 1936, as we shall see.
2. Possible Solutions
This article began with a mere sketch of the Liar argument using
sentence (L). To appreciate the various proposed solutions to the
paradox, and the central role of (T), we need to examine more than
just a sketch of the argument. The argument actually requires the
following assumptions in addition to (T):
(2) Any declarative sentence "S" says that S.
(3) The Liar Sentence L is a legitimate declarative sentence.
That is, it is well formed.
(4) A legitimate declarative sentence is either true or else false.
(5) The usual naming convention holds so that
the phrase "This sentence" in L refers to L, and
(L) = "This sentence is false".
tarskiTarski added precision to convention (T) and to these other
assumptions by focusing not on English directly but on a classical
formal language capable of expressing arithmetic. Here the
difficulties produced by the Liar argument became much clearer; and,
very surprisingly, he was able to prove that the assumptions lead to
semantic incoherence. Tarski pointed out that the crucial,
unacceptable assumption is (3). Tarski reasoned this way: For there to
be a legitimate Liar Sentence in the language, there must be a
definable notion of "is true" which holds for the true sentences and
fails to hold for the other sentences. If there were such a "global
truth predicate," then the predicate "is a false sentence" would also
be definable and [here is where we need the power of arithmetic] a
Liar Sentence would exist, namely ∃x(Qx & ~Tx), where T is the
monadic, global truth predicate and Q is a monadic predicate satisfied
only by the name [Gödel number] of the Liar sentence. But if so, then
from (T), (2), (3), (4) and (5), one can deduce a contradiction.
This deduction of Tarski's is a formal analog of the informal Liar
Argument. The contradictory result tells us that the argument began
with a false assumption. Because (T), (2), (4), and (5) are essential
to what we call a "classical formal language," the mistaken assumption
is (3), and the only possible problem here is the assumption that the
global truth predicate "is a true sentence" can be defined. So, Tarski
has proved that truth is not definable in a classical language–thus
the name "Undefinability Theorem." Tarski's theorem establishes that
classically interpreted languages capable of expressing arithmetic
cannot contain a global truth predicate. A language containing its own
global truth predicate is said to be semantically closed. Tarski's
Theorem implies that classical formal languages with the power to
express arithmetic cannot be semantically closed. This suggests that
English itself may not be semantically closed, or, if English is
closed, then it is self-contradictory. This shocking result indicates
to some that our thought about our thoughts is incoherent. That's the
conclusion Tarski himself reached, so he quit trying to find the
coherent structure underlying natural languages and concentrated on
developing systems of formal languages that did not allow the
deduction of the contradiction. Many other philosophers of logic have
not drawn Tarski's pessimistic conclusion from his theorem.
For these optimists, there are four, main, detailed and coherent ways out.
1. The Liar Sentence L is meaningless, so the Liar argument can't
even get started because its main assumption (that the Liar Sentence
exists or is meaningful) is faulty. Natural language is incoherent,
and its underlying sensible structure is that of an infinite hierarchy
of levels. Because the Liar Sentence would have to reside on more than
one level simultaneously, it's not really a meaningful sentence. This
way out of the paradox is taken by Russell in his ramified theory of
types and, following Tarski, by Quine in his hierarchy of
meta-languages. For Russell, the referential phrase "This sentence" in
L is the culprit because the phrase is not allowed to refer to the
sentence in which the phrase itself occurs. For Quine, instead, the
culprit is the phrase "is false" in L because the phrase must be
satisfied by sentences in a language lower in the hierarchy and not by
the very sentence in which the phrase occurs.
2. Kripke, on the other hand, retains the intuition that the Liar
Sentence is meaningful, but argues that it is neither true nor false.
It lacks a classical truth value as does the odd sentence "The present
king of France is bald." Kripke trades infinite syntactic complexity
for infinite semantic complexity. He rejects the infinite hierarchy of
meta-languages underlying English in favor of one formal object
language having an infinite hierarchy of partial interpretations. The
truth predicate is the formal language's only basic
partially-interpreted predicate. Each step in the semantic hierarchy
is an interpretation of the language, and in these interpretations all
the basic predicates except one must have their interpretations
already fixed in the base level from which the first step is taken.
This one exceptional predicate is intended to be the truth predicate
for the previous level. It becomes a truth predicate for its own level
when the inductive interpretation building reaches the so-called
'fixed point'. Each step uses all the sentences which had their truth
values fixed at the lower steps in order to help fix the truth values
of semantically more complex sentences, for example, to fix the truth
value of sentences with even longer chains of nested truth predicates.
The Liar sentence, even up at the infinite semantic height of the
lowest fixed point, still isn't true or false. But at the fixed point,
the language satisfies Tarski's Convention (T).
3. The third way out says the Liar Sentence is meaningful and is
true or else false, but one step of the argument in the Liar Paradox
is incorrect (such as the move from the falsehood of the Liar Sentence
to its truth). Prior, following the informal suggestions of Buridan
and Peirce, takes this way out and concludes that the Liar Sentence is
simply false. So do Barwise and Etchemendy. They accuse the Liar
argument of equivocating by not paying careful attention to the
ambiguity of the Liar sentence, namely that it can be interpreted as
being the negation of itself or the denial of itself. If it is the
negation of itself, then it is simply false, and the Liar argument
cannot successfully show that it is true. If it is the denial of
itself, then it is simply true. Neither interpretation allows an
argument to conclude that the sentence is both true and false.
4. A fourth and more radical way out of the paradox is to argue
that semantic incoherence is not necessarily caused by letting the
Liar Sentence be both true and false. This solution embraces the
contradiction, then tries to limit the damage that is ordinarily a
consequence of that embrace. This way out of the paradox uses a
paraconsistent logic.
There are many suggestions for how to deal with the Liar Paradox, but
most are never developed to the point of giving a formal, symbolic
theory. Some give philosophical arguments for why this or that
conceptual reform is plausible as a way out of paradox, but then don't
show that their ideas can be carried through in a rigorous way.
Usually it appears that a formal treatment won't be successful. Other
attempts at solutions will take the formal route and then require
changes in standard formalisms so that a formal analog of the Liar
Paradox's argument fails, but then the attempted solution offers no
philosophical argument to back up these formal changes. A decent
theory of truth showing the way out of the Liar Paradox requires both
a coherent formalism (or at least a systematic theory of some sort)
and a philosophical justification backing it up. The point of the
philosophical justification is an unveiling of some hitherto unnoticed
or unaccepted rule of language for all sentences of some category
which has been violated by the argument of the paradox.
It is to the credit of Russell, Quine, Kripke, Barwise and Etchemendy
(among others) that they provide a philosophical justification for
their solutions while also providing a formal treatment in symbolic
logic that shows in detail both the character and implications of
their proposed solution. Kripke's elegant and careful treatment of L
stumbles on the Strengthened Liar and reveals why that argument
deserves its name. The theories of Russell-Tarski-Quine do "solve" the
Strengthened Liar, but at the cost of assigning "levels" to the
relevant sentences. Barwise and Etchemendy avoid these problems but
require accepting the idea that no sentence can be used to say
anything about the whole world. In the formal, symbolic tradition,
other important researchers in the last quarter of the 20th century
were Burge, Gupta, Herzberger, McGee, Routley, Skyrms, van Fraassen,
and Yablo. Martin and Woodruff created the same solution as Kripke,
though a few months earlier. Dowden and Priest first showed how to
embrace contradiction, although Priest provided the most systematic
development of this way out.
Leading solutions to the Liar Paradox all have a common approach, the
"systematic approach." The solutions agree that the Liar Paradox
represents a serious challenge to our understanding the logic of
natural language, and they agree that we must go back and
systematically reform or clarify some of our original beliefs in order
to solve the paradox. The solution must be presented systematically
and be backed up by an argument about the general character of our
language. In short, there must be both systematic evasion and
systematic explanation. Also, when it comes to developing this
systematic approach, the goal of establishing a logical basis for a
consistent semantics of natural language is much more important than
the goal of explaining the naive way most speakers use the terms
"true" and "not true." As Vann McGee expresses this point, "The
problem of giving voice to our pre-analytic intuitions about truth is
comparatively less important, just as understanding popular
misconceptions about space and time is comparatively less important
than understanding the actual geometry of space-time."
This "systematic approach" has been seriously challenged by
Wittgenstein. He says one should try to overcome "the superstitious
fear and dread of mathematicians in the face of a contradiction." The
proper way to respond to any paradox is by an ad hoc reaction and not
by any systematic treatment designed to cure both it and any future
ills. Symptomatic relief is sufficient. It may appear legitimate, at
first, to admit that the Liar Sentence is meaningful and also that it
is true or false, but the Liar Paradox shows that one should retract
this admission and either just not use the Liar Sentence in any
arguments, or say it is not really a sentence, or at least say it is
not one that is either true or false. Wittgenstein is not particularly
concerned with which choice is made. And, whichever choice is made, it
needn't be backed up by any theory that shows how to systematically
incorporate the choice. He treats the whole situation cavalierly and
unsystematically. After all, he says, the language can't really be
incoherent because we've been successfully using it all along, so why
all this "fear and dread"? Most logicians want systematic removal of
the paradox, but Wittgenstein is content to say that we may need to
live with this paradox and to agree never to utter the Liar sentence,
especially if it seems that removal of the contradiction could have
worse consequences.
Influenced by Wittgenstein, P. F. Strawson has argued that the proper
way out of the Liar Paradox is to re-examine how the term "truth" is
really used by speakers. When we say some proposition is true, we
aren't making a statement about the proposition. We are not ascribing
a property to the proposition–such as the property of correspondence,
or coherence, or usefulness. When we call a proposition "true" we are
approving it, or praising it, or admitting it, or condoning it. We are
performing an action. Similarly, when we say to our sister, "I promise
to pay you fifty dollars," we aren't ascribing some property to the
proposition, "I pay you fifty dollars." Rather, we are performing the
act of promising. For Strawson, when speakers utter the Liar Sentence,
they aren't saying something true or false; they are attempting to
praise something that isn't there, as if they were saying "Ditto" when
no one has spoken. The person who utters the Liar Sentence is making a
pointless utterance. The Liar Sentence is grammatical, but it isn't
being used to express a proposition and so is not something from which
a contradiction can be derived.
3. Overview of the Solutions
Some of the solutions to the Liar Paradox require a revision in
classical logic, the formal logic in which sentences of a formal
language have exactly two possible truth values (TRUE, FALSE), and in
which the usual rules of inference allow one to deduce anything from
an inconsistent set of assumptions. Kripke's revision uses a 3-valued
logic with the truth values TRUE, FALSE and NEITHER. Some logicians
argue that classical logic is not the incumbent which must remain in
office unless an opponent can dislodge it, although this is gospel for
other philosophers of logic (probably because of the remarkable
success of two-valued logic in expressing most of modern mathematical
inference). Instead, the office has always been vacant for natural
language.
Other philosophers object to revising classical logic merely to find a
way out of the Paradox. They say that philosophers shouldn't build
their theories by attending to the queer cases. There are more
pressing problems in the philosophy of logic and language than finding
a solution to the Paradox, so any treatment of it should wait until
these problems have a solution. From the future resulting theory which
solves those problems, one could hope to deduce a solution to the Liar
Paradox. However, for those who believe the Paradox is not a minor
problem but one deserving of immediate attention, there can be no
waiting around until the other problems of language are solved.
Perhaps the investigation of the Liar Paradox will even affect the
solutions to these other problems.
4. References and Further Reading
For further reading on the Liar Paradox that provides more of an
introduction to the area while not presupposing a strong background in
symbolic logic, the author recommends the article below by Mates, plus
the first chapter of the Barwise-Etchemendy book, and then chapter 9
of the Kirkham book. The rest of this bibliography is a list of
contributions to research on the Liar Paradox, and nearly all require
the reader to have significant familiarity with the techniques of
symbolic logic.
* Barwise, Jon and Etchemendy, John. The Liar: An Essay in Truth
and Circularity, Oxford University Press, 1987. For a brief overview
of their original solution to the Liar, see Kirkham, 1992, pp.
298-306.
* Burge, Tyler. "Semantical Paradox," Journal of Philosophy, 76
(1979), 169-198.
* Dowden, Bradley. A Theory of Truth: The Liar Paradox and
Tarski's Undefinability Theorem, Ph.D. Dissertation, Stanford
University, 1979.
* Dowden, Bradley. "Accepting Inconsistencies from the Paradoxes,"
Journal of Philosophical Logic, 13 (1984), 125-130.
* Gupta, Anil. "Truth and Paradox," Journal of Philosophical
Logic, 11 (1982), 1-60. Reprinted in Martin (1984), 175-236.
* Herzberger, Hans. "Paradoxes of Grounding in Semantics," Journal
of Philosophy, 68 (1970), 145-167.
* Kirkham, Richard. Theories of Truth: A Critical Introduction,
MIT Press, 1992.
* Kripke, Saul. "Outline of a Theory of Truth," Journal of
Philosophy, 72 (1975), 690-716. Reprinted in Martin (1984).
* Martin, Robert. The Paradox of the Liar, Yale University Press,
Ridgeview Press, 1970. 2nd ed. 1978.
* Martin, Robert. Recent Essays on Truth and the Liar Paradox,
Oxford University Press, 1984.
* Martin, Robert. and Woodruff, Peter. "On Representing
'True-in-L' in L," Philosophia, 5 (1975), 217-221.
* Mates, Benson. "Two Antinomies," in Skeptical Essays, The
University of Chicago Press, 1981, 15-57.
* McGee, Vann. Truth, Vagueness, and Paradox: An Essay on the
Logic of Truth, Hackett Publishing, 1991.
* Priest, Graham. "The Logic of Paradox," Journal of Philosophical
Logic, 8 (1979), 219-241; and "Logic of Paradox Revisited," Journal of
Philosophical Logic, 13 (1984), 153-179.
* Priest, Graham, Routley, Richard and Norman, J. (eds.),
Paraconsistent Logic: Essays on the Inconsistent, Philosophia-Verlag,
1989.
* Prior, Arthur. "Epimenides the Cretan," Journal of Symbolic
Logic, 23 (1958), 261-266; and "On a Family of Paradoxes," Notre Dame
Journal of Formal Logic, 2 (1961), 16-32.
* Quine, W. V. "The Ways of Paradox," in his The Ways of Paradox
and Other Essays, rev. ed., Harvard University Press, 1976.
* Russell, Bertrand. "Mathematical Logic as Based on the Theory of
Types," American Journal of Mathematics, 30 (1908), 222.
* Skyrms, Brian. "Return of the Liar: Three-valued Logic and the
Concept of Truth," American Philosophical Quarterly, 7 (1970),
153-161.
* Strawson, P. F. "Truth," in Analysis, 9, (1949).
* Tarski, Alfred. "The Concept of Truth in Formalized Languages,"
in Logic, Semantics, Metamathematics, pp. 152-278, Clarendon Press,
1956.
* Van Fraassen, Bas. "Truth and Paradoxical Consequences," in Martin (1970).
* Woodruff, Peter. "Paradox, Truth and Logic Part 1: Paradox and
Truth," Journal of Philosophical Logic, 13 (1984), 213-231.
* Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics,
Basil Blackwell, 3rd edition, 1978.
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