The Liar Paradox is an argument that arrives at a contradiction by reasoning about a Liar Sentence. The most familiar Liar Sentence is the following self-referential sentence:

(1) 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 (1) is true, then (1) is false. On the other hand, if (1) is false, then it is true to say (1) is false, but because the Liar Sentence is saying precisely that (namely that it is false) (1) is true. So (1) is true if and only if it is false. Since (1) is one or the other, it is both.

The argument depends upon a few more assumptions and steps, but these are apparently as uncontroversial as those above.

The contradictory result throws us into the lion's den of semantic incoherence. For example, the Liar Sentence can be put to devious uses. In the late medieval period, Buridan did this 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.

Table of Contents (Clicking on the links below will take you to that part of this article)

• History of the Paradox and Possible Solutions
• Bibliography

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 number 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 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 it leads to the Liar Paradox. The longer answer refers to Tarski's Undefinability Theorem of 1936.

We began this discussion with a mere sketch of the Liar argument using sentence (1). To provide the details, we need (T) plus the following assumptions that also are apparently acceptable:

(2) Any declarative sentence "S" says that S.

(3) The Liar Sentence, (1), is a legitimate declarative sentence.

(4) A declarative sentence is either true or else false.

(5) The usual naming convention holds so that

the phrase "This sentence" in (1) refers to (1), and

(1) = "This sentence is false".

Tarski added precision to convention (T) and these other assumptions by focussing not on English directly but on a classical formal language capable of expressing arithmetic. Here the difficulties became much clearer; and, very surprisingly, he was able to prove that the assumptions lead to semantic incoherence. Tarski pointed out that the crucial assumption is (3). 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. But if so, then from (T), (2), (3), (4) and (5) [but not (1) because the Liar Sentence is not an assumption in the Liar Argument], one could deduce a contradiction. Tarski's deduction is a formal analog of the 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. Most other philosophers of logic have not drawn Tarski's pessimiistic conclusion.

For these optimists, there are four main detailed and coherent ways out.

(1) The Liar Sentence 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 (1) 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 (1) 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 truth value as does the odd sentence "The present king of France is bald." He rejects the infinite hierarchy of meta-languages underlying English in favor of one formal object language having a hierarchy of partial interpretations, one of which (his lowest fixed-point) assigns an interpretation to all the basic (atomic) predicates of the language except for the truth predicate. The truth predicate is the only partial predicate, and the formal analog of the Liar Sentence is assigned neither the value True nor the value False in the fixed point. Under the fixed point interpretation of the formal language that is the coherent structure within English, the language satisfies Tarski's Convention (T); both S and "S"-is-true have the same truth conditions for any sentence S.

(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 (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.

(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.

Although there are many suggestions for how to deal with the Liar Paradox, 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. Some other solutions require changes in formalisms so that one or another formal analog of the Liar Paradox's argument fails, but then they give no philosophical argument to back up their 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, and Kripke 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 (1) stumbles on the Strengthened Liar and reveals why it deserves its name. The theories of Russell-Tarski-Quine do 'solve' the Strengthened Liar. In the formal, symbolic tradition, other important researchers in the last quarter of the 20th century are Barwise, Burge, Etchemendy, 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.

Principal 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 preanalytic 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 plus 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"?

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 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 Sentence is grammatical but it's not a proposition and so is not something from which a contradiction can be derived.

The most serious challenges to Strawson's solution have attempted to show that this general analysis of truth is incorrect. If we say, "I hope what she will say is true," I am not performing the act of praising what she says; I don't even know what she will say.

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 independently. Perhaps the investigation of the Liar Paradox will even affect the solutions to these other problems.

Bibliography

For an essay 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 Benson Mates, 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 items require the reader to have significant familiarity with the techniques of symbolic logic.

Bradley Dowden
Email: dowden@csus.edu
California State University Sacramento