Liar paradox

In philosophy and logic, the liar paradox encompasses paradoxical statements such as:

A discussion of the liar paradox

The problem of the paradox is that it seems to show that our most cherished common beliefs about truth and falsity actually lead to a contradiction. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar and semantic rules. Consider the simplest version of the paradox, the sentence This statement is false. If we suppose that the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So the hypothesis that it is true leads to the contradiction that it is true and false. Yet we cannot conclude that the sentence is false for that hypothesis also leads to contradiction. If the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either hypothesis, we end up concluding that the statement is both true and false. But it has to be either true or false (or so our common intuitions lead us to think), hence there seems to be a contradiction at the heart of our beliefs about truth and falsity.

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However, the fact that the liar sentence can be shown to be true if it is false and false if it is true has led some to conclude that it is neither true nor false. This response to the paradox is, in effect, to reject one of our common beliefs about truth and falsity: the claim that every statement has to be one or the other. This common belief is called the Principle of Bivalence.

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The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:

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:This statement is not true.

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If it is neither true nor false, then it is not true, which is what it says; hence it's true, etc.

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This again has led some, notably Graham Priest, to posit that the statement is both true and false (see paraconsistent logic).

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A. N. Prior claims that there is nothing paradoxical about the Liar paradox. His claim (which he attributes to Charles S. Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement "It is true that two plus two equals four" contains no more information than the statement "two plus two is four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...". Thus the statement This statement is false is said to be equivalent to

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:This statement is true and this statement is false.

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The latter is a simple contradiction of the form "A and not A", and hence is false. There is no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction.

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But this analysis does not provide a solution to versions of the paradox that don't use direct self-reference, such as the two-sentence version:

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:The next sentence is false.

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:The preceding sentence is true.

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On Prior's analysis these would be equivalent to:

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:This whole sentence is true and the next sentence is false.

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:This whole sentence is true and the preceding sentence is true.

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Neither of these is by itself contradictory, but there is no way to assign truth values to them consistently, so we still have a paradox.

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Saul Kripke points out that whether or not a sentence is paradoxical can depend upon contingent facts. Suppose that the only thing Smith says about Jones is

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:A majority of what Jones says about me is false.

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Now suppose that Jones says only these three things about Smith:

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:Smith is a big spender.

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:Smith is soft on crime.

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:Everything Smith says about me is true.

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If the empirical facts are that Smith is a big spender but he is not soft on crime, then Smith's remark about Jones and Jones's last remark about Smith are both paradoxical. Kripke proposes a solution in the following manner:

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If a statement's truth value is ultimately tied up in some evaluable fact about the world, call that statement "grounded." If not, call that statement "ungrounded." Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.

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Jon Barwise and John Etchmendy propose that the liar sentence (which they interpret as synonymous with the Strengthened Liar) is ambiguous. They base this conclusion on a distinction they make between a denial and a negation. If the liar means It is not the case that this statement is true then it is denying itself. If it means This statement is not true then it is negating itself. They go on to argue, based on their theory of "situational semantics" that the "denial Liar" can be true without contradiction while the "negation Liar" can be false without contradiction.

Related Topics:
Jon Barwise - John Etchmendy

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