These statements are paradoxical because there is no way to assign them consistent truth values. Consider that if "This statement is false" is true, then what it says is the case; but what it says is that it is false, hence it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.
Note that the paradox does not depend on the fact that the above sentences refer directly to their own truth values. In fact, the paradox arises when one constructs the following sentences:
However, it is arguable that this reformulation is little more than a syntactic expansion. The idea is that neither sentence accomplishes the paradox without its counterpart.
In the sixth century B.C.E. the philosopher-poet Epimenides, himself a Cretan, reportedly wrote:
The Epimenides paradox is often considered as an equivalent or interchangeable term for the "liar paradox" but they are not the same at least in its origin. First, it is unlikely that Epimenides intended his words to be understood as a kind of liar paradox. They were probably only understood as such much later in history. Second, that fact that this statement is paradoxical depends on contingent facts unlike the examples of liar paradox given above, for this sentence is not a paradox either when it false (because no proof exists that all Cretans really are liars) or when Epimenides is not Cretan.
The oldest known version of the liar paradox is instead attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century B.C.E. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. Eubulides reportedly said:
The Apostle Paul's letter to Titus in the New Testament refers to this quote in the first century AD.
Alfred Tarski discussed the possibility of a combination of sentences, none of which are self-referential, but become self-referential and paradoxical when combined. As an example:
Paradox of this kind was problematic to Tarski's project of giving a precise definition of truth, since, with the paradox, there is no way to give consistent truth-value assignments. To avoid the problem, he argued that, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the object languag, while the referring sentence is considered to be a part of a meta-language with respect to the object language. It is legitimate for sentences in languages higher on the semantic hierarchy to refer to sentences lower in the language hierarchy, but not the other way around. Tarski restricted his definition of truth to the languages with the hierarchy and accrodingly avoid the self-referential statements.
The problem of the liar paradox is that it seems to show that a naive conception of truth and falsity—i.e. every sentence is either true or false—actually lead to a contradiction. Sentences can be constructed that cannot consistently be assigned truth values even though they are completely in accord with grammar and semantic rules.
Consider the simplest version of the paradox, the sentence:
Suppose that the statement is true. Then, since the statement asserts that it is itself false, it must be false. Thus, 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. However, this contradicts the naive conception of truth that it has to be either true or false.
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 the common beliefs about truth and falsity: the claim that every statement has to abide by the principle of bivalence, a concept related to the law of the excluded middle.
Nonetheless, this conception of truth is also plagued by the following version of the liar paradox:
If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true and so one is led to another paradox.
This result has led some, notably Graham Priest, to posit that the statement follows paraconsistent logic and is both true and false (See Dialetheism below). Nevertheless, even Priest's analysis is susceptible to the following version of the liar:
If (C) is both true and false then it must be true. This means that (C) is only false, since that's what it says, but then it can't be true, and so one is led to another paradox.
Another variation is:
In this version, the writer of the statement cannot verify it to be true, because doing so makes it false, but at the same time cannot verify it to be false, as this would make it true. Anybody else except the writer, however, can easily see and verify the statement's truth.
A. N. Prior asserts 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 following two statements are equivalent:
The latter is a simple contradiction of the form "A and not A," and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction.
Saul Kripke points out that whether a sentence is paradoxical or not can depend upon contingent facts. Suppose that the only thing Smith says about Jones is
Now suppose that Jones says only these three things about Smith:
If the empirical facts are that Smith is a big spender but he is not soft on crime, then both Smith's remark about Jones and Jones's last remark about Smith are paradoxical.
Kripke proposes a solution in the following manner. 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.
Jon Barwise and John Etchemendy 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.
Graham Priest and other logicians have proposed that the liar sentence should be considered to be both true and false, a point of view known as dialetheism. In a dialetheic logic, all statements must be either true, or false, or both. Since dialetheism recognizes the liar paradox, an intrinsic contradiction, as being true, it must discard the long-recognized principle of ex falso quodlibet. This principle asserts that any sentence whatsoever can be deduced from a true contradiction. Thus, dialetheism only makes sense in systems that reject ex falso quodlibet. Such logics are called a paraconsistent logic.
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