Liar Paradox - Explanation of The Paradox and Variants

Explanation of The Paradox and Variants

The problem of the liar paradox is that it seems to show that 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.

The simplest version of the paradox is the sentence:

This statement is false. (A)

If (A) is true, then "This statement is false" is true. Therefore (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then "This statement is false" is false. Therefore (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.

However, 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, the rejection of the claim that every statement has to be either true or false, also known as the principle of bivalence, a concept related to the law of the excluded middle.

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

This statement is not true. (B)

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. Since initially (B) was not true and is now true, another paradox arises.

Another reaction to the paradox of (A) is to posit, as Graham Priest has, that the statement is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:

This statement is only false. (C)

If (C) is both true and false, then (C) is only false. But then, it is not true. Since initially (C) was true and is now not true, it is a paradox.

There are also multi-sentence versions of the liar paradox. The following is the two-sentence version:

The following statement is true. (D1) The preceding statement is false. (D2)

Assume (D1) is true. Then (D2) is true. This would mean that (D1) is false. Therefore (D1) is both true and false.

Assume (D1) is false. Then (D2) is false. This would mean that (D1) is true. Thus (D1) is both true and false. Either way, (D1) is both true and false - the same paradox as (A) above.

The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements (wherein the last statement asserts the truth/falsity of the first statement), provided there are an odd number of statements asserting the falsity of their successor; the following is a three-sentence version, with each statement asserting the falsity of its successor:

E2 is false. (E1) E3 is false. (E2) E1 is false. (E3)

Assume (E1) is true. Then (E2) is false, which means (E3) is true, and hence (E1) is false, leading to a contradiction.

Assume (E1) is false. Then (E2) is true, which means (E3) is false, and hence (E1) is true. Either way, (E1) is both true and false - the same paradox as with (A) and (D1).

There are many other variants, and many compliments, possible. In normal sentence construction, the simplest version of the compliment is the sentence:

This statement is true. (F)

If F is assumed to bear a truth value, then it presents the problem of determining the object of that value. But, a simpler version is possible, by assuming that the single word 'true' bears a truth value. The analogue to the paradox is to assume that the single word 'false' likewise bears a truth value, namely that it is false. This reveals that the paradox can be reduced to the mental act of assuming that the very idea of fallacy bears a truth value, namely that the very idea of fallacy is false: an act of misrepresentation. So, the symmetrical version of the paradox would be:

The following statement is false. (G1) The preceding statement is false. (G2)