Liar Paradox - Logical Structure of The Liar Paradox

Logical Structure of The Liar Paradox

For a better understanding of the liar paradox, it is useful to write it down in a more formal way. If "this statement is false" is denoted by A and its truth value is being sought, it is necessary to find a condition that restricts the choice of possible truth values of A. Because A is self-referential it is possible to give the condition by an equation.

If some statement, B, is assumed to be false, one writes, "B = false". The statement (C) that the statement B is false would be written as "C = 'B = false'". Now, the liar paradox can be expressed as the statement A, that A is false:

"A = 'A = false'"

This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the boolean domain "A = false" is equivalent to "not A" and therefore the equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical approach to make the equation solvable is the dialetheistic approach, in which case the solution is A being both "true" and "false". Other resolutions mostly include some modifications of the equation; Arthur Prior claims that the equation should be "A = 'A = false and A = true'" and therefore A is false. In computational verb logic, the liar paradox is extended to statement like, "I hear what he says; he says what I don't hear", where verb logic must be used to resolve the paradox.