Monday, December 31, 2012

On a formal Liar

A lot of formal work is done on the Liar paradox, so note that a formal logic is no more than a mathematical model of correct reasoning.
......E.g. Richard Heck was tempted by his formal version of the Liar paradox (Thought 1(1): 36–40) ‘to conclude that there can be no truly satisfying, consistent resolution of the Liar paradox’ (p. 39). And he did have a strong model because he assumed little more than two very weak logical principles, his equations 3 and 4 (p. 38). But it was still only a model.
......Heck’s informal illustration of equation 3 was: ‘It cannot be both that snow is white and that “snow is not white” is true’ (p. 38). That is unobjectionable because ‘snow is not white’ just means that it is not the case that snow is white. Insofar as snow is white, the claim made by ‘snow is not white’ is not true. And equation 4 was similar, e.g. it cannot be both that snow is not blue and that ‘snow is not blue’ is not true.
......Heck had a model of the Liar paradox because he had already introduced a term, λ, defined by his equation 2 (p. 36), which was a formal version of such definitions as the following: ‘L’ names the self-referential claim made by ‘L is not true’. It follows from that informal definition that insofar as L is true, it is true that L is not true. And L should conform to the logic behind equation 3, so insofar as L is true, it is not true that L is not true. But it does not follow logically that L is not true, because L may well be as true as not.
......Formally, equations 2 and 3 rule out T(λ). And similarly, equations 2 and 4 rule out ¬T(λ). But for a solution to Heck’s problem to be truly satisfying, it need only stay true to the underlying purpose of one’s formal logic. If we want to include terms like λ in our formal language, then we will need a better model of truth than T, but when deducing theorems from axioms we won’t normally need to allow for the possibility of terms like λ. And for such purposes as A.I. paraconsistent logic is sufficiently consistent.

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