Liars and Formalities

A lot of formal work is being done on the Liar paradox, raising the question of why it is. A formal logic is a mathematical model of correct reasoning, a logical paradox is a prima facie problem with correct reasoning, and a scientific theory of something is a mathematical model of it; so prima facie, that formal work may well be part of a scientific attempt to solve the logical problem of the liar paradox.
......Richard Heck, for example, was recently 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).
......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’.
......Informally, it follows from that definition of L 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. Nevertheless, 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 his formal logic. Heck had little to say about that.
......Heck’s problem shows that if we want to include terms like λ in our formal language, then we will need a better model of truth than T, raising the question of why we would want to include terms like λ in a formal language.
......Perhaps it will help if, much as we distinguish between statistics as mathematics and the uses and abuses of statistics, we distinguish between formal logic as mathematics and the uses and abuses of formal logic.
......Perhaps, for example, logicians are trying to find out what correct reasoning is. A lot of what they say does seem, prima facie, to support that view. But if they did not already know how to reason correctly, how could they hope to use their mathematical models scientifically? Still, maybe logicians are trying to find out how computers could be better able to help us to reason correctly about very complicated matters. Some computer scientists are interested in formal logic, and there are many philosophers who take the mind to be no more than a biological computer. But of course, this is all very speculative and vague. A bit of clarity when it comes to the purposes of a formal logic might go a long way.
......Logicians make their definitions very precise, but then they are, as a rule, very vague about how their terms relate to reality. Where in the world is their precision? How is it logic that is being made precise? I have for years been wondering What is Logic? and Classical Logic: how is it correct?