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?

The no-no Paradox

Roy Sorensen (2001: Vagueness and Contradiction, Oxford: Clarendon Press, 169) considered the following pair of sentences:
......The neighbouring italicized sentence is not true.
......The neighbouring italicized sentence is not true.
While it is logically possible that one of those sentences – or rather, one of the claims made by those sentences – is true and the other false, those two tokens of that sentence-type should have the same truth-value because there are no significant contextual differences between them. It is therefore plausible that each is as true as not. For each token, were the claim expressed by it as true as not, the other claim would be as untrue as not, which clearly coheres with it too being as true as not.

The Unsatisfied Paradox

In this month's issue of The Reasoner (page 185), Peter Eldridge-Smith gave the following informal description of his Unsatisfied paradox:

My favourite predicate just happens to be 'does not satisfy my favourite predicate'. Crete satisfies 'does not satisfy my favourite predicate' iff Crete does not satisfy my favourite predicate. Therefore, Crete satisfies my favourite predicate iff Crete does not satisfy my favourite predicate.

And not just Crete, there is no thing that satisfies Peter's favourite predicate, and no thing that fails to satisfy it without it also not being the case that it fails to satisfy it. Nevertheless, Peter's favourite predicate could be as true as not of Crete, or anything else. Predicates can do that, e.g. 'is blue' is as true as not of an object that is as blue as not, and some predicates apply equally to all things, e.g. 'is a thing' is true of all things.

The Pinocchio Paradox

Suppose that Pinocchio's nose grows if, and only if, he says something that is not true, and that he says "My nose is growing". Then his nose is growing if and only if it is not growing. (This paradox originated with Veronique Eldridge-Smith.) According to Peter Eldridge-Smith:

The Pinocchio scenario is not going to arise in our world, so it is not a pragmatic issue. It seems though that there could be a logically possible world in which Pinocchio’s nose grows if and only if he is saying something not true. However, there cannot be such a logically possible world wherein he makes the statement ‘My nose is growing’.

In the world in which Pinocchio's nose grows and shrinks in such a way, suppose that he says, of various uniformly coloured objects, that they are blue. What happens if the object is as blue as not? (There must be such colours, because otherwise some colour that is blue is the same colour as some colour that is not blue.) Well, whatever happens, that could also be what happens when he says "My nose is growing". It is, for example, possible that Pinocchio's nose is in a quantum-mechanically entangled state, as much growing as not. That seems to be a logically possible world.