Moore's Paradox

"We cannot talk" is, arguably, a close relative of Moore's 'paradox': "It is raining, but I do not believe it is raining".

There are lots of detailed explications of this very odd statement, but it seems to me that there is a distinction that could help us to understand (and 'solve') it - between self-contradictory statements (e.g. the Liar) and statements which are incoherent with their own status as statements.

"I cannot talk" is clearly an example of the latter. It makes perfect sense to say "X cannot talk" where X is a third party - a mute, an animal or an object; but not where we substitute the author of the statement into X. If you can't talk, you can't say so either; so if you utter these words, you either cannot be saying what they are normally taken to mean (i.e. that potential interpretaion is barred - no charitable principle can rescue it) or you are not meaning to talk seriously - perhaps you are doing a bit of contentless verbalising, or playing word games.

In the traditional formulation, "It's raining" and "I don't believe it's raining" from the same speaker are not directly contradictory, but they do challenge each other's intelligibility. Making both statements is so grossly inconsistent with the context of assertion combined what we normally mean by "believe" that they are rendered unintelligible. An interlocutor who makes these statements, as serious contributions to a conversation, is either revealing an inability to converse or seeking to undermine the game.

Meaning, validity, the Liar ...

Liar paradoxes seem unavoidable in languages that allow us to attribute validity or truth to our own judgements in an unrestricted way.

It seems likely that only languages that allow us to attribute validity or truth can also allow us to attribute meaning: can allow us to say when a gesture or signal is being appropriately ('truly') used, or - more important - can allow us to state the conditions under which a gesture would be being appropriately used.

Is this a radical suggestion: that a gesture or signal has meaning if it can be translated into a language that can be used to (at least partly) say what its meaning is? Does this link having meaning too closely to being able to attribute meaning? Some reflection on the use of 'to mean' makes me think not, but ...

So we have behaviour; then we have games - in which behaviour may or may not be within the rules of the game; then we have language games, in which we have truth and falsehood - which have some relationship with the rules of the game, but which may not just reflect them; then we have language games in which truth and falsehood can be explicitly attributed, and in which concepts like 'true' and 'meaningful' have a role.

In this third group, liar paradoxes can be formulated, open question arguments arise, and we can do philosophy: e.g. by making statements like those in this post.

Godel, Arithmetic and Validity

Godel's proofs show that any axiomatic system to which arithmetic could be reduced must be either incomplete or self-contradictory.

The proof doesn't depend on any interpretation of the symbolism that he uses, so his paradoxes are, in a sense, syntactical rather than semantic. What he shows is that any system which could encode arithmetical calculations will also be able to encode computations which can be interpreted as tests of the validity of its own theorems, and which are therefore self-referential (since the outcomes of these validity tests are also theorems).

We don't normally think of arithmetic as being intrinsically about truth-telling, or validity. Why is it impossible to do arithmetic without generating valdity tests? If Godel is right, it can't be just an accident. His method depends on being able (in prinicple) to arithmetise all logical validity testing - it seems to make logical validity testing and an essential subset of arithmetical computation isomorphic.

Numbers are truth and truth is numbers.

Version 11

Intelligibility ... I could be wandering into a bog here.