Logic is Fallible

Last post I explained Haskell Curry’s paradox. I promised a discussion. I lied. Some paradoxes (eg.Zeno’s) have well-defined, satisfactory resolutions. Not this one. There are resolutions too complicated for any of us to follow but they probably don’t warm the logician’s heart.

This is a problem for logic. If our laws of logic can produce nonsense, then something’s gone wrong. Now, I’m pretty confident that one day this paradox will be resolved properly. But this would obviously require changes to of our logic. This has wider implications.

A lot of people think maths/logic are somehow a higher level of knowledge: eternal, stable, necessarily true. In reality, a legitimate debate rages on: are laws of logic/maths discovered or invented? I think Curry’s paradox shows it’s a bit of both.

Maths/logic have elements of empirical science. We define concepts partly based on how they conform to the actual world. This is why we’re unsettled by Curry’s paradox: it countradicts our experience (we know X can’t be true for all X). But if we check our logic against reality, it means it’s not nearly as pristine as a naive view would hold. Some people think quantum physics contradicts our laws of logic altogether, since an electron goes into the left slit AND the right slit AND both AND neither. This description of the electron’s behaviour isn’t accepted as accurate by a majority, but IF true, our laws of logic need a “slight” revision. If that doesn’t make logic an empirical science I don’t know what would.

Believe it or not, this stuff can influence actual conversation. How many times have I heard creationists claim that if there isn’t a God then the universe must have come from nothing “which would be [logically] absurd”? Yet what seemed a logical absurdity for millenia is coming to question. Phantom particles are those which pop in and and out of existence — from nothing to nothing.

The armchair is needed, but it’s not enough. The mysteries of space/time/causality must be discovered. They’re not divine concepts that we can just think of. This shows that we’ve very fallible. And that the world is messy. We better get used to it.