How to express the concept "most" in first order logic

Question

I'd like to express the concept most as in "Most Fs are Gs" in General Predicate Logic with Identity. Most means more than half. Is the reason that it is inexpressible because there is no way to express more than half in GPLI?

Thank you!

Answer 1

That's exactly right! There is no ready function that tells you how many Fs there are in the domain.

You could try a conjunction like:

'if there is exactly one F, then that one is a G, and if there are exactly two Fs, then both are G's, and if there are exactly three Fs, then at least two of them are Gs, and ...'

However, while each of these component claims is expressible, this does not work for two reasons:

1. even if the domain is finite, this conjunction still needs to be infinitely long, which is not allowed in GPLI

2. This approach fails to deal with infinite domains.

Now, just because this approach fails does not prove that there is no other way to do it (there is a proof, but it is tricky) but at least you now have an idea as to why something like this is impossible.