Consistency vs Inconsistency in a set of sentences: which is more common

Question

I'm curious whether there is any research in the "probability" that a set of sentences in a first-order logic is consistent. Obviously, there are an infinite number of inconsistent sets and an infinite number of consistent sets. In sentential logic, you can come up with a "probability" by taking n atoms and the conjunction of the sentences and, if we pretend that the truth table column for this conjunction is equally likely to be true or false, we have $2^n$ ways to fill the column, only one of which is inconsistent. Does anyone know of something like this for predicate logics? Is there a way to fix my above argument so it doesn't pretend true and false are equally likely in the column of the conclusion?

Answer 1

It depends a bit how you count / distinguish sets of sentences,

If you assume that sets are closed under deduction then there is only one inconsistent set (the set of all sentences) all other sentences are consistent so the chance that you have the inconsistent set is almost nil.

If you don't assume closure it is the opposite, every consistent set has many ways to make it inconsistent so the chance that you have an consistent set is (again) almost nil.