In first-order logic, it is possible for a sentence $S$ and its negation $\lnot S$ to both be invalid. That is, it is possible for there to not exist a proof of $S$, and also not of $\lnot S$.
For instance, consider the following sentence $S$, followed by its negation:
- $\exists x \exists y \lnot (x = y)$
- $\forall x \forall y (x = y)$
$S$ states that there exist two objects that are not equal, and $\lnot S$ states that all objects are equal. Neither of these sentences are valid, because first-order logic says nothing about the size of the universe. If the universe has only one object, the sentence is true, and if not, it is false.
My question: the problem is that $S$ can hold only in some models of first-order logic, and $\lnot S$ can also hold only in some models. If we restrict the cardinality of the set of objects in our logic, then do we end up in a situation where a sentence either holds (e.g. are "valid" for those models), or where its negation holds?
If not, how does this fail?
Note that this is different than asking whether we can prove every sentence or its negation within some theory, which we know is not true because there can be more than one model of a theory with the same cardinality. I am asking about proving the validity of a first-order sentence in general, or proving its negation, if the cardinality of the universe is somehow specified.