So I just realised that in all the work I have seen (which is literally a drop of the ocean) on First Order Logic, I never really saw anything on expressiveness of FOL with arbitrary number of variables and unary predicates. Intuitively it seems to me that unary predicates are only as expressive as propositions, but is it true ? are there formulas in First Order Logic with unary predicate and finite number of variables which cannot be expressed in First Order Logic with two variables ? (I am strictly talking of a language which is function free and has only unary predicates and finite set of variables)
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$\begingroup$Does the identity predicate count as part of the logic? In which case "there are exactly two $F$s" can be expressed using three variables but not two.
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