First-order logic arity defines decidability?
Logic with unary predicates (not operators), is called monadic. The thing that is called propositional logic only has nullary predicates, i.e., constants true and false, and no quantifiers.

The undecidability of predicate logic follows because predicate logic (with at least one binary predicate) is powerful enough to describe how a Turing machine works, and so we can express in the logic sufficiently complicated statements about Turing machines to get undecidability. In contrast, in monadic logic, where we only have unary predicates, we cannot describe the working of a Turing machine. I am being deliberately vague here to give an idea without getting bogged down in technicalities.