Logic

Is modal logic first-order logic or second-order logic or higher-order logic? What makes a logical system fall into any of those categories? Is it based on expressive power?

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The usual story is roughly this. The quantifiers of a first-order logic (ordinary universal and existential quantifiers; or perhaps fancier dyadic quantifiers) range over objects in some given domain. A second-order logic, as well as having those first-order quantifiers, has quantifiers ranging over properties of those objects in the given domain -- or what may or may not come to the same thing, ranging over sets of objects from the domain. A third-order logic adds quantifiers ranging over properties of properties of things in the domain -- or what may or may not come to the same thing, ranging over sets of sets of objects from the domain. And so it goes.

It's not for nothing, then, that people do often say, following Quine, that higher order logics are just fragments of set theory in disguise. But be that as it may. Where does e.g. propositional modal logic fall into the picture?

Well, on the one hand, propositional modal logic has no quantifiers. So a fortiori it doesn't have first-order quantifiers. So, in one good sense, it is isn't first-order logic (or higher-order logic either).

On the other hand, propositional modal logic does map onto a fragment of a certain first-order theory, by -- as a first shot -- translating "necessarily p" to "(for all worlds w)(p is true at w)". And there are some nice theorems characterizing that first-order fragment (e.g. the van Bentham Characterization Theorem, if I remember right). So in another sense, I suppose you can say standard modal logic is a fragment of first-order logic in disguise.

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