From First-Order to Higher-Order Modal Logic

Abstract

First-order logic permits quantification into name position. Second-order logic permits quantification into predicate or sentence position too. Higher-order logic takes the generalization even further. The growth of higher-order modal logic is traced, starting with Lewis and Langford’s quantification into sentence position in propositional modal logic, and on to the higher-order modal logics of Barcan Marcus, Carnap, Montague, Gallin, and others. Higher-order modal logic is proposed as a suitably general setting in which to assess fundamental issues in modal metaphysics. However, there are difficulties in interpreting higher-order quantification, since it lacks adequate paraphrases in natural language. Although Boolos’s paraphrase of quantification into monadic predicate position in terms of plural quantification works well in non-modal settings, for many purposes it is unsuitable in modal settings since plurals are modally rigid. Nevertheless, we can hope to reach a suitable understanding of irreducibly higher-order quantification by the direct method, without paraphrase.