Lecture Notes on First-Order Reductions of First-Order Modal Logic 15-816: Modal Logic

First-order modal logic is clearly more than classical first-order logic, because modalities can be used to phrase more advanced necessity and possibility properties. At the same time, first-order modal logic and non-modal first-order logic are not entirely different either. Modalities quantify over all ( ) or some (♦) worlds. Quantifiers also quantify over all (∀) or some (∃) objects (yet not usually over worlds). So all we need to do to relate firstorder modal with first-order non-modal logic is to make worlds and their accessibility relation “accessible” in the object language. Indeed, quantified modal logic can be embedded in first-order modal logic in the following sense. Here we assume constant domain for simplicity. For every first-order modal formula φ, we construct a reduced formula φ[(s) in non-modal first-order logic with one extra free variable s. To every signature Σ of first-order modal logic, we associate a reduced signature Σ[. Furthermore, to every first-order Kripke structure K (we assume constant domain), we define a corresponding first-order structure K[ of first-order