Necessities and Necessary Truths: A Prolegomenon to the Use of Modal Logic in the Analysis of Intensional Notions∗

In philosophical logic necessity is usually conceived as a sentential operator rather than a predicate. An intensional sentential operator does not allow one to express quantified statements such as ‘There are necessary a posteriori propositions’ or ‘All laws of physics are necessary’ in first-order logic in a straightforward way, while they are readily formalised if necessity is formalised by a predicate. Replacing the operator conception of necessity by the predicate conception, however, causes various problems and forces one to reject many philosophical accounts involving necessity that are based on the use of operator modal logic. We argue that the expressive power of the predicate account can be restored if a truth predicate is added to the language of first-order modal logic, because the predicate ‘is necessary’ can then be replaced by ‘is necessarily true’. We prove a result showing that this substitution is technically feasible. To this end we provide partial possible-worlds-semantics for the language with a predicate of necessity and perform the reduction of necessities to necessary truths. The technique applies also to many other intensional notions that have been analysed by means of modal operators.