Free logics, philosophical issues in

The expression ‘free logic’ is a contraction of the more cumbersome ‘logic free of existence assumptions with respect to both its general terms (predicates) and its singular terms’. Its most distinctive feature is the rejection of the principle of universal specification, a principle of classical predicate logic which licenses the logical truth of statements such as ‘If everything rotates then (the planet) Mars rotates’. If a free logic contains the general term ‘exists’, this principle is replaced by a restricted version, one which licenses the logical truth only of statements such as ‘If everything rotates then Mars rotates, provided that Mars exists’. If the free logic does not contain the general term ‘exists’, but contains the term ‘is the same as’, the principle is replaced by a version which licenses only statements such as ‘If everything rotates then Mars rotates, provided that there is an object the same as Mars’.

Most free logicians regard the restricted version of universal specification as simply making explicit an implicit assumption, namely, that Mars exists. Indeed, free logic is the culmination of a long historical trend to rid logic of existence assumptions with respect to its terms. Just as classical predicate logic purports to be free of the hidden existence assumptions which pervaded the medieval theory of inference with respect to its general terms, so free logic rids classical predicate logic of hidden existence assumptions with respect to its singular terms.

There are various kinds of free logic, with many interesting and novel philosophical applications. These cover a wide range of issues from the philosophy of mathematics to the philosophy of religion. In addition to the issue of how to analyse singular existence statements, of the form ‘3 + 7 exists’ and ‘That than which nothing greater can be conceived exists’, of special importance are issues in the theory of definite descriptions, set theory, the theory of reference, modal logic and the theory of complex general terms.