Leibnizian intensional semantics for syllogistic reasoning

Venn diagrams are standardly used to give a semantics for Syllogistic reasoning. This interpretation is extensional. Leibniz, however, preferred an intensional interpretation, according to which a singular and universal sentence is true iff the (meaning of) the predicate is contained in the (meaning of) the subject. Although Leibniz’s preferred interpretation played a major role in his philosophy (in Leibniz [16] he justifies his metaphysical ‘Principle of Sufficient Reason’ in terms of it) he was not able to extend his succesfull intensional interpretation (making use of characteristic numbers) without negative terms to one where also negative terms are allowed. The goal of this paper is to show how syllogistic reasoning with complex terms can be given a natural set theoretic ‘intensional’ semantics, where the meaning of a term is not defined in terms of individuals. We will make use of the ideas behind van Fraassen’s [6, 7] hyperintensional semantics to account for this.