The Modal Object Calculus and its Interpretation


Edward N. Zalta


Advances in Intensional Logic, Maarten de Rijke (ed.), Dordrecht: Kluwer, 1997, pp. 249--279


The modal object calculus is the system of logic which houses the (proper) axiomatic theory of abstract objects. The calculus has some rather interesting features in and of itself, independent of the proper theory. The most sophisticated, type-theoretic incarnation of the calculus can be used to analyze the intensional contexts of natural language and so constitutes an intensional logic. However, the simpler second-order version of the calculus couches a theory of fine-grained properties, relations and propositions and serves as a framework for defining situations, possible worlds, stories, and fictional characters, among other things. In the present paper, we focus on the second-order calculus. The second-order modal object calculus is so-called to distinguish it from the second-order modal predicate calculus. Though the differences are slight, the extra expressive power of the object calculus significantly enhances its ability to resolve logical and philosophical concepts and problems.

There are two objectives in this paper: (1) to recast the intended interpretation of the modal object calculus in a new and interesting way, based on a reconception of logic and model theory, and (2) to describe a new interpretation of the modal object calculus. In Section 1, the modal object calculus is defined and the basic ideas underlying the definitions are explained. In Section 2, applications of the calculus are sketched. In Section 3, the intended interpretation of the calculus is recast in terms of the reconception of logic and model theory. In Section 4, a new interpretation of the calculus is defined by developing, in a modal setting, a suggestion due to Peter Aczel. Finally, Section 5 contains some observations about the ideas that have been presented and offers a brief outline of how to extend those ideas to produce a model of the theory of abstract objects.

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