## The Modal Object Calculus and its Interpretation

### Author

Edward N. Zalta
### Reference

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

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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