A Classically-Based Theory of Impossible Worlds


Edward N. Zalta


Notre Dame Journal of Formal Logic, 38/4 (Fall 1997): 640-660 (Special Issue, Graham Priest, Guest Editor)


The appeal to possible worlds in the semantics of modal logic and the philosophical defense of possible worlds as an essential element of ontology have led philosophers and logicians to introduce other kinds of `worlds' in order to study various philosophical and logical phenomena. The literature contains discussions of `non-normal worlds', `non-classical worlds', `non-standard worlds', and `impossible worlds'. These atypical worlds have been used in the following ways: (1) to interpret unusual modal logics, (2) to distinguish logically equivalent propositions, (3) to solve the problems associated with propositional attitude contexts, intentional contexts, and counterfactuals with impossible antecedents, and (4) to interpret systems of relevant and paraconsistent logic. However, those who have attempted to develop a genuine metaphysical theory of such atypical worlds tend to move too quickly from philosophical characterizations to formal semantics.

In this paper, I derive a metaphysical theory of impossible worlds from an axiomatic theory of abstract objects. The axiomatic theory is couched in a language with just a little more expressive power than a classical modal predicate calculus. The logic underlying the theory is classical. This system (language, logic, and proper theory) is reviewed in the first section of the paper. Impossible worlds are not taken to be primitive entities but rather characterized intrinsically using a definition that identifies them with, and reduces them to, abstract objects. The definition is given at the end of the second section. In the third section, the definition is shown to be a good one. We discuss consequences of the definition which take the form of proper theorems and which assert that impossible worlds, as defined, have the important characteristics that they are supposed to have. None of these consequences, however, imply that any contradiction is true (though contradictions can be `true at' impossible worlds). This classically-based conception of impossible worlds provides a subject matter for paraconsistent logic and demonstrates that there need be no conflict between the laws of paraconsistent logic (when properly conceived) and the laws of classical logic, for they govern different kinds of worlds. In the fourth section of the paper, I explain why the resulting theory constitutes a theory of genuine impossible worlds, and not a theory of ersatz impossible worlds. The penultimate section of the paper examines the philosophical claims made on behalf of impossible worlds, to see just exactly where such worlds are required and prove to be useful. We discover that whereas impossible worlds are not needed to distinguish necessarily equivalent propositions or for the treatment of the propositional attitudes, they may prove useful in other ways. The final section of the paper contains some observations and reflections about the material in the sections that precede it.

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