## A Classically-Based Theory of Impossible Worlds

### Author

Edward N. Zalta
### Reference

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

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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