Worlds and Propositions Set Free

The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory in the definition of worlds. Next the authors show that an object-theoretic analysis of the Kaplan paradox reveals that there is no genuine paradox at all, as the central premise of the paradox is simply a logical falsehood and hence can be rejected on the strongest possible grounds — not only in object theory but for the very framework of propositional modal logic in which Kaplan frames his argument. The authors close by fending off a possible objection that object theory avoids the Russell paradox only by refusing to incorporate set theory and, hence, that the object-theoretic solution is only a consequence of the theory’s weakness. ∗Copyright c © 2013 by Otávio Bueno, Christopher Menzel, and Edward N. Zalta. This paper is published in Erkenntnis, 79 (2014): 797–820. The 2nd and 3rd authors would like to express their gratitude to Hannes Leitgeb and the Munich Center for Mathematical Philosophy (MCMP), where both spent time as visiting fellows during the 2011–12 academic year, and to the Alexander von Humboldt Foundation, for its support of the MCMP. All three authors wish to express their indebtedness to two anonymous reviewers for detailed and perceptive comments on an earlier version of this paper. Their comments led to substantive changes. O. BUENO, C. MENZEL, AND E. ZALTA 2