The Reduction of Necessity to Essence

In ‘Essence and Modality’, Kit Fine (1994) proposes that for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all objects. Call this view Fine’s Thesis. This paper is a study of Fine’s Thesis in the context of Fine’s logic of essence (LE). Fine himself has offered his most elaborate defence of the thesis in the context of LE. His defence rests on the widely shared assumption that metaphysical necessity obeys the laws of the modal logic S5. In order to get S5 for metaphysical necessity, he assumes a controversial principle about the nature of all objects. I will show that the addition of this principle to his original system E5 leads to inconsistency with an independently plausible principle about essence. In response, I develop a theory that avoids this inconsistency while allowing us to maintain S5 for metaphysical necessity. However, I conclude that our investigation of Fine’s Thesis in the context of LE motivates the revisionary conclusion that metaphysical necessity obeys the principles of the modal logic S4, but not those of S5. I argue that this constitutes a distinctively essentialist challenge to the received view that the logic of metaphysical necessity is S5.

A CORRECT POLYNOMIAL TRANSLATION OF S4 INTO INTUITIONISTIC LOGIC

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.

COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS

We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

On modal components of the S4-logics

We consider the representation of each extension of the modal logic S4 as sum of two components. The first component in such a representation is always included in Grzegorczyk logic and hence contains "modal resources" of the logic in question, while the second one uses essentially the resources of a corresponding intermediate logic. We prove some results towards the conjecture that every S4-logic has a representation with the least component of the first kind.

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

In the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

STRONG COMPLETENESS OF S4 FOR ANY DENSE-IN-ITSELF METRIC SPACE

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space.