Counterparts, Essences and Quantified Modal Logic

It is commonplace to formalize propositions involving essential properties of objects in a language containing modal operators and quantifiers. Assuming David Lewis’s counterpart theory as a semantic framework for quantified modal logic, I will show that certain statements discussed in the metaphysics of modality de re, such as the sufficiency condition for essential properties, cannot be faithfully formalized. A natural modification of Lewis’s translation scheme seems to be an obvious solution but is not acceptable for various reasons. Consequently, the only safe way to express some intuitions regarding essential properties is to use directly the language of counterpart theory without modal operators.

Assertibility and Paradox

Antinomies involving the assertibility modal arise when we consider what are called “syntactic interpretations” of the assertibility operator A, translation schemes in which the operator is interpreted in terms of a monadic predicate in the nonmodal fragment of the language. Such a translation scheme, of course, is nothing more than a systematic attempt to replace the assertibility modality by a concept, and trouble tends to arise precisely when the concept is too truthlike. In this chapter, a number of paradoxical syntactic interpretations of the assertibility modality are explored. In each case, the paradox consists in an inconsistency affecting the syntactic transcription of certain apparently unexceptionable combinations of modal principles concerning the assertibility operator. It is argued that the modal construal of the notion of assertibility is fundamental, and a standard model theoretic semantics is presented for it. I finally suggest that an assertibility predicate for a language L with the assertibility operator be recovered at the metalinguistic level: a sentence p will be said to be assertible in L if and only if the sentence Ap is true in L, with the result that the alternative forms for a theory of the assertibility property in L exactly mirror the alternative forms for a theory of truth for L. Some consequences of these observations for the philosophy of mathematics are explored.

ON WEAK GROUND

Though the study of grounding is still in the early stages, Kit Fine, in ”The Pure Logic of Ground”, has made a seminal attempt at formalization. Formalization of this sort is supposed to bring clarity and precision to our theorizing, as it has to the study of other metaphysically important phenomena, like modality and vagueness. Unfortunately, as I will argue, Fine ties the formal treatment of grounding to the obscure notion of a weak ground. The obscurity of weak ground, together with its centrality in Fine’s system, threatens to undermine the extent to which this formalization offers clarity and precision. In this paper, I show how to overcome this problem. I describe a system, the logic of strict ground (LSG) and demonstrate its adequacy; I specify a translation scheme for interpreting Fine’s weak grounding claims; I show that the interpretation verifies all of the principles of Fine’s system; and I show that derivability in Fine’s system can be exactly characterized in terms of derivability in LSG. I conclude that Fine’s system is reducible to LSG.