The goal of this post is investigate how the formal systems in Zalta's book 'Abstract Objects' might be expressed in topos theory. In this book Zalta gives us a series of successively stronger axiomatic-deductive systems (the framework is called Object Theory) which are applied to formalizations of key aspects of various historical philosophical systems following a chronological order: Plato, Leibniz, Frege and Meinong. The first two systems are, broadly speaking, specialized kinds of second-order logic while the last ones are a kind of type theory or higher order logic. The details of the syntax and deductive systems (as well as the proposed semantics) are somewhat complex; and Zalta and his collaborators have since refined these systems in significant ways. One of the most important and novel features of all these systems is the encoding relation together with the distinction between abstract and concrete objects. In this paper we focus on Zalta's second system, a form of second-order modal logic, but considered as naturally embedded in the natural higher order logic of a special kind of topos. We will present (and argue for) a general topos-theoretic interpretations of the modal operator and then define a special class of toposes (Zalta toposes) which can express the encoding relation. We then show how some axioms and theorems in 'Abstract Objects' relating to the formalization of Plato's theory of forms and Leibniz's theory of possible worlds can be mathematically obtained working directly in a Zalta topos.
No comments:
Post a Comment