Doctor of Philosophy
Bell, John L.
The history of philosophy is rich with theories about objects; theories of object kinds, their nature, the status of their existence, etc. In recent years philosophical logicians have attempted to formalize some of these theories, yielding many fruitful results. My thesis intends to add to this tradition in philosophical logic by developing a second-order logical system that may serve as a groundwork for a multitude of theories of objects (e.g. concrete and abstract objects, impossible objects, fictional objects, and others). Through the addition of what we may call sortal quantifiers (i.e. quantifiers that bind individual variables ranging over objects of three unique sorts), a groundwork for a logic that captures concrete and non-concrete objects will be developed. We then extend this groundwork by the addition of a single new operator and the modal operators of a Priorian temporal logic. From this extension, our formal system can represent and define concrete, abstract, fictional, and impossible objects as well as formally axiomatize informal theories of them.
Summary for Lay Audience
First order formal languages pick out individual objects with constant symbols (e.g. c, d, c10, e) and variables (e.g. x, y, v10, z). Constants name individual objects and variables are assigned to individual objects (this assigning to a variable x some object, is similar to determining what a pronoun like ‘it’ denotes by previously naming the object denoted or otherwise indicating it in a conversation). The objects in our formal system are ‘described’ using predicate symbols (e.g. B, L, P2, S). If a constant b names Bertrand Russell and the predicate P indicates a philosopher, then Pb is interpreted as, ‘Bertrand Russell is a philosopher’. Now, what happens when we want our formal language to represent an object like a square circle? We could name the square circle s, and indicate squareness and circularity with the predicates S and C (respectively) and have Ss & Cs mean (roughly) ‘the square circle is both square and circular’, but notice that, as a matter of fact, square things are not circular and circular things are not square. The square circle is impossible for this reason. This also means that to adequately represent the square circle in our formal language it would be implied that Cs & ØCs (where the ‘Ø’ symbol is read ‘it is not the case that’). This is to say that ‘the square circle is circular and it’s not the case that the square circle is circular. This is a contradiction and it essentially ruins our formal language by ensuring that the logic of it can prove everything. It is the aim of this thesis project to develop a formal language and logic that can represent impossible objects like the square circle, but others too that are of interest to philosophers. By adding a few new symbols, we can save ourselves from contradiction and keep our logic useful. Ideally, all important kinds of objects will be representable in the proposed language, as well as important statements about them. From this formal groundwork, theories of objects can be formalized.
Winters, David, "A Groundwork for A Logic of Objects" (2019). Electronic Thesis and Dissertation Repository. 6621.