The Algebraic Structure of Propositions

This dissertation is about various puzzles that arise when we attempt to construct general theories of propositions, properties and relations. The goal of the dissertation is to make some progress on these puzzles by employing a broadly abductive approach to them. On my view, one feature of an abductive approach is working out the overall algebraic structure of the objects in question. On grounds of simplicity and elegance, theories admitting of more straightforward algebraic description are favored, at least initially. Theories not admitting of such a description must make up for this lack of elegance. This provides us with some guidance when our evidence and intuition falls short.

Chapter 1 concerns the relationship between theories of propositional fineness of grain and theories of belief. I argue that we should accept a relatively coarse grained account of propositions, Booleanism, and deny the principle of Distribution, according to which believing a conjunction implies believing both conjuncts. The principle of Distribution turns out to be surprisingly strong: for many theories of propositional fineness of grain, conjoining that theory with Distribution has implausible consequences. Indeed I will tentatively suggest that no theory of propositional fineness of grain can accommodate Distribution without some collateral damage.

Chapter 2 concerns the relationship between indefinite extensibility and theories of propositional explanation. In particular, I will argue that we cannot overcome problems with the principle of sufficient reason by adopting a view on which contingent truth is indefinitely extensible. Given our best theories of indefinite extensibility, even if contingent truth turns out to be indefinitely extensible, the principle of sufficient reason still entails that all truths are necessary, given plausible background principles.

Chapter 3 concerns a puzzle of logical form that has been put forward recently by Kit Fine (2017). This chapter develops a solution to the puzzle by outline two notions of form and arguing that our intuitive notion of logical form does not privilege one over the other. The puzzle is then answered by showing that it rests on a conflation of these two notions of form.

Chapter 4 puts forward a theory of the representational properties of propositions that goes some way towards answering worries relating to the unity of the proposition. Contra many recent accounts in this vicinity, I will argue that the best theory here is a primivitist one. The theory makes use a novel operation I call application. I will argue that with this primitive, a very general theory of the representational properties of various sorts of abstract objects can be developed.