What Can You Say? Measuring the Expressive Power of Languages

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power. In doing so, I show how these investigations help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language./nWhen we attempt to describe the world, we are trying to distinguish the way things are from all the many ways things could have been—in other words, we are trying to locate ourselves within a region of logical space. According to this picture, languages can be thought of as ways of carving logical space or, more formally, as maps from sentences to classes of models. For example, the language of first-order logic is just a mapping from first-order formulas to model-assignment pairs that satisfy those formulas. Almost all formal languages discussed in metaphysics and logic, as well as many of those discussed in natural language semantics, can be characterized in this way./nUsing this picture of language, I analyze two different approaches to defining expressive power, each of which is motivated by different roles a language can play in a debate. One role a language can play is to divide and organize a shared conception of logical space. If two languages share the same conception of logical space (i.e., are defined over the same class of models), then one can compare the expressive power of these languages by comparing how finely they carve logical space. This is the approach commonly employed, for instance, in debates over tense and modality, such as the primitivism-reductionism debate./nBut a second role languages can play in a debate is to advance a conception or theory of logical space itself. For example, consider the debate between perdurantism, which claims that objects persist through time by having temporal parts located throughout that time, and endurantism, which claims that objects persist through time by being wholly present at that time. A natural thought about this debate is that perdurantism and endurantism are simply alternative but equally good descriptions of the world rather than competing theories. Whenever the endurantist says, for instance, that an object is red at time t, the perdurantist can say that the object’s temporal part at t is red. On this view, one should conceive of perdurantism and endurantism not as theories picking out disjoint regions of logical space, but as theories offering alternative conceptions of logical space: one in which persistence through time is analogous to location in space and one in which it is not. A similar distinction applies to other metaphysical debates, such as the mereological debate between universalism and nihilism./nIf two theories propose incommensurable conceptions of logical space, we can still compare their expressive power utilizing the notion of a translation, which acts as a correlation between points in logical space that preserves the language’s inferential connections. I build a formal theory of translation that explores different ways of making this notion precise. I then apply this theory to two metaphysical debates, viz., the debate over whether composite objects exist and the debate over how objects persist through time. This allows us to get a clearer picture of the sense in which these debates can be viewed as genuine.

Advisor: Holliday, Wesley H
Yalcin, Seth