# Richard Lawrence

## Research

My research engages with contemporary issues in the philosophy of language, philosophical logic, and linguistics. But my interest in these issues stems from an interest in a set of older and more foundational questions, which have played a central role in the philosophy of language and analytic philosophy more generally. These are such questions as: what is it for an expression to mean something? How are the meanings of expressions related to things in the world and to our ways of knowing about them? How should we go about giving a systematic theory of meaning, and what should the central concepts in such a theory be?

Broadly speaking, my answers to these questions belong to the pragmatist tradition. I believe that linguistic meaning is instituted by social practices. Linguistic expressions mean what they do because social practices set standards for their correct use. We learn to participate in such practices, and this training mediates the relationship between our language and the world. Accordingly, semantic theory should ultimately connect the meanings of particular linguistic expressions with things that human beings can learn to do. Such a theory will answer puzzling questions about the things we speak about by revealing what we are doing with our language for them. It will tell us, for example, how it is possible to refer to numbers and other abstracta, by making plain how expressions like ‘3’ or ‘the smallest inaccessible cardinal’ are used in mathematical practice. My research aims at developing this pragmatist approach to semantic theorizing without sacrificing formal precision or expressive adequacy.

My dissertation begins a study of a specific type of practice that I call an investigation. An investigation is the activity of asking, and answering, a particular question. Carrying out investigations is something we learn to do in different domains by different methods, in accordance with a common practical and epistemological structure. Because of its generality, this type of practice is interesting for a variety of philosophical disciplines, including logic, epistemology, and the philosophy of science. It is thus a particularly fundamental and illuminating example of a practice around which to develop a pragmatist theory of meaning.

In the near term, my research will focus on continuing the program begun in my dissertation, developing the concept of an investigation and applying it in the theory of meaning for natural language. Here are two projects in this program that I plan to publish in the immediate future:

##### “What is an investigation?”

An investigation is structured by a question: it is the activity of seeking an answer to that question. The process of solving an elementary algebra problem provides a particularly clear example of an investigation, so I take it as a case study. The equations in an algebraic derivation often have a puzzling feature: the equation used to state the problem (say, $$x^2 - 6x + 9 = 0$$) is materially equivalent to the one used to state its solution ($$x = 3$$); yet the latter gives the solution, because it gives the values of a variable, while the former does not. So I ask what it means to ‘give’ the values of a variable, arguing that four different features are required. The variable must be in the scope opened by the problem statement; the values given must be in the range of the variable, which is determined by the problem; the statement giving the values must represent a complete solution; and it must be in a canonical form. I argue that these features characterize the ends of investigation more generally: to answer a question and conclude an investigation is to find a statement that has these four features.

##### “Specification and investigation”

A specificational sentence is a copular sentence like “What I’d like to do is relax with a novel”, “The reason she left was that the conversation took a nasty turn”, or “One way to improve your health is to exercise”. Drawing on ideas from both the linguistics literature and from literature on the logic of questions, I defend an analysis of specificational sentences as question-answer pairs: the pre-copular phrase expresses a question to which the post-copular phrase supplies the answer. This analysis is interesting because it implies that specificational sentences play a special role in investigations. Asserting a specificational sentence marks the final move in an investigation, by explicitly connecting an open question with its answer. To show how the meaning of specificational sentences relates to their role in investigatory practice, I use a game-theoretical semantics to represent their truth conditions. On my analysis, the bound variables introduced in the first part of a specificational sentence induce an investigation during a semantic game; when the second part displays a successful strategy for completing that investigation, the sentence is true. This semantics explains many of the interesting features of specificational sentences, including their sensitivity to questions in discourse, the difference between specificational and equative sentences, and their logical relationship to non-specificational paraphrases.

In longer-term research, I will connect this work in the theory of meaning with my interests in other areas, particularly in logic, metaphysics, and the philosophy of science. I am especially interested in the relationship between investigatory practice and the systems of categories we use to describe and reason about the world. How should an understanding of investigations inform the basic categories we deploy in logic, ontology, and scientific explanations? My research into this question will examine both historical and contemporary discussions of systems of categories and the logical tools required to represent them. Here are some further projects that I plan to undertake in this area:

##### Concepts and objects in game-theoretical semantics

Frege’s distinction between objects and concepts is a foundational distinction in modern logic. A standard reading construes Frege’s distinction as a distinction between two ontological categories, but this reading faces serious interpretive challenges. I argue that a better interpretation conceives objects and concepts as epistemological roles: something belongs to one category or the other depending on how it is apprehended in scientific thought. I spell out this idea using game-theoretical semantics, arguing that we can understand the distinction between concepts and objects in terms of their different roles in a semantic game that defines truth in a model. Objects are what players seek, find, and specify when executing quantifier moves in this game. Concepts are what guides a player as she makes such moves: they distinguish strategic choices of objects from non-strategic ones. Because such quantifier moves involve investigations over the model’s domain, this characterization of objects and concepts describes them in terms of their roles in investigations. This account has the virtue of ontological neutrality, offering a clear sense in which numbers, colors, and planets all count as objects, without committing us to finding anything common in their ontologies.

##### Abstraction principles and specification

Inspired by Frege’s program of reducing mathematics to logic and definitions, the neo-logicist program in philosophy of mathematics aims to give a foundation for mathematics via abstraction principles. Abstraction principles provide identity conditions for mathematical objects using purely logical resources. Neo-logicists have struggled, however, to articulate the conditions that abstraction principles must satisfy if they are to ground our actual mathematical knowledge. I argue that what has been missing in the debate over abstraction principles is attention to how mathematical objects are specified. To ground our knowledge of the objects in a certain mathematical domain, we need more than an abstraction principle which determines whether or not two such objects are identical; we must also determine which statements of identity count as giving such objects to us, in the sense of specifying them in a canonical form. I show that careful attention to our actual practices of specifying natural numbers can adjudicate between candidate abstraction principles in number theory. At the same time, this insight presents a new challenge to neo-logicism in domains like set theory or real analysis, where our practices of specifying mathematical objects are less determinate.

##### Model theory for specificational sentences

Frege remarked that “Jupiter has four moons” expresses the same content as “The number of Jupiter’s moons is four”, but these two sentences represent two different ways of ‘carving up’ that content. In making this remark, Frege was putting his finger on an important idea: in one sense, a specificational sentence like “The number of Jupiter’s moons is four” is equivalent to a non-specificational paraphrase (by having the same content); but in another sense, they are not equivalent (they carve up that content differently). I show how to represent this idea in a model-theoretic setting. The representation simultaneously captures two facts which at first appear difficult to reconcile. First, a single non-specificational form (“Jupiter has four moons”) may be equivalent to several distinct specificational forms (“The number of Jupiter’s moons is four”, “The planet with four moons is Jupiter”). Second, these distinct specificational forms seem to require interpretations with different domains: it makes no sense to say “The number of moons is Jupiter” or “The largest planet is four”. Both facts can be represented by thinking of the models for the language as equipped with multiple categories or sorts, and thinking of the move from a non-specificational form to a specificational one as the move from such a model to the induced structure, for a restricted language, on one of its sorts. This approach provides a precise, well-understood apparatus for thinking about what it means to ‘recarve’ content.