Frege on Definitions: A Case Study of Semantic Content

Frege on Definitions: A Case Study of Semantic Content

Frege on Definitions: A Case Study of Semantic Content

Frege on Definitions: A Case Study of Semantic Content


In this short monograph, John Horty explores the difficulties presented for Gottlob Frege's semantic theory, as well as its modern descendents, by the treatment of defined expressions.

The book begins by focusing on the psychological constraints governing Frege's notion of sense, or meaning, and argues that, given these constraints, even the treatment of simple stipulative definitions led Frege to important difficulties. Horty is able to suggest ways out of these difficulties that are both philosophically and logically plausible and Fregean in spirit. This discussion is then connected to a number of more familiar topics, such as indexicality and the discussion of concepts in recent theories of mind and language.

In the latter part of the book, after introducing a simple semantic model of senses as procedures, Horty considers the problems that definitions present for Frege's idea that the sense of an expression should mirror its grammatical structure. The requirement can be satisfied, he argues, only if defined expressions--and incomplete expressions as well--are assigned senses of their own, rather than treated contextually. He then explores one way in which these senses might be reified within the procedural model, drawing on ideas from work in the semantics of computer programming languages.

With its combination of technical semantics and history of philosophy, Horty's book tackles some of the hardest questions in the philosophy of language. It should interest philosophers, logicians, and linguists.


Every book has a history, and this short book has a long history—much of it, in fact, was written twenty years ago, as part of a dissertation on meaning in mathematical language. After completing this dissertation, I turned at once to an entirely separate line of research, centered around logic and artificial intelligence, and particularly nonmonotonic logic. One thing led to another, and as it turns out, I did not think about language or meaning again for nearly two decades, until a chance encounter with some of the more recent literature convinced me that the problems I had worried about had not been resolved, or even addressed in any detail, during the intervening period.

It is an interesting experience working with material that was originally drafted so long ago, and it is not an experience that I would recommend to a friend. Nevertheless, I do not propose to use distance from the material as any sort of an excuse. Although parts of the book are old, other parts are new. The older parts have been thoroughly rewritten, and I stand by both the arguments and the conclusions presented here.

As the title suggests, this book is very much a case study, entirely focused on the treatment of defined expressions in Frege’s own semantic theory, although I hope the discussion will have some bearing on those contemporary theories with their roots in Frege’s work. At various points, Frege considers a number of more contentious styles of definition, such as contextual or implicit definition, but I concentrate here only on the simplest and most pedestrian—ordinary stipulative definition. This form of definition can be illustrated by the introduction of the new predicate ‘is prime’, for example, into an arithmetical language through the stipulation that a number x is prime just in case any number that divides x is either equal to x or equal to 1.

The book is organized around two problems posed for Frege’s theory, and others like it, by languages that allow for the introduction of defined expressions like this.

Frege’s semantic theory is based, first of all, on a notion of sense, or meaning . . .

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