The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

4
Logical Consequence: Models and Modality
STEWART SHAPIRO

My long-standing interest in the notion of logical consequence became urgent when I was working on my recent book on higher-order logic ( 1991). The most common complaint is that second-order logic is out of bounds, as logic, because its consequence relation is not effective. Even though higher- order logic is squarely within the prevailing model-theoretic tradition, its set of logical truths is not recursively enumerable--it is not even in the Kleene hierarchy. Advocates of second-order logic have been accused of such absurdities as attributing occult powers to the mind and rejecting Church's thesis (see, for example, Burgess ( 1993)). The force of this argument depends crucially on what logical study is supposed to accomplish, our present topic. A common slogan is that logic is to codify the pretheoretic norms of correct reasoning, the notion of logical consequence. This is surely correct, as far as it goes, but it does not go very far. What is this pre-theoretic notion of logical consequence, and what is it to codify something? Detailed answers to these questions are rare, even in works on philosophical logic. For example, in two widely read monographs both entitled Philosophy of Logic ( Quine ( 1986) and Putnam ( 1971)), the question of what logical theory is all about is barely asked.

Several intriguing and complex questions lie in the vicinity: questions about the relationship between formal languages and natural languages, questions about logical form, questions about logical and nonlogical terminology, and general matters of epistemology and metaphysics. The purpose of this paper is to motivate the problems and provide the main outlines of solutions. The upshot is that when properly adjusted, the model-theoretic formulation of consequence provides a reasonable mathematical model of

____________________
This paper benefited considerably from the discussion at the conference, 'Philosophy of Mathematics Today'. The participants, who are the authors of the other articles in this volume, were most generous. I would also like to thank Nuel Belnap, Jill Dieterle, John Etchemendy, Ivor Grattan-Guinness, Robert Kraut, David McCarty, Penelope Maddy, Gregory Moore, Michael Scanlan, Barbara Scholz, Gila Sher, Mark Steiner, Neil Tennant, and Mark Wilson for reading an earlier version of this project and providing useful criticism. I apologize for any omissions from the list. I also thank the students who participated in two extremely productive seminars on this and related topics: John Chaplin, Karen Cordrick, Morgan Jackson, Jill Dieterle, Ty Lightner, and Pierluigi Miraglia.

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