Artificial Intelligence Center
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A major theme of Johan's Exploring Logical Dynamics and other recent work is the move from using the tools of logic to analyze the products of cognitive processes to using those tools, including perhaps some new ones, to analyze the processes themselves. Along with this has gone a focus on more complex cognitive products, products whose very complexity brings to the fore the fact that there are complex processes underlying their production and understanding. One kind of cognitive product is a grammatically well-formed sentence of a natural language apt for statement-making. Another, much more complex product, is a multi-person discourse or conversation, thought of as a complex structure of some sort, some of whose components may be statement-making sentences. Of course, a discourse isn't simply a set or even a sequence of such things; it has a structure that results both from the content of the components and from a variety of other structural features. This particular pair of products suggests another: a statement of mathematics and a proof (or derivation) of that statement. The latter, too, is a complex structure; it isn't simply a set or even a sequence of statements. Is there more to this suggestion or is it merely the result of more than free association? This is a question worth thinking---even thinking out loud---about. And, of course, since it is a question worth thinking about, Johan van Benthem has thought about it. So herein is my contribution to what I hope will be a continuing dialogue.
Dedicated, with great joy and fondest regards to Johan van Benthem on his fiftieth birthday, and with deep sadness to the memory of my friend and colleague, Megumi Kameyama.