Intensional Logic in Context ∗

What exactly is “Intensional programming”? Every year, at every ISLIP, we try to answer this question. And the answers keep getting better. There is, of course, an easy answer, namely “programming in a language based on intensional logic”. But this answer raises another, more fundamental question: “What is intensional logic?” Logicians have been trying to answer that question for about 2500 years, and their answers are also getting better. In this tutorial I will present some of these answers to the second question and explain how they help answer the first. 1 Intension vs Extension The term “intensional” itself is relatively recent—Carnap introduced it in the 1930s, based on Frege’s distinction between the “sense” and the “denotation” of an expression. In Frege’s terminology the denotation of an expression is just that—the particular object it in fact denotes (or currently denotes). (This is what Carnap and most modern logicians call the extension). On the other hand, the “sense” (what we now call the intension) is the entire concept it represents—what we, at some level, intend when we write it. For example, the expression “the President of France” denotes currently Jacques Chirac; but no one would claim that M Chirac somehow sums up the whole concept of the French presidency. Intensional logic is therefore the logic of expressions in which the intension of subexpressions (and not just their extensions) have to be taken into account. These are very common in natural language. The French constitution specifies that the French President is directly elected; and this is not the same as specifying that M Chirac be directly elected. 2 The mysteries of intensionality Many famous paradoxes are based on the observation that intensional expressions seem to violate the basic law of substitution of equals for equals. For example, it is certainly true that The number of planets = 9 Furthermore Kepler, the famous astronomer, was well aware of the basic rules of arithmetic; we can be sure that Kepler knew that 9 is a perfect square But if we substitute equals for equals in this latter assertion, we derive Kepler knew that the number of planets is a perfect square ∗Published in Intensional Programming II, Panos Rondogiannis and Manolis Gergatsoulis, editors, pp. 1–13, World-Scientific, 2000. Invited talk given at the 12th International Symposium on Intensional Programming, National Center for Scientific Research “Demokritos”, Athens, Greece, 28–30 June 1999.