The intended interpretation of intuitionistic logic

We present an overview of the unintended interpretations of intuitionistic logic that arose after Heyting formalized the “observed regularities” in the use of formal parts of language, in particular, first-order logic and Heyting Arithmetic. We include unintended interpretations of some mild variations on “official” intuitionism, such as intuitionistic type theories with full comprehension and higher order logic without choice principles or not satisfying the right choice sequence properties. We conclude with remarks on the quest for a correct interpretation of intuitionistic logic. §1. The Origins of Intuitionistic Logic Intuitionism was more than twenty years old before A. Heyting produced the first complete axiomatizations for intuitionistic propositional and predicate logic: according to L. E. J. Brouwer, the founder of intuitionism, logic is secondary to mathematics. Some of Brouwer’s papers even suggest that formalization cannot be useful to intuitionism. One may wonder, then, whether intuitionistic logic should itself be regarded as an unintended interpretation of intuitionistic mathematics. I will not discuss Brouwer’s ideas in detail (on this, see [Brouwer 1975], [Heyting 1934, 1956]), but some aspects of his philosophy need to be highlighted here. According to Brouwer mathematics is an activity of the human mind, a product of languageless thought. One cannot be certain that language is a perfect reflection of this mental activity. This makes language an uncertain medium (see [van Stigt 1982] for more details on Brouwer’s ideas about language). In “De onbetrouwbaarheid der logische principes” ([Brouwer 1981], pp. 253–259; for English translations of Brouwer’s work on intuitionism, see [Brouwer 1975]) Brouwer argues that logical principles should not guide but describe regularities that are observed in mathematical practice. The Principle of Excluded Third, A∨ ¬A, is an example of a logical principle that has become a guide for mathematical practice instead of simply describing it: the Principle of Excluded Third is observed in verifiable “finite” situations and generalized to a rule of mathematics. But according to Brouwer mathematics is not an experimental science, in which one only has to repeat an experiment sufficiently often to establish a law, so the Principle of Excluded Third should be discarded. All his life Brouwer avoided the use of a formal language or logic, perhaps because of its unreliability, perhaps because of his personal style (see [Brouwer 1981a], p. xi). This does not imply that he did not believe in the possibility of a useful place for logic in intuitionistic mathematics, but rather that Brouwer would not himself resort to a formal language. This attitude was detrimental to the 1