Brains, Persons, and Society *** ABSTRACTS
Cervelli, Persone e Società ***ABSTRACTS
Does a Kripke Semantics Vindicate the Adoption of Intuitionistic Logic in Set Theory?
Jonathan Lear (1977) presents an argument for the adoption of intuitionistic logic in our settheoretic
reasoning. The argument consists of three steps: one from the open-endedness of the
concept of set to cognitivism, the thesis that we never quantify over all sets, but only over the
extension of ‘set’ at a certain time; one from cognitivism to Kripke semantics; one from Kripke
semantics to intuitionistic logic. In two recent articles (Paseau 2001; 2003), Alexander Paseau has
criticised both the step from cognitivism to Kripke semantics and that from Kripke semantics to
intuitionistic logic. In this paper I show that Paseau's criticism of the step from Kripke semantics to
intuitionistic logic is flawed and that the adoption of a Kripke semantics in set theory does sanction
the use of intuitionistic logic.
Paseau challenges the step from Kripke semantics to intuitionistic logic on the basis of a
model-theoretic result first noticed by Hazen (1982). Hazen observes that if a Kripke model is
directed, it validates φ v ¬¬φ, which is generally not intuitionistically valid. Paseau argues that the
nodes of the Kripke model, interpreted as indexing either actual or possible knowledge states, are
linearly ordered and, since linearity is a property stronger than directedness, that the Kripke model
validates φ v ¬¬φ on both interpretations. Paseau argues that possible knowledge states are linearly
ordered by combining the Comparability of cardinals with Zermelo’s Categoricity Theorem. He
then tries to justify the use of the Comparability of Cardinals, given its equivalence to the Axiom of
Choice. As I show, however, the argument does not need the Axiom of Choice, since the levels of
the hierarchy, rather than being indexed by cardinals, are indexed by ordinals, and the ordinals can
be shown to be comparable without the Axiom of Choice.
Nevertheless, I claim, Paseau’s argument fails because of its use of Zermelo's Categoricity
Theorem, a theorem of second-order set theory, which we are not entitled to use in this context. For
to argue that the open-endedness of the concept of set does not lead to the adoption of intuitionistic
logic by assuming that we can grasp the concept well-defined property in such a comprehensive
way that we can quantify over all properties would not be a neutral assumption if this concept were
to exhibit the same kind of open-endedness as the concept set. But several considerations, I argue,
point in just this direction. First, the concept of well-defined property is indefinitely extensible:
once we have specified a language and recognised its well-defined properties, it is always possible
to specify, by reference to the expressions of that language, a further well-defined property which is
not expressible in it. Second, second-order set theory decides the Continuum Hypothesis: there is a
sentence in the language of second-order logic which is true if and only if the Continuum
Hypothesis is true. This, however, has not led anyone to try to settle the Continuum Hypothesis by
reflecting on the acceptability of this sentence as a logical truth.
I conclude by providing an example of a possible evolution of knowledge states that
branches as different conceptions of the width and height of the cumulative hierarchy are combined
and by offering some final considerations on the significance of the failure of Paseau’s argument.
Hazen, A., 1982, On a possible misinterpretation of Kripke’s semantics for intuitionistic logic,
Analysis, 42, 128-133.
Lear, J., 1977, Sets and Semantics, Journal of Philosophy, 74(2), 86-102.
Paseau, A., 2001, Should the logic of set theory be intuitionistic?, Proceedings of the Aristotelian
Society, 101, 369-378.
Paseau, A., 2003, The open-endedness of the set concept and the semantics of set theory, Synthese,