Review of Language in Action: Categories, Lambdas, and Dynamic Logic, by J. van Benthem


Israel, D. J. Review of Language in Action: Categories, Lambdas, and Dynamic Logic, by J. van Benthem. Artificial Intelligence Journal, vol. 63, no. 1-2, Oct 1993.


For about two thousand years, logic and logicians were stuck on the problem of multiple quantification. Aristotle and his successors had systematized large parts of monadic quantification theory (as we now call it), but very little progress had been made on extending the Aristotelian account to sentences involving more than one quantified noun phrase, as in “Every logician admires someone,” (This happens to follow from the truth of “Every logician admires Johan.”) Such sentences will involve relational expressions; and relations in general remained enough of a mystery that many denied their reality. In the nineteenth century, DeMorgan, Schr6der, and Peirce instituted the algebraic study of relations, in particular binary relations. But it fell to Frege to solve the problem of multiple quantification; thus it is Frege who gets the credit for producing, more or less whole, modern quantification theory.

A crucial part of Frege’s approach was to abandon or ignore the traditional decomposition of sentences into subject and predicate. These were notions that had been given no precise mathematico-logical content. In place of this decomposition, Frege attempted systematically to think of sentences and other complex expressions as constructed by way of applying syntactic functions to expressions. […]

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