Juliette Kennedy (University of Helsinki) – How first order is first order logic?

Fundamental to the practice of logic is the dogma regarding the first order/second order logic distinction, namely that it is ironclad. Was it always so? The emergence of the set theoretic paradigm is an interesting test case. Early workers in foundations generally used higher order systems in the form of type theory; but then higher order systems were gradually abandoned in favor of first order set theory—a transition that was completed, more or less, by the 1930s.

As for logic in general, the concept of a logic being first order is not only about whether the variables range over the elements of a given domain, or over sets of elements, or over sets of sets of elements, and so on; it is also, I suggest, about the context.

Of course, set theory is a theory and second order logic is a logic, at least that is the common understanding. However if one cares to view set theory as a logic—and if we do think of set theory as a logic, it is a logic with the cumulative hierarchy 𝑉 as its standard (class) model—then set theory turns out to be a stronger logic than second order logic. This is perhaps as it should be, given that the latter restricts the domain of quantifiable objects to those generated by (at most) a single iteration of the power set operation, while set theory allows for arbitrary iterations of the power set operation.

This talk is based on the forthcoming paper "How first order is first order logic?" by J. Kennedy and Jouko Väänänen for The Oxford Handbook of Philosophy of Logic. Editors: Elke Brendel, Massimiliano Carrara, Filippo Ferrari, Ole Hjortland, Gil Sagi, Gila Sher, Florian Steinberger, Oxford University Press.