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Maximality of first-order logic without identity

Hold Date
2021-10-12 12:00〜2021-10-12 13:00
Object person
Guillermo Badia (The University of Queensland, Australia, and Johannes Kepler University Linz, Austria)

First-order (or elementary) logic is typically regarded as a paradigm logical system for applications in mathematics and philosophy since it strikes a nice balance between expressivity and (what is seen by many as) desirable technical properties. Examples of the latter include the fact that the validities of the logic are computable, that the only infinite domain one ever needs to consider in checking the validity of a first-order sentence is the natural numbers, and that to show that a set of sentences can be made true in some domain we only need to check that every finite subset of the set can be made true in some domain. Shockingly, in the 1960s, Per Lindström established through a series of theorems that first-order logic cannot be improved while still striking the aforementioned balance: any expressive improvement comes with a technical price. One issue with Lindström's approach is that it assumes that first-order logic comes equipped with an identity symbol "=", to be interpreted as numerical equality. Although Lindström's proofs heavily use this fact, up to the 1960s, most logic textbooks would only add identity a posteriori to their presentation of first-order logic, as it was thought that "pure" logic didn't need to involve a mathematical notion like numerical equality. Moreover, plenty can still be formalized without identity, e.g. foundational theories such as ZF set theory, where "=" is definable. In this talk, I'll show that Lindström style results can be obtained in an identity-free context by non-trivial modifications of the arguments in ways that are not available in the presence of identity. Hence, proponents of first-order logic without identity have at least as strong a case for it being the paradigm logical system as those of first-order logic with identity.

This is joint work with Xavier Caicedo and Carles Noguera.