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William Oakcraft (Sep 24 2026 at 15:57):Good morning!
Lawvere's program in categorical logic seems a natural continuation of the so called "classical algebraic logic" (pre-Rasiowa/Blog-Pigozzi) tradition (on cylindric algebras, polyadic algebras, relation algebras), he even says so in Adjointness in Foundations.
I have always questioned myself: is there actual historical communication of the former algebraic logic tradition with categorical logic, at least an acknowledgement? (especially from Tarski). It seems acknowledgement and appraisal of the former by the latter is much more common (although I still would like to find actually unified comparisons and presentations of both styles), but I've never seen no one of the classical algebraic logic tradition even mentioning categorical logic (even though some of their most prominent members lived up to the end of the 20th century).
Of course we have contemporary representatives of the old tradition still living and active and interacting a lot more with categorical logic (Hajnal Andreka, Istvan Nemeti, T. S. Ahmed), but virtually nothing from the "founding fathers". Has anyone ever seen something in this direction?
Todd Trimble (Sep 27 2026 at 01:13):Not really. Both Tarski and Halmos have been dead for quite a while, and neither found categorical thinking very congenial; if either nodded at Lawvere's program, that fact is quite unknown to me. The connection between polyadic algebras and hyperdoctrines seems pretty unmistakeable, but in traditional algebra, the emphasis was always on one-sorted structures, so a polyadic algebra is a single set equipped with a bunch of operations, whereas a hyperdoctrine can have many sorts, and is more general (and I believe more flexible). I don't think they had any notion of how existential quantification is akin to induced representations, but hyperdoctrines and their semantics makes that quite clear.