A couple of my introductory logic books appeal to modal and set-theoretic notions in building up first-order logic. (They explicitly acknowledge these connections and say, for example, that validity is a modal notion, or they speak of “sets” of propositions, using the term in an expressly technical sense.) How do these different logical systems relate to each other? Do modal logic and set theory “flow out of” first-order logic? Or is set theory the foundation of all of logic? Or are they somehow or other interdefinable?
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1Great question to my mind, as is your other one, where I tried to give some kind of answer. As you write, I think most books/authors do not examine these problems/issues very carefully. In that sense, one can maybe say that they are more practical than theoretical in nature. If you really want to discover and investigate these topics it seems to me that you are heading towards the philosophy of mathematics and the philosophy of language, and maybe also others fields of study. Perhaps you might be interested in reading this https://math.stanford.edu/~feferman/papers/whatrests.pdf – Ettore Feb 17 '24 at 10:46
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1Thank you kindly for the suggestion. I think you’re right that these questions deal a lot with the philosophy of mathematics and language. Proof Theory is definitely on my woefully long list of things to explore. Robert Brandom has interesting work on the nature of meaning, and I’ve heard it said that inferentialism has connections with proof theory. – inkd Feb 17 '24 at 19:42
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1Thanks a lot for your comment too! I just had the time to quickly read through a few paragraphs on wikipedia about Brandom and Inferentialism, very interesting, for sure worth a study in its own right.. – Ettore Feb 23 '24 at 16:42