I was wondering, can Second Order Logic (SOL) be a fundamental logic?

I am trying to gather some opinions from both sides to see what others might think.


Echoing what has been said in the comments on your question, the idea of "fundamental" seems a little bit ill-defined. However, what it seems like you are asking is something about the strength of second order logic and its ability to be the foundational system that can solve everything (well, at any rate everything that is solvable). We have, via our tools of metalogic, ideas such as completeness and compactness that can define a logic, and we would be well within our rights to say that second order does have some strong properties. However, second order logic is no where as strong as first order logic, in terms of these properties. First order logic itself isn't even able to be used as a completely whole foundation for logic! This is because of Gödel's incompleteness theorems. Even if we accept the results of the first and argue away that it doesn't make a difference if there are unprovable truths, since we can look and see that they are true, we still cannot prove the consistency of first order logic within itself! The second incompleteness theorem also applies to second order logic so we would be in the same boat. If first order logic, which is stronger, isn't enough to form the foundation for logic then how could second order success when it is much less strong?

Lindström's theorem is a result in metalogic that showed first order logic is the "strongest" logic, due to possessing the Löwenheim-Skolem theorem as well as the compactness theorem. The metalogical results of second order logic show that second order logic (with full semantics) show that it does not possess these properties. Even worse, as Quine (1970) pointed out, second order logic doesn't even have a complete proof system! This is very bad news for second order logic. Ultimately, no axiomatic will be strong enough to be used as a "fundamental" logic and this is brought to us with special thanks in part by Tarski and Gödel. That being said, first order logic is a much better pick than second order logic for us to make due with.

I am sure that the deadline for your essay has passed but I hope this has provided a (somewhat) useful answer for anyone who has a similar question.

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