After reading Gödel's incompleteness theorem, I wonder if there are any systems whose axioms are sound and complete. I realize that Gödel's arguments (at least from my sources) only apply to arithmetic logic. So, I wonder if it applies to propositional logic as well.

If propositional logic is not sound and complete, are there any systems of axioms which are? I find it strange because then the system would almost become circular in a way. I hope I made some sense, I apologize if I misrepresented some theorems or terms.

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    Yes, so is predicate calculus (without additional axioms), Presburger arithmetic (with no multiplication), and Tarski's elementary Euclidean geometry, see Wikipedia. – Conifold Aug 8 '18 at 18:53
  • Maybe you aren't aware, but there are two different uses of the term "complete" in regards to logic that sometimes get confused (e.g. you can find many questions of people asking how Gödel's completeness and incompleteness theorems don't contradict each other, they're talking about two different things.) The issue is that soundness is usually talked about together with the completeness that Gödel proved, that semantic validity implies provability. That's what we mean when we say "a system of logic and it's deductive system are sound and complete". – Not_Here Aug 8 '18 at 19:04
  • But in your title, you seem to be talking about the other form of completeness, the one that Gödel proved his incompleteness theorems about. I just want to make sure that you're clear on the concepts you're using, because again "soundness and completeness" almost always refers to deductive systems, and the other type of completeness which I think you're asking about is different and doesn't really have anything to do with soundness in the way the other use of the word does. – Not_Here Aug 8 '18 at 19:05
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    Distinguish between two kinds of completeness: semantic completeness and syntactic completeness. Semantic completeness means that given a formal logic and a proof system, if a sentence A is true in every model of the system, then A is provable from the deduction rules of that system. This is what Gödel's completeness theorem proves for first order logic, it is also the converse of soundness. Syntactic completeness means, for a given formal theory, if A is a sentence in the language of the theory, then either A or ~A is provable from that theory. This is what Gödel's incompleteness theorem uses – Not_Here Aug 8 '18 at 19:14

Yes, you can give axioms and rules of inference so that all and only the tautologies are derivable. Propositional calculus is better behaved than first order logic in that sense. That doesn't mean you can find the proofs quickly though, unless P=NP.

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