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So this sounds like this might lead to an undecidable theory but I thought I would give it a try and ask about it after I found nothing on the subject. I am somewhat interested in finding functions wh …

The argument goes like this: Say you have a sound and complete logic for proving the termination of computer programs. The rules of the logic make the proofs recursively enumerable as well. … In practice we work with what we think are sound systems of logic and we work with recursively enumerable sets of proofs as well. …

Say you tried to solve $f(A, g(A)) = f(B, B)$ after applying $A \to B$ you'd then have $f(A, g(A)) = f(A, A)$ and you'd have to unify $A = g(A)$ as a sub problem.

So I wasn't sure weather or not this counted as "research level" or not but I figured it wasn't so I decided to post it here. There is a paper by S. Bellantoni et al. called "Higher Type Recursion, Ra …

or we directly find a model for the logic and show that the given statement is false in the model. … I don't remember exactly how this works nor am I really clear that I could extend a result from first order constrictive logic (the logic of constructive set theory). How do we know this? …

So propositional logic (PL) is efficiently (in P) decidable because I can convert formulas to an equisatisifiable CNF-formula, negate and convert (efficiently, by De Morgans laws) to DNF. … in bullets are there any logics that have the following properties: Decision procedure for membership in set of theorems is efficent More expressive than PL (but not nearly as expressive as Tarski's logic …