Computational complexity logic

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Intuitionistic logic only proves theorems for which there exists a corresponding algorithm. Is there a logic which only proves theorems for which there exists a polynomial-time algorithm?

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Not sure I understand your question, but Horn-satisfiability for propositional clauses is P-Complete. That is, any problem in P can be reduced to propositional Horn satisfiability. Have a look at the other P-Complete problems in the Wikipedia article cited above to see if they're closer to what you're looking for.