Abstract

Despite initial appearance, paradoxes in classical logic, when comprehension is unrestricted, do not go away even if the law of excluded middle is dropped, unless the law of noncontradiction is eliminated as well, which makes logic much less powerful. Is there an alternative way to preserve unrestricted comprehension of common language, while retaining power of classical logic? The answer is yes, when provability modal logic is utilized. Modal logic NL is constructed for this purpose. Unless a paradox is provable, usual rules of classical logic follow. The main point for modal logic NL is to tune the law of excluded middle so that we allow for a sentence and its negation to be both false in case a paradox provably arises. Curry's paradox is resolved differently from other paradoxes but is also resolved in modal logic NL. The changes allow for unrestricted comprehension and naive set theory, and allow us to justify use of common language in formal sense.