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general theory of intensional logic

Alternative logics

It has been shown that this logic can be interpreted in terms of the same kind of modal logic serving as a system of epistemic logic. In the light of its purpose to consider only the known, this isomorphism is suggestive. The avowed purpose of the intuitionist is to consider only what can actually be established constructively in logic and in mathematics—i.e., what can actually be known. Thus, he refuses to consider, for example, “Either A or not-A” as a logical truth, for it does not actually help one in knowing whether A or not-A is the case. This does not close, however, the philosophical problem about intuitionism. Special problems arise from intuitionists’ rejection (in effect) of the nonepistemic aspects of logic, as illustrated by the fact that only a part of epistemic logic is needed in this translation of intuitionistic logic into epistemic logic.Other new logics are obtained by modifying the rules of those games that are involved in the game-theoretical interpretation of first-order logic mentioned above. The logician may reject, for instance, the assumption that he possesses perfect information, an assumption that characterizes classical first-order logic. One may also try to restrict the strategy sets of the players—to recursive strategies, for example.