@MISC{Hirsch98gamesin, author = {Robin Hirsch and Ian Hodkinson}, title = {Games in Algebraic Logic}, year = {1998} }

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Abstract

11.52> A is A, to non-contradiction, Nothing both A and not-A, and to excluded middle, Everything either A or not-A. These three principles, it is affirmed, dictate all the forms of inference, and evolve all the canons of syllogism. I am not prepared to deny the truth of either of these propositions, at least when A is not self-contradictory, but I cannot see how, alone, they are competent to the functions assigned. I see that they distinguish truth from falsehood: but I do not see that they, again alone, either distinguish or evolve one truth from another. Every transgression of these laws is an invalid inference: every valid inference is not a transgression of these laws. But I cannot admit that every thing which is not a transgression of these laws is a valid inference. And I cannot make out how just the only propositions which are true