Games in Algebraic Logic

In 1860, Augustus de Morgan published DeM60], thereby launching an investigation into the algebra of relations. This developed into the subject now called Algebraic Logic, though in the 19th century it was simply thought of as mathematical logic. This work, along with Frege's quantiier logic, became the foundation of modern logic and model theory. In De Morgan's writing there is no sharp separation of philosophy and mathematics and the central problem for him was to unveil the laws of rational thought. Of course the word`rational' is critical and problematic here: he did not wish to consider how an insane person might think, nor the eeect of one's mood on the thought process, nor any subjective features of thinking. Rational thought is considered here as a purely objective process, quite independent from the real cognitive process | a strange concept admittedly, but a pervasive idea in the philosophy of mathematics. De Morgan was particularly interested in discovering the principles of everyday thinking and of mathematical argument. Two authors innuenced him greatly: Aristotle and Boole. Aristotle's syllogism had held sway for 2000 years. Indeed Kant Kan72] had argued that Since Aristotle's time Logic has not gained much in extent, as indeed nature forbids it should. .. .Aristotle has omitted no essential point of the understanding; we have only become more accurate, methodical, and orderly. But De Morgan was among a number of philosophers who found the Aristotelian syllogism inadequate to model the laws of thought. Consider the following quote from De Morgan Accordingly, all logical relation is aarmed to be reducible to identity 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 aarmed, 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 …