/ 07 03 27 6 v 1 3 0 M ar 2 00 7 An exercise in “ anhomomorphic logic ” ⋆

A classical logic exhibits a threefold inner structure comprising an algebra of propositions A, a space of “truth values” V , and a distinguished family of mappings φ from propositions to truth values. Classically A is a Boolean algebra, V = Z2, and the admissible maps φ : A→Z2 are homomorphisms. If one admits a larger set of maps, one obtains an anhomomorphic logic that seems better suited to quantal reality (and the needs of quantum gravity). I explain these ideas and illustrate them with three simple examples. From a certain point of view, the phrase “classical logic” should be used in the plural, not the singular, because the things with which logic deals depend on the “domain of discourse”, and this can vary both with time and with the “system” one has in mind. To each such domain corresponds its own Boolean algebra, namely the algebra A of all “questions” one may ask about the system. 1 But a domain of questions together with rules for combining them via and, xor, not, etc, is not all there is to a logic. In addition, one ⋆ To appear in a special volume of Journal of Physics, edited by L. Diosi, H-T Elze, and G. Vitiello, and devoted to the Proceedings of the DICE2006 meeting, held September 2006, in Piombino, Italia. arxiv quant-ph/0703276 1 Instead of “question” one also says “predicate”, “proposition” or “event”. I will use these terms interchangeably, and will sometimes refer to A as the “event algebra”, for lack of a better term.