Lukasiewicz logic: algebras and sheaves

Abstract

Classical logic arose from the need to study forms and laws of the human reasoning. But soon, it came out the di culties of classical logic to formalize uncertain events and vague concepts, for which it is not possible to assert if a sentence is true or false. In order to overcome these limits, at the beginning of the last century, non classical logics were introduced. In these logic it fails at least one among the basic principles of classical logic. For example, cutting out the principle of truth functionality (the true value of a sentence only depends on the truth values of its component more simpler sentences), we obtain modal logics for which the truth value of a sentence depends on the context where we are. In this case, the context is seen as a possible world of realization. Cutting out the principle of bivalence, we obtain many-valued logics instead. The rst among classical logician not to accept completely the principle of bivalence was Aristotele, who is, however, considered the father of classical logic. Indeed, Aristotele presented again the problem of futuri contingenti1 introduced by Diodorus Cronus as exception to the principle of bivalence (see Chapter 9 in his De Intepretatione). The \futuri contingenti" are sentences talking about future events for which it is not possible to say if they are true or false. However, Aristotele didn't make up a system of many-valued logic able to overcome classical logic's limits. [edited by author]