## 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]