What is a logic? Towards axiomatic emptiness
J.-Y. B eziau · 2010
We rst recall the original Greek sense of the word logic and how logic was developed on the one hand as an e cient way of reasoning by the use of reduction to the absurd and on the other hand as a useless system of logic by Aristotle. Then we discuss the changes of the modern conception of logic: the rejection of the principle of noncontradiction considered as fundamental by Aristotle and the structuralist move breaking the Aristotelian accident/essence dichotomy. Finally we explain why and how in universal logic like in universal algebra axiomatic emptiness prevails: a logical structure is a structure obeying no axioms. 1 In the beginning was the Logos What is logic? is a di cult question. In this paper we will tackle the issue through a simpler question What is a logic? . We will explain why we can consider that a logic is a structure without axioms similarly to the case of universal algebra and giving a example of a simple logic, anti-classical logic, obeying none of the standard axioms that however can reasonably be considered as a logic. But it is important to keep in mind the more general issue of what logic is and to see how our answer to the simpler question is related and directed by this broader perspective. Logic nowadays is generally the study of some logical systems related with mathematics, computer science, arti cial intelligence or philosophy. These logical systems are sometimes called formal systems and their construction and study is developed using some mathematical tools at various degrees. Most of the time people have lost the more profound idea of what logic is. It is good to remember that the word logic came from the Greek word Logos , a key concept of Greek culture. The word Logos has Work supported by a grant DCR (Funcap/CNPq) and within the LogComp research project (CNPq). Thanks to all members of Log Comp.