Logic in Opposition

It is claimed hereby that, against a current view of logic as a theory of consequence, opposition is a basic logical concept that can be used to define consequence itself. This requires some substantial changes in the underlying framework, including: a non-Fregean semantics of questions and answers, instead of the usual truth-conditional semantics; an extension of opposition as a relation between any structured objects; a definition of oppositions in terms of basic negation. Objections to this claim will be reviewed

Two Squares of Opposition: for Analytic and Synthetic Propositions

In the paper I prove that there are two squares of opposition. The unconventional one is built up for synthetic propositions. There a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation

Questions and Answers about Oppositions

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth

An Arithmetization of Logical Oppositions

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers

Inductive Logic

Inductive Logic is a ‘thematic compilation’ by Avi Sion. It collects in one volume many (though not all) of the essays, that he has written on this subject over a period of some 23 years, which all demonstrate the possibility and conditions of validity of human knowledge, the utility and reliability of human cognitive means when properly used, contrary to the skeptical assumptions that are nowadays fashionable