Inside Classical Logic: Truth, Contradictions, Fractionality

Abstract Fractional semantics provides a multi-valued interpretation of a variety of logics, governed by purely proof-theoretic principles. This approach employs a method of systematic decomposition of formulas through a well-disciplined sequent calculus, assigning a fractional value that measures the “quantity of identity” (intuitively, “quantity of truth”) within a sequent. A key consequence of this framework is the breakdown of the traditional symmetry between truth and contradiction. In this paper, we explore the ramifications of this novel perspective on classical logic. Specifically, we (i) introduce an alternative paraconsistent consequence relation, and (ii) show how the gradual character of contradictions induces a corresponding characterization of tautologies, thereby obtaining a full-fledged informational refinement of classical logic.