Many-valued logic: beyond algebraic semantics

Abstract

The last three decades have witnessed major advances in many-valued logic and related fields. The theory of Łukasiewicz logic and Chang’s MV-algebras has flourished, establishing profound connections with other fields of mathematics; Petr Hajek’s framework for mathematical fuzzy logic has met with remarkable success, bringing into focus the central role of residuation; and the algebraic analysis of substructural logics via residuated lattices has proved itself most fruitful, providing an illuminating bridge between algebra and proof theory. Many-valued logic is here to stay—or so it seems today. Algebra was a key ingredient of these developments: indeed, many-valued logic has been traditionally investigated through its algebraic semantics. And yet, logic is not just algebra. A distinctive feature of logic is its concern with the relationship between syntax (language) and semantics (the world). From this point of view, algebra belongs to the syntactic aspects of the subject. One may go as far as saying that the locution ‘‘algebraic semantics’’, though established, is in fact oxymoronic. While Heyting algebras are the equivalent algebraic semantics of the intuitionistic propositional calculus, a genuinely semantic understanding of intuitionism must rely on relational, Kripke-style structures. When compared on these grounds, many-valued logic appears to lag behind inuitionistic and modal logics. This special issue of Soft Computing is intended as a contribution to the major research project of developing a deeper understanding of the meaning of many-valued logical systems, reaching beyond algebraic semantics. The collection of essays we present here is a follow-up to the third instalment of the biennial conference series ManyVal, held in Varese, Italy, during May 2010. The meeting gathered about 40 researchers with varied backgrounds, but all sharing a keen interest in the semantics of many-valued logics. Although the papers in this special issue range from the foundational to the technical, it will transpire that at some level they are all related to the theme of the conference. Two papers in the present collection address issues of broad significance explicitly. Didier Dubois’s contribution discusses an embedding of Kleene and Belnap many-valued logics—sporting three and four truth values, respectively—into a system of epistemic logic. This is in line with the origins of these many-valued systems, which were conceived by their inventors as logics of knowledge and contradiction. Beyond the specific results it illustrates, the paper may be construed as a thought-provoking essay within the major debate on the relationship between vagueness (many-valued logic) and ignorance (epistemic logic). By contrast, the paper by Thomas Vetterlein focusses on theories of vagueness in themselves. Using as testing grounds such classical topics as the sorites paradox and the syllogism, the author advances and defends his own view that a central challenge in connection with reasoning under vagueness is to find models for the varying levels of granularity relevant to a particular context. He argues that this leads to rejecting the view that any single formal S. Aguzzoli Dipartimento di Scienze dell’Informazione, Universita degli Studi di Milano, Milan, Italy e-mail: aguzzoli@dsi.unimi.it