Two, Three, Four, Infinity: The Path to the Four-Valued Logic and Beyond

Abstract

I give a kind of intellectual history of the so-called “Belnap-Dunn Four-valued Logic,” examining its evolution: the 4-element De Morgan lattice of Antonio Monteiro, and related work of Bialynicki – Birula and Helena Rasiowa, and John Kalman; Timothy Smiley’s 4-element matrix for Belnap’s Tautological Entailment; Dunn’s interpretation in terms of “aboutness;” Bas van Fraassen’s semantics for Tautological Entailment using “facts;” and Dunn’s interpretation in terms of how a sentence can be assigned both true and false, or neither true nor false, as well as the usual two values simply true, or simply false. Of course I discuss Nuel Belnap’s viewing the four values as elements in a “bi-lattice” and his famous use of this interpretation for “How a Computer Should think.” I also examine relationships to Richard Routley and Valerie Routley’s “star semantics.” Moreover, I discuss extension of the 4-valued semantics to the whole system R (allowing nested relevant implications), focusing especially on Edwin Mares’ work.I then examine later adaptations and extensions of the Four-valued Logic, including work by Yaroslav Shramko, Tatsutoshi Takenaka, Dunn, Heinrich Wansing, and Hitoshi Omori on “trilattices.” I also explain my recent extension to an infinite valued “Opinion Tetrahedron” (extending Audun Jøsang’s Opinion Triangle) which has the four values as its apexes.I end by acknowledging that ideas involving the 4-values date back to classical Indian logic (Sanjay’s “Four Corners”), prior to the sixth century B.C.E.

We of course borrow our title from George Gamow’s [35], One, Two, Three...Infinity. This books was a great influence on my intellectual development in high school and likely had a lot to do with developing my interest in logic.