Joe Dunn

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley–Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing a general conception of conditionality that may unify the three given conceptions.

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.

Classically, an argument A therefore B is ‘valid’ (or A is said to ‘entail’ B) if and only if (iff) each situation (model) is such that either A is false or B is true. This fits well with so-called ‘tableau’ methods for showing that A entails B by working out the mutual inconsistency of A and ~B. But both the classical notion of validity and the corresponding tableau methods allow that A may entail B because of some feature of A alone, irrespective of B, and vice versa. Thus if A is a contradiction, then each situation is such that A is false, and so a fortiori is such that A is false or B is true. And if A is a contradiction, then a tableau construction will show that A is inconsistent, and so a fortiori that A and ~B are inconsistent. Of course, the same points can be made dually when B is a logical truth.

This paper presents an overview of the pioneering contributions of Larisa Maksimova to relevance logics. She is one of the first researchers who set out to methodically study systems of relevance logics, initially, focusing on Ackermann’s $$\varPi '$$ Π ′ of “Rigorous Implication,” and then extending her work to Anderson and Belnap’s systems E of Entailment and R of Relevant Implication, and other related logics. Not only did she develop an algebraic semantics for E, but also we find that a semantic definition of entailment via a ternary accessibility relation appears — the very first time — in an abstract by her.

The challenge of giving a semantics for relevance logic in terms of worlds or situations intrigued several logicians. As a solution, Fine gave a two-sorted semantics. We overview the semantics as well as some further work of Fine in the area of relevance logic. Then we show that beyond supplying technical results such as soundness, completeness and the finite model property (fmp) for many logics, the operational–relational semantics provides footing for an informal interpretation and it naturally leads to an interpretation in terms of relevance.

We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic.

This paper combines two classes of generalized logics, one of which is the class of weakly implicative logics introduced by Cintula and the other of which is the class of gaggle logics introduced by Dunn. For this purpose we introduce implicational tonoid logics. More precisely, we first define implicational tonoid logics in general and examine their relation to weakly implicative logics. We then provide algebraic semantics for implicational tonoid logics. Finally, we consider relational semantics, called Routley–Meyer–style semantics, for finitary those logics.

Arnon Avron has written: “Dunn-McCall logic RM is by far the best understood and the most well-behaved in the family of logics developed by the school of Anderson and Belnap.” I agree. There is the famous saying: “Do not let the perfect become the enemy of the good.” I might say: “good enough.” In this spirit, I will examine the logic R-Mingle, exploring how it is only a “semi-relevant logic” but still a paraconsistent logic. I shall discuss the history of RM, and compare RM to Anderson and Belnap’s system R of relevant implication and to classical two-valued logic. There is a “consumer’s guide,” evaluating these logics as “tools,” in the light of my recent work on “Humans as Rational Toolmaking Animals.”

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity"

In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people working in the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Both I and Belnap, motivated the "Belnap-Dunn 4-valued Logic" by talk of the reasoner being simply "told true" (T) and simply "told false" (F), which leaves the options of being neither "told true" nor "told false" (N), and being both "told true" and "told false" (B). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this on a daily basis with the Web. But the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Jøsang's Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty— that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Jøsang's "Opinion Triangle" becomes an "Opinion Tetrahedron" and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical "probability" to non-classical logic.