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This book celebrates and expands on J. Michael Dunn’s work on informational interpretations of logic. Dunn, in his Ph.D. thesis (1966), introduced a semantics for first-degree entailments utilizing the idea that a sentence can provide positive or negative information about a topic, possibly supplying both or neither. He later published a related interpretation of the logic R-mingle, which turned out to be one of the first relational semantics for a relevance logic. An incompatibility relation between information states lends itself to a definition of negation and it has figured into Dunn's comprehensive investigations into representations of various negations. The informational view of semantics is also a prominent theme in Dunn’s research on other logics, such as quantum logic and linear logic, and led to the encompassing theory of generalized Galois logics (or "gaggles"). Dunn’s latest work addresses informational interpretations of the ternary accessibility relation and the very nature of information. The book opens with Dunn’s autobiography, followed by a list of his publications. It then presents a series of papers written by respected logicians working on different aspects of information-based logics. The topics covered include the logic R-mingle, which was introduced by Dunn, and its applications in mathematical reasoning as well as its importance in obtaining results for other relevance logics. There are also interpretations of the accessibility relation in the semantics of relevance and other non-classical logics using different notions of information. It also presents a collection of papers that develop semantics for various logics, including certain modal and many-valued logics. The publication of this book is well timed, since we are living in an "information age.” Providing new technical findings, intellectual history and careful expositions of intriguing ideas, it appeals to a wide audience of scholars and researchers.

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 (in the terminology of Avron) 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.”

Prominent philosophers (Bar-Hillel and Carnap, Popper, Floridi) have argued that contradictions contain either too much or too little information to be useful. We dispute this with what we call the “Paradox of the Two Firefighters.” Suppose you are awakened in your hotel room by a fire alarm. You open the door. You see three possible ways out: left, right, straight ahead. You see two firefighters. One says there is exactly one safe route and it is to your left. The other says there is exactly one safe route and it is to your right. While the two firemen are giving you contradictory information, they are also both giving you the perhaps useful information that there is a safe way out and it is not straight ahead. We give two analyses. The first uses the “Opinion Tetrahedron,” introduced by Dunn as a generalization of Audun Josang’s “Opinion Triangle.” The Opinion Tetrahedron in effect embeds the values of the “Belnap-Dunn 4-valued Logic” (Truth, Falsity, Neither, Both) into a context of subjective probability generalized to allow for degrees of belief, disbelief, and two kinds of uncertainty—that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflict). Josang had only a single value for uncertainty. We also present an alternative solution, again based on subjective probability but of a more standard type. This solution builds upon “linear opinion pooling.” Kiefer had already developed apparatus for assessing risk using expert opinion, and this influences the second solution. Finally, we discuss how these solutions might apply to “Big Data” and the World Wide Web.

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.

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. Josang’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). Josang’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.

I explore the question of whether the concept of relevance is relevant to the study of what Anderson and Belnap call “relevance logic.” The answer should be “Of course!” But there are some twists and turns, as is shown by the fact that it has taken over 50 years to get here. Despite protests by R. K. Meyer that the concept of relevance is not part of what he calls “relevant logic,” I suggest and defend interpreting the Routley–Meyer ternary accessibility relation using information states a, b, c, so Rabc means “in the context a, b is relevant to c.” Motivations are provided from Sperber and Wilson’s work in linguistics on relevance.

We introduce an abstract algebraic structure — a lattice defi ned on a generalized truth value space of constructive logic. For background one can refer to the idea of ‘under-determined’ and ‘over-determined’ valuations (Dunn), a ‘useful four-valued logic’ (Belnap), and the notion of a bilattice (Ginsberg). We consider within one general framework the notions of constructive truth and constructive falsity, as well as the notions of non-constructive truth and non-constructive falsity. All possible combinations of th e basic truth values give rise to an interesting ‘16-valued logic’. It appears that these 16 truth values constitute what we call a trilattice — a natural mathematical structure with three partial orderings that represent respectively a n increase in information, truth and constructivity. The presentation of the paper is essentially conceptual: the stress is laid on introducing new concepts and structures as well as on their general interpretation.

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), 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 Łukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) 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".

Applications philosophiques de la logique de la predication pertinente, mise au point par l'A., aux proprietes intrinseques et aux relations internes. Les proprietes intrinseques sont des predicats monadiques pertinents et les relations internes, des predicats polyadiques pertinents. L'A. compare sa theorie de la predication a celle de D. Lewis

I owe much to my thesis director Nuel Belnap, but I wonder if I can blame him for my recent idealist tendencies. The novelist Colin Wilson (1967) has a theory of ideas as infectious agents, much like viruses and needing hosts. It is interesting to speculate how the idealist infection was transmitted from Blanshard to Anderson and Belnap at Yale, and from them to me as their graduate student at Pittsburgh. Perhaps rather than being an infectious disease it is a hereditary one, linked in some complicated way with the intellectual genes for relevance logic, Platonism, and logical humor.

The validity of an entailment has nothing to do with whether or not the components are true, false, necessary, or impossible; it has to do solely with whether or not there is a necessary connection between antecedent and consequent. Hence it is a mistake (we feel) to try to build a sieve which will “strain out” entailments from the set of material or strict “implications” present in some system of truth-functions, or of truth-functions with modality. Anderson and Belnap (1962, p. 47).

Although this paper does not directly reference work done by Ed Gettier, I feel that it is an appropriate contribution to this volume in that it deals with two of the great intellectual loves of Ed’s life, higher-order logic and modal logic, and it deals with them in a ‘logic-chopping’ way that I rightly or wrongly think would please Ed.

I The basic formal ideas of this interpretation were in a certain sense anticipated by E. W. Beth with his "reduced logic" in [3]. We are grateful to Professor Bas C. van Fraassen for bringing this to our attention. The idea has also been employed by P. T. Geach in his "Quantification theory and the problem of identifying objects of reference" in Acta Philosophica Fennica, 1963, and by Henry S. Leonard in his "Essences, attributes, and predicates" in the Proceedings of the American Philisophical Association, 1963-64. Further, H. Leblanc has studied the interpretation in a pair of forthcoming papers, "A simplified strong completeness proof for QC=," and "A simplified account of validity and implication for quantification logic."

This paper highlights the importance of a strategy for semantic analysis initiated by J. Michael Dunn, known in the literature as the “American Plan.” The key insight of the plan relies on allowing under-determined and over-determined logical valuations, which prove to be essential for a logical analysis of information structures. The main directions in the development of this fundamental idea are explained, and an implementation of the possible generalization thereof is briefly reviewed, culminating in the notion of a multi-consequence logic.

This paper reformulates and decides a certain conjecture in Dunn's ‘Relevant Predication 1: The Formal Theory’ (Journal of Philosophical Logic16, 347–381, 1987). This conjecture of Dunn's relates his object-language characterisation of a property's being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in ‘Relevant Implication and Relevant Functions’ (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et al.) In the course of the investigation this paper also investigates Kit Fine's semantics for quantified relevance logics, which appears in his appropriately titled ‘Semantics for Quantified Relevance Logics’