
287On the Ternary Relation and ConditionalityJournal of Philosophical Logic 41 (3). 2012.One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the RoutleyMeyer 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 o…Read more

156Kripke models for linear logicJournal of Symbolic Logic 58 (2): 514545. 1993.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 operat…Read more

155Relevance logics and relation algebrasReview of Symbolic Logic 2 (1): 102131. 2009.Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the firstdegree fragment of relevance logics. These results show that some core ideas are sha…Read more

127Contradictory Information: Too Much of a Good Thing (review)Journal of Philosophical Logic 39 (4). 2010.Both I and Belnap, motivated the "BelnapDunn 4valued 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 …Read more

105Canonical Extensions and Relational Completeness of Some Substructural LogicsJournal of Symbolic Logic 70 (3). 2005.In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion

87Partiality and its dualStudia Logica 66 (1): 540. 2000.This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the fourvalued 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 threevalued logic, Kleene's threevalued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the firstdegree fragment of the Dun…Read more

72Algebraic Methods in Philosophical LogicOxford University Press. 2001.This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects

71Editors' Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing (review)Journal of Philosophical Logic 42 (3): 443459. 2013.

70Symmetric generalized galois logicsLogica Universalis 3 (1): 125152. 2009.Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topologic…Read more

66Fourvalued LogicNotre Dame Journal of Formal Logic 42 (3): 171192. 2001.Fourvalued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of t…Read more

59Negation in the Context of Gaggle TheoryStudia Logica 80 (2): 235264. 2005.We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn 's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn 's original one. Ku is the minimal logic that has a characteristic semantics. We also show that…Read more

51New Consecution Calculi for R→tNotre Dame Journal of Formal Logic 53 (4): 491509. 2012.The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzenstyle calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\m…Read more

37On the decidability of implicational ticket entailmentJournal of Symbolic Logic 78 (1): 214236. 2013.The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implic…Read more

32Extracting BB′IW Inhabitants of Simple Types From Proofs in the Sequent Calculus $${LT_\to^{t}}$$ L T → t for Implicational Ticket EntailmentLogica Universalis 8 (2): 141164. 2014.The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by combinators over the base { B, B′, I, W } is decidable too. Typeassignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a combinator for a theorem of ${T_\to}$. Here we describe an algorithm to extract an inhabitant from a s…Read more

27RMingle is Nice, and so is Arnon AvronIn Ofer Arieli & Anna Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of NonClassical Logics, Springer Verlag. pp. 141165. 2021.Arnon Avron has written: “DunnMcCall logic RM is by far the best understood and the most wellbehaved 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 RMingle, exploring how it is only a “semirelevant logic” but still a paraconsistent logic. I shall discuss the history of RM, and compare RM to Anderson and Beln…Read more

26Implicational Tonoid Logics: Algebraic and Relational SemanticsLogica Universalis 15 (4): 435456. 2021.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 semanti…Read more

24Contradictory Information: Better Than Nothing? The Paradox of the Two FirefightersIn Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer Verlag. pp. 231247. 2019.Prominent philosophers 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. Whi…Read more

24Chapter VIII. Ackermann's strenge implikationIn J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 129141. 2017.

23Algebraic Methods in Philosophical LogicOxford University Press UK. 2001.This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

23St. Alasdair on Lattices EverywhereIn Ivo Düntsch & Edwin Mares (eds.), Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, Springer Verlag. pp. 323346. 2021.Urquhart works in several areas of logic where he has proved important results. Our paper outlines his topological lattice representation and attempts to relate it to other lattice representations. We show that there are different ways to generalize Priestley’s representation of distributive lattices—Urquhart’s being one of them, which tries to keep prime filters in the representation. Along the way, we also mention how semilattices and lattices figured into Urquhart’s work.

22Generalized Galois Logics: Relational Semantics of Nonclassical Logical CalculiCenter for the Study of Language and Inf. 2008.Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat lessfamiliar and…Read more

21Chapter X. proof theory and decidabilityIn J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 267391. 2017.

20FrontmatterIn J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. 2017.

19Two extensions of the structurally free logic LCLogic Journal of the IGPL 6 (3): 403424. 1998.The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system , but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a RoutleyMeyer style semantics for the distributive extension, including some new definitions an…Read more

17Implicational Partial Galois Logics: Relational SemanticsLogica Universalis 15 (4): 457476. 2021.Implicational tonoid logics and their relational semantics have been introduced by Yang and Dunn. This paper extends this investigation to implicational partial Galois logics. For this, we first define some implicational partial gaggle logics as special kinds of implicational tonoid logics called “implicational partial Galois logics.” Next, we provide Routley–Meyerstyle relational semantics for finitary those logics.

17Canonical extensions and relational completeness of some substructural logicsJournal of Symbolic Logic 70 (3): 713740. 2005.In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.