•  27
    Nuel Belnap: Doctoral students
    with Carlos Giannoni, Robert Meyer, Peter Woodruff, James Garson, Kent Wilson, Dorothy Grover, Ruth Manor, Alasdair Urquhart, and Garrel Pottinger
    In J. Dunn & A. Gupta (eds.), Truth or Consequences: Essays in Honor of Nuel Belnap, Kluwer Academic Publishers. 1990.
  •  13
    Canonical extensions and relational completeness of some substructural logics
    with Mai Gehrke and Alessandra Palmigiano
    Journal of Symbolic Logic 70 (3): 713-740. 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.
  •  4
    Fine’s Semantics for Relevance Logic and Its Relevance
    In Federico L. G. Faroldi & Frederik Van De Putte (eds.), Kit Fine on Truthmakers, Relevance, and Non-classical Logic, Springer Verlag. pp. 125-149. 2023.
    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 naturall…Read more
  •  40
    Combinators and structurally free logic
    with R. Meyer
    Logic Journal of the IGPL 5 (4): 505-537. 1997.
    A 'Kripke-style' semantics is given for combinatory logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for relevance logic. We prove by algebraic means a completeness theorem for combinatory logic, by proving a representation theorem for 'combinatory posets.' A philosophical interpretation is given of the models, showing that an element of a combinatory poset can be understood simultaneously as a set of states and as a set of actions on states. This …Read more
  •  20
    Algebraic Methods in Philosophical Logic
    Oxford 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.
  •  1
    Modern Uses of Multiple-Valued Logic (edited book)
    with G. Epstein
    D. Reidel. 1977.
  •  19
    Truth-Value Semantics
    Journal of Symbolic Logic 43 (2): 376-377. 1978.
  •  39
    Truth or Consequences: Essays in Honor of Nuel Belnap (edited book)
    with A. Gupta
    Kluwer Academic Publishers. 1990.
    This collection of essays was compiled for the occasion of Nuel Belnap's 60th birthday.
  •  15
    Is Existence a (Relevant) Predicate?
    Philosophical Topics 24 (1): 1-34. 1996.
  •  18
    Contradictory Information: Better Than Nothing? The Paradox of the Two Firefighters
    with Nicholas M. Kiefer
    In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer Verlag. pp. 231-247. 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
  •  28
    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. Type-assignment 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
  •  320
    On the Ternary Relation and Conditionality
    with Jc Beall, Ross T. Brady, A. P. Hazen, Edwin D. Mares, Robert K. Meyer, Graham Priest, Greg Restall, David Ripley, John Slaney, and Richard Sylvan
    Journal of Philosophical Logic 41 (3). 2012.
    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 o…Read more
  •  13
    Truth or Consequences: Essays in Honor of Nuel Belnap
    with L. R. S. and A. Gupta
    Philosophical Quarterly 43 (172): 399. 1993.
  •  6
    R-Mingle is Nice, and so is Arnon Avron
    In Ofer Arieli & Anna Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of Non-Classical Logics, Springer Verlag. pp. 141-165. 2021.
    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 Beln…Read more
  •  3
    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 semi-lattices and lattices figured into Urquhart’s work.
  •  3
    This is a collection of invited papers from the 1975 International Sym posium on Multiple-valued Logic. Also included is an extensive bib liography of works in the field of multiple-valued logic prior to 1975 - this supplements and extends an earlier bibliography of works prior to 1965, by Nicholas Rescher in his book Many-Valued Logic, McGraw-Hill, 1969. There are a number of possible reasons for interest in the present volume. First, the range of various uses covered in this collection of pape…Read more
  •  6
    Entailment: The Logic of Relevance and Necessity Vol. 2 (edited book)
    with Alan Ross Anderson and Nuel D. Belnap
    Princeton University Press. 1992.
  •  14
    Implicational Partial Galois Logics: Relational Semantics
    Logica Universalis 15 (4): 457-476. 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–Meyer-style relational semantics for finitary those logics.
  •  19
    Implicational Tonoid Logics: Algebraic and Relational Semantics
    Logica Universalis 15 (4): 435-456. 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
  •  20
    A ternary relation is often used nowadays to interpret an implication connective of a logic, a practice that became dominant in the semantics of relevance logics. This paper examines two early manuscripts --- one by Routley, another by Meyer --- in which they were developing set-theoretic semantics for various relevance logics. A standard presentation of a ternary relational semantics for, let us say, the logic of relevant implication R is quite illuminating, yet the invention of this semantics …Read more
  •  45
    Four-valued Logic
    Notre Dame Journal of Formal Logic 42 (3): 171-192. 2001.
    Four-valued 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
  •  15
    Frontmatter
    with Nuel D. Belnap and Alan Ross Anderson
    In J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. 2017.
  •  9
    Chapter XII. Applications and discussion
    with Nuel D. Belnap and Alan Ross Anderson
    In J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 488-564. 2017.
  •  14
    Chapter XI. functions, arithmetic, and other special topics
    with Nuel D. Belnap and Alan Ross Anderson
    In J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 392-487. 2017.
  •  12
    Index of names
    with Nuel D. Belnap and Alan Ross Anderson
    In J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 711-718. 2017.
  •  3
    Chapter IX. semantics
    with Nuel D. Belnap and Alan Ross Anderson
    In J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 142-266. 2017.
  •  3
    Chapter VII. Individual quantification
    with Nuel D. Belnap and Alan Ross Anderson
    In J. Michael Dunn, Nuel D. Belnap & Alan Ross Anderson (eds.), Entailment, Vol. Ii: The Logic of Relevance and Necessity, Princeton University Press. pp. 70-128. 2017.