•  119
    Relevance logics and relation algebras
    with Katalin Bimbó and Roger D. Maddux
    Review of Symbolic Logic 2 (1): 102-131. 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 first-degree fragment of relevance logics. These results show that some core ideas are sha…Read more
  •  102
    The substitution interpretation of the quantifiers
    with Nuel D. Belnap
    Noûs 2 (2): 177-185. 1968.
  •  98
    Contradictory Information: Too Much of a Good Thing (review)
    Journal of Philosophical Logic 39 (4). 2010.
    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 …Read more
  •  94
    Curry's Paradox
    with Robert K. Meyer and Richard Routley
    Analysis 39 (3). 1979.
  •  93
    Dualling: A critique of an argument of Popper and Miller
    British Journal for the Philosophy of Science 37 (2): 220-223. 1986.
  •  82
    Star and perp: Two treatments of negation
    Philosophical Perspectives 7 331-357. 1993.
  •  68
    Quantum Logic as Motivated by Quantum Computing
    with Tobias J. Hagge, Lawrence S. Moss, and Zhenghan Wang
    Journal of Symbolic Logic 70 (2). 2005.
  •  67
    Kripke models for linear logic
    with Gerard Allwein
    Journal of Symbolic Logic 58 (2): 514-545. 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
  •  66
    Quantum Mathematics
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980. 1980.
    This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Thi…Read more
  •  65
    Relevant predication 1: The formal theory (review)
    Journal of Philosophical Logic 16 (4). 1987.
  •  60
    Partiality and its dual
    Studia Logica 66 (1): 5-40. 2000.
    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 Dun…Read more
  •  55
    Canonical Extensions and Relational Completeness of Some Substructural Logics
    with Mai Gehrke and Alessandra Palmigiano
    Journal 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
  •  52
    Positive modal logic
    Studia Logica 55 (2). 1995.
    We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.
  •  49
    Negation in the Context of Gaggle Theory
    with Chunlai Zhou
    Studia Logica 80 (2-3): 235-264. 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
  •  45
  •  44
    Algebraic Methods in Philosophical Logic
    Oxford 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
  •  42
    Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic…Read more
  •  41
  •  39
    Symmetric generalized galois logics
    Logica Universalis 3 (1): 125-152. 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
  •  39
    A sieve for entailments
    Journal of Philosophical Logic 9 (1). 1980.
    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)
  •  32
    Editors' Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing (review)
    with Lawrence S. Moss and Zhenghan Wang
    Journal of Philosophical Logic 42 (3): 443-459. 2013.
  •  31
    A modification of Parry's analytic implication
    Notre Dame Journal of Formal Logic 13 (2): 195-205. 1972.
  •  30
    Completeness of relevant quantification theories
    with Robert K. Meyer and Hugues Leblanc
    Notre Dame Journal of Formal Logic 15 (1): 97-121. 1974.
  •  28
    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
  •  27
    Relevant predication 3: essential properties
    In J. Dunn & A. Gupta (eds.), Truth or Consequences, Kluwer Academic Publishers. pp. 77--95. 1990.
  •  26
    E, R, and $gamma$
    with Robert K. Meyer
    Journal of Symbolic Logic 34 (3): 460-474. 1969.