•  170
    Relevant predication 1: The formal theory (review)
    Journal of Philosophical Logic 16 (4): 347-381. 1987.
  •  160
    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 (t…Read more
  •  121
  • Dual combinators bite the dust
    with R. K. Meyer and K. Bimbó
    Bulletin of Symbolic Logic 4 463-464. 1998.
  •  60
    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 less-familiar and…Read more
  •  84
    Quantification and RM
    Studia Logica 35 (3): 315-322. 1976.
  •  69
    Drange's paradox lost
    Philosophical Studies 18 (6): 94-95. 1967.
  •  125
  •  92
  •  363
    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
  •  105
    Relevant Robinson's arithmetic
    Studia Logica 38 (4): 407-418. 1979.
    In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g.,…Read more
  •  141
    Negation in the Context of Gaggle Theory
    Studia Logica 80 (2): 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
  •  56
    A truth value semantics for modal logic
    Journal of Symbolic Logic 42 (2): 87--100. 1973.
  •  91
    Algebraic Completeness Results for Dummett's LC and Its Extensions
    with Robert K. Meyer
    Mathematical Logic Quarterly 17 (1): 225-230. 1971.
  •  235
    E, r, and γ
    with Robert K. Meyer
    Journal of Symbolic Logic 34 (3): 460-474. 1969.
  •  100
    New Consecution Calculi for R→t
    Notre Dame Journal of Formal Logic 53 (4): 491-509. 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 Gentzen-style 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 $\mat…Read more
  •  156
    Quantum Mathematics
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980 512-531. 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
  •  51
    Generalized Ortho Negation
    In Heinrich Wansing (ed.), Negation, De Gruyter. pp. 3-26. 1996.
  •  114
    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
  •  236
    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
  •  35
    Entailment: The Logic of Relevance and Necessity, Vol. II
    with Alan Ross Anderson and Nuel D. Belnap
    Princeton University Press. 1992.
  •  286
    Star and perp: Two treatments of negation
    Philosophical Perspectives 7 331-357. 1993.
  •  175
    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