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341On 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 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
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248Conditional assertion and restricted quantification: Abstracts of commentsNoûs 4 (1): 13. 1970.
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172Relevant predication 2: Intrinsic properties and internal relationsPhilosophical Studies 60 (3): 177-206. 1990.
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153Relevance logics and relation algebrasReview 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
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149Kripke models for linear logicJournal 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
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140Dualling: A critique of an argument of Popper and MillerBritish Journal for the Philosophy of Science 37 (2): 220-223. 1986.
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126Intuitive semantics for first-degree entailments and 'coupled trees'Philosophical Studies 29 (3): 149-168. 1976.
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125Contradictory 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
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103Canonical 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
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100Positive modal logicStudia 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.
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97A Kripke-style semantics for R-Mingle using a binary accessibility relationStudia Logica 35 (2). 1976.
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96Relevant predication 1: The formal theory (review)Journal of Philosophical Logic 16 (4): 347-381. 1987.
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95Quantum MathematicsPSA: 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
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85Partiality and its dualStudia 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
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83Negation in the Context of Gaggle TheoryStudia 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
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76A consecutive calculus for positive relevant implication with necessityJournal of Philosophical Logic 9 (4): 343-362. 1980.
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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
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71A theorem in 3-valued model theory with connections to number theory, type theory, and relevant logicStudia Logica 38 (2). 1979.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
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70Relevant predication 3: essential propertiesIn J. Dunn & A. Gupta (eds.), Truth or Consequences: Essays in Honor of Nuel Belnap, Kluwer Academic Publishers. pp. 77--95. 1990.
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69Editors' Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing (review)Journal of Philosophical Logic 42 (3): 443-459. 2013.
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69Symmetric generalized galois logicsLogica 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
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65Four-valued LogicNotre 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
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52A modification of Parry's analytic implicationNotre Dame Journal of Formal Logic 13 (2): 195-205. 1972.
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51A sieve for entailmentsJournal 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)
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51Algebraic completeness results for r-Mingle and its extensionsJournal of Symbolic Logic 35 (1): 1-13. 1970.
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Computing and Information |
Areas of Interest
Philosophy of Mind |
Philosophy of Cognitive Science |
Philosophy of Mathematics |