"Anomalous Objects"

**Abstract**: Objects that appear to
violate the rules of mathematics and logic appear at the borderline of
mathematics and
other subjects, particularly physics. A well-known example of such an
anomalous object is the Dirac delta
function. In many cases, such objects have been eventually incorporated
into the standard set-theoretical
universe. The present talk discusses strategies for incorporation, and
whether we have reasons to believe that
such objects show the standard set-theoretical universe to be inadequate.

Compiled by

Eddie Yeghiayan

"Completeness of Weak Implication."

*Theoria*(1971), 37(3):274-282.

**Abstract**: "Church's weak theory of implication is provided with semantics for which completeness is proved. The basic idea is that just as intuitionistic logic may be described, following Kripke, by valuations over a relational structure, so weak implication may be described by appropriate valuations over a semilattice. This semilattice is interpreted informally as a lattice of pieces of information."Review of Robert L. Martin, ed.,

*The Paradox of the Liar*.*Dialogue*(December 1971), 10(4):823-825.- (with Nicholas Rescher.)
*Temporal Logic*. Library of Exact Philosophy, 3. Vienna & New York: Springer, 1971.

"General Theory of Implication." [Abstract]

*Journal of Symbolic Logic*(June 1972), 37(2):443."Semantics for Relevant Logics."

*Journal of Symbolic Logic*(March 1972), 37(1):159-168.

**Abstract**: "A Semantical treatment is given of certain systems of entailment, including logics akin to the systems e and r of Anderson and Belnap. The leading idea is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in entailment logics validity may be defined in terms of valuations on a semilattice-interpreted informally as the semilattice of possible pieces of information."

Text of the article.

"Free Distributive Pseudocomplemented Lattices."

*Algebra Universalis*(1973), 3:13-15."Free Heyting Algebras."

*Algebra Universalis*(1973), 3:94-97."An Interpretation of Many-Valued Logic."

*Zeitschrift für Mathematische Logik und Grundlagen der Mathematik*(1973), 19(2):111-114."A Semantical Theory of Analytic Implication."

*Journal of Philosophical Logic*(April 1973), 2(2):212-219.

**Abstract**: "The system of analytic implication of Parry is based on the idea that for a to imply B every concept occurring in B must also occur in A. Dunn (Notre Dame Journal Formal Logic 13 (1972)) proves an algebraic completeness result for an extension of Parry's system. In the present paper a Kripke-style completeness result is given for a modal extension of Dunn's system."

"Implicational Formulas in Intuitionistic Logic." [Abstract]

*Journal of Symbolic Logic*(June 1974), 39(2):390."Implicational Formulas in Intuitionistic Logic."

*Journal of Symbolic Logic*(December 1974), 39(4):661-664.

**Abstract**: "This paper gives an alternate and shorter proof of Diego's result that there are only finitely many non-equivalent formulas in N variables in positive implication logic. A characterization in terms of a poset of prime theories is provided for the corresponding algebra of formulas. The proofs use Kripke models as an analytical technique in investigating the free Algebras."

Text of the review.- "Proofs, Snakes and Ladders."
*Dialogue*(December 1974), 13(4):723-731.

**Abstract**: "The main idea in this article is to provide a kind of visualization of natural deduction proofs." Review of R.L. Goodstein's

*The Development of Mathematical Logic*.*Historia Mathematica*(May 1974), 1:212-214.

"Popper's Logical Conceptions."

*Communication & Cognition*(1975), 8(2-4):237-277.Review of Richard Montague's

*Formal Philosophy: Selected Papers*.*Canadian Journal of Philosophy*(March 1975), 4(3):573-578.

"Elementary Classes in Infinitary Logic." [Abstract]

*Journal of Symbolic Logic*(June 1976), 41(2):552.Review of Jaako Hintikka's

*Time and Necessity: Studies in Aristotle's Theory of Modality*.*Dialogue*(December 1976), 15(4):696-699.

- "A Finite Matrix whose Consequence Relation is Not Finitely
Axiomatizable."
*Reports on Mathematical Logic*(1977), 9:71-73.

**Abstract**: "A finite many-valued logic is exhibited which has the property that the consequence relation determined by its characteristic five-valued matrix is not equivalent to a consequence relation axiomatized by a finite set of basic sequents, together with the rules of weakening, substitution and cut." Review of Alan Ross Anderson and Nuel D. Belnap, Jr.'s

*Entailment, Vol 1: The Logic of Relevance and Necessity*.*Canadian Journal of Philosophy*(June 1977), 7(2):405-411.Review of Raymond Balbes and Philip Dwinger's

*Distributive Lattices*.*Journal of Symbolic Logic*(December 1977), 42(4):587-588.

Text of the review.

Review of Casimir Lewy's

*Meaning and Modality*.*Journal of Philosophy*(August 1978), 75(8):438-446.

Text of the review."A Topological Representation Theory for Lattices."

*Algebra Universalis*(1978), 8:45-58.

"Distributive Lattices with a Dual Homomorphic Operation."

*Studia Logica*(1979), 38(2):201-209.

**Abstract**: "The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail."

"Decidability and the Finite Model Property."

*Journal of Philosophical Logic*(August 1981), 10(3):367-370.

**Abstract**: "I show that there is a recursively axiomatized modal logic which has the finite model property and is undecidable. This provides a counter-example to several published claims that all such logics are decidable.""The Decision Problem for Equational Theries."

*Houston Journal of Mathematics*(1981), 7(4):587-589."Distributive Lattices with a Dual Homomorphic Operation II."

*Studia Logica*(1981), 40(4):391-404.

**Abstract**: "An Ockham lattice is defined to be a distributive lattice with "0" and "1" which is equipped with a dual homomorphic operation. In this paper we prove: (1) the lattice of all equational classes of Ockham lattices is isomorphic to a lattice of easily described first-order theories and is uncountable, (2) every such equational class is generated by its finite members. In the proof of (2) a characterization of orderings of "w" with respect to which the successor function is decreasing is given.""Projective Distributive Para-Algebras."

*Journal of the Australian Mathematical Society Series B: Applied Mathematics*(1981), 24(2):269-275.

"Equational Classes of Distributive Double P-Algebras."

*Algebra Universalis*(1982), 14(2):235-243."Intensional Languages Via Nominalization."

*Pacific Philosophical Quarterly*(April 1982), 63(2):186-192.

**Abstract**: "This is an informal discussion of problems relating to the syntactical interpretation of systems of non-classical logic. I discuss: (1) syntactical interpretations of the Lewis systems of modal logic; (2) interpretations of so.5; (3) syntactical interpretations of combinatory logic."Review of Benson Mates'

*Skeptical Essays*.*Canadian Philosophical Reviews*(February 1982), 2(1):29-31.

"Decidability and the Finite Model Property." [Abstract]

*Journal of Symbolic Logic*(June 1983), 48(2):516."Relevant Implication and Projective Geometry."

*Logique et Analyse*(September 1983), 26 [103-104]:345-357.

**Abstract**: "This is an informal exposition of recently discovered connections between the areas of the title. I show how to construct a model for the logic of relevant implication from a geometry. This construction leads to an undecidability proof for the propositional logics R and E, and many other relevant logics."Review of Hermann Dishkant's "The First Order Predicate Calculus Based on the Logic of Quantum Mechanics," G. N. Georgacarakos' "Orthomodularity and Relevance," G. N. Georgacarakos' "Equationally Definable Implication Algebras for Orthomodular Lattices," R. J. Greechie and S. P. Gudder's "Is a Quantum Logic a Logic?", Gary M. Hardegree and C. A. Hooker's "The Conditional in Abstract and Concrete Quantum Logic," Gary M. Hardegree's "Material Implication in Orthomodular (and Boolean) Lattices," J. M. Jauch, C. Piron, P. G. O. Freund, C. J. Goebel, Y. Nambu's "What is "Quantum-Logic"? Jerzy Kotas' "An Axiom System for the Modular Logic," P. Mittelstaedt's "On the Interpretation of the Lattice of Subspaces of the Hilbert Space as a Propositional Calculus," and J. Jay Zeman's "Generalized Normal Logic."

*Journal of Symbolic Logic*(March 1983), 48(1):206-208.

Text of the review.Review of Krister Segerberg's

*Classical Propositional Operators: An Exercise in the Foundations of Logic*.*Canadian Philosophical Reviews*(December 1983), 3(6):306-308.Review of M. Richard Diaz'

*Topics in the Logic of Relevance*.*History and Philosophy of Logic*(1983), 4(1):115 .

Critical Notice of Jon Barwise, and others, eds.,

*Handbook of Mathematical Logic*.*Canadian Journal of Philosophy*(December 1984), 14(4):675-682."The Undecidability of Entailment and Relevant Implication." [Abstract]

*Journal of Symbolic Logic*(March 1984), 49(1):326."The Undecidability of Entailment and Relevant Implication."

*Journal of Symbolic Logic*(December 1984), 49(4):1059-1073.

Text of the article.

**Abstract**: "I prove in this paper that the propositional logics of relevant implication (R) and entailment (E) are undecidable. I also prove a large family of other logics undecidable. The proof uses a geometrical coordinatization theorem derived from the work of Von Neumann on continuous geometry."

Review of Charles Parsons'

*Mathematics in Philosophy: Selected Essays*.*History and Philosophy of Logic*(1985), 6(2):239-241.Review of Patrick Suppes'

*Probablistic Metaphysics*.*Canadian Philosophical Reviews*(December 1985), 5(10):478-480.Review of V.B. Sehtman's "An Undecidable Superintuitionistic Propositional Calculus," and S. V. Popov's "Nondecidable Intermediate Calculus."

*Journal of Symbolic Logic*(December 1985), 50(4):1081-1083.

Text of the review.

"The Complexity of Gentzen Systems for Propositional Logic." [Abstract]

*Journal of Symbolic Logic*(June 1986), 51(2):510."Many-Valued Logic." In Dov M. Gabbay and Franz Guenthner, eds.,

*Handbook of Philosophical Logic: Volume III, Alternatives to Classical Logic*, pp. 71-116. Dordrecht: Reidel, 1986.

**Abstract**: "This is a technical and philosophical survey of many-valued logic. The classical systems of many-valued logic are discussed, as well as more recent work on topics such as fuzzy logic. The philosophical significance of such systems is discussed critically."

(with Steve Giambrone and Robert K. Meyer.) "A Contractionless Semilattice Semantics."

*Journal of Symbolic Logic*(June 1987), 52(2):526-529.

**Abstract**: "This paper presents an alternative semantics for u RW $$subscript$$+ (the "semilattice" version of the positive fragment of the relevant logic R without the contraction axiom, (a(-. a(-b)(-. a(-b.). The original semantics was obtained by dropping the idempotence condition on the underlying semilattice model structures, thus yielding commutative monoids. Here we show that the semilattice structures can be retained provided that they are restricted to distributive semilattices and an appropriate "disjointness" condition is placed on the interpretation of implication."

Text of the article.Critical Notice of Dov M. Gabay and Felix Guenther, eds.,

*Handbook of Philosophical Logic, Vol. III: Extensions of Classical Logic*.*Canadian Journal of Philosophy*(June 1987), 17(2):483-489."Hard Examples for Resolution."

*Journal of the Association for Computing Machinery*(1987), 34(1):209-219.(with Steve Giambrone.) "Proof Theories for Semilattice Logics."

*Zeitschrift für Mathematische Logik und Grundlagen der Mathematik*(1987), 33(5):433-439.Review of Hao Wang's

*Beyond Analytical Philosophy: Doing Justice to What We Know*.*Canadian Journal of Philosophy*(June 1987), 17(2):477-482.

(with Robert K. Meyer, Errol P. Martin, and Steve Giambrone.) "Further Results on Proof Theories of Semilattice Logics."

*Zeitschrift für Mathematische Logik und Grundlagen der Mathematik*(1988), 34(4):301-304.- Review of Ermanno Bencivenga, Karel Lambert and Bas C. Van Fraasen's
*Logic, Bivalence and Denotation*.*Canadian Philosophical Reviews*(April 1988), 8(4):121-123. Review of Rachel Wallace Garden's

*Modern Logic and Quantum Mechanics*.*Journal of Symbolic Logic*(June 1988), 53(2):648-649.

Text of the review.Review of Richard Routley, Robert K. Meyer, Val Plumwood, and Ross T. Brady's

*Relevant Logics and their Rivals: Part 1. The Basic Philosophical and Semantical Theory*.*Studia Logica*(June 1988), 47(2):168-171."Russell's Zigzag Path to the Ramified Theory of Types."

*Russell: The Journal of the Bertrand Russell Archives*(Summer-Winter 1988), 8(1-2):82-91.

**Abstract**: "This is a brief survey of the foundational schemes which were considered by Russell from 1902 to 1907. Among the main themes are logic as universal language, Russell's surprising reluctance to adopt type distinctions."

"The Complexity of Gentzen Systems for Propositional Logic."

*Theoretical Computer Science*(August 2, 1989), 66(1):87-97.Review of J. Michael Dunn's "Relevant Predication 1: The Formal Theory."

*Journal of Symbolic Logic*(June 1989), 54(2):615-616.

The text of the review.."Russell's Zigzag Path to the Ramified Theory of Types." In Ian Winchester and Kenneth Blackwell, eds.,

*Antinomies and Paradoxes: Studies in Russell's Early Philosophy*, pp. 82-91. Hamilton, Ontario: McMaster University Library Press, 1989."What is Relevant Implication?" In Jean Norman and Richard Sylvan, eds.,

*Directions in Relevant Logic*, pp. 167-174. Reason and Argument, 1. Dordrecht & Boston: Kluwer, 1989.

"The Complexity of Decision Problems in Relevance Logic." In J. Michael Dunn and Anil Gupta, eds.,

*Truth or Consequences: Essays in Honor of Nuel Belnap*, pp. 61-76. Dordrecht & Boston: Kluwer, 1990."Functional Interpretations of Feasibly Constructive Arithmetic." In Samuel R. Buss and Philip J. Scott, eds.,

*Feasible Mathematics: A Mathematical Sciences Institute Workshop, Ithaca, New York, June 1989*, pp. 97-98. Progress in Computer Science and Applied Logic, 9. Boston: Birkhauser, 1990."The Logic of Physical Theory." In A.D. Irvine, ed.,

*Physicalism in Mathematics*, pp. 145-154. The University of Western Ontario Series in Philosophy of Science, 45. Dordrecht & Boston: Kluwer, 1990.Review of A. G. Dragalin's "Matematiceskij Intuicionizm. Vvedenie v Teoriu Dokazatel'stv," and D. Deutsch's "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer."

*Journal of Symbolic Logic*(September 1990), 55(3):1309-1310.

Text of reviewReview of A.S. Troelstra and D. Van Dalen's

*Constructivism in Mathematics: An Introduction, Volume 1*.*Studia Logica*(March 1990), 49(1):151-152.Review of Jon Barwise's

*The Situation in Logic*.*Canadian Philosophical Reviews*(March 1990), 10(3):96-98.Review of Stephen Read's

*Relevant Logic: A Philosophical Examination of Inference*.*History and Philosophy of Logic*(1990), 11(1):98-99.

"A French Perspective on Russell." Review of

*Hermès*, No. 7: "Bertrand Russell de la logique à la politique."*Russell*(Summer 1991), 11(1):103-105.Review of A.S. Troelstra and D. Van Dalen's

*Constructivism in Mathematics, Volume 2*.*Studia Logica*(June 1991), 50(2):355-356.Review of Stuart Shapiro, ed.,

*Intensional Mathematics*.*Studia Logica*(March 1991), 50(1):160-162.

(with Stephen Bellantoni and Toniann Pitassi.) "Approximation and Small-depth Frege Proofs."

*SIAM Journal on Computing*(December 1992), 21(6):1161-1179."Complexity of Proofs in Classical Propositional Logic." In Y.N. Moschovakis, ed.,

*Logic from Computer Science: Proceedings of a Workshop held November 13-17, 1989, Berkeley, California*, pp. 597-608. Mathematical Sciences Research Institute Publications, 21. New York: Springer, 1992."The Relative Complexity of Resolution and Cut-free Gentzen Systems."

*Annals of Mathematics and Artificial Intelligence*(1992), 6(1-3):157-168.Review of Arnon Avron's "Relevance and Paraconsistency--A New Approach," "Relevance and Paraconsistency--A New Approach. Part II: The Formal Systems," "Relevance and Paraconsistency--A New Approach. Part III: Cut-Free Gentzen-Type Systems."

*Journal of Symbolic Logic*(December 1992), 57(4):1481-1482.

Text of the reviewReview of Michael Detlefsen, ed.,

*Proof and Knowledge in Mathematics*.*Canadian Philosophical Reviews*(August 1992), 12(4):237-238."#47: Semilattice Semantics for Relevance Logics. #65: The Undecidability of All Principal Relevance Logics." In Alan Ross Anderson, Nuel D. Belnap, Jr., with contributions by J. Michael Dunn, Kit Fine and Alasdair Urquhart,

*Entailment: The Logic of Relevance and Necessity*, Volume II, pp. 142-155, 348-375. Princeton, New Jersey: Princeton University Press, 1992.

"Failure of Interpolation in Relevant Logics."

*Journal of Philosophical Logic*(October 1993), 22(5):449-479.

**Abstract**: "Craig's interpolation theorem fails for the propositional logics "E" of entailment, "R" of relevant implication and "T" of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids."(with Stephen Cook.) "Functional Interpretations of Feasibly Constructive Arithmetic."

*Annals of Pure and Applied Logic*(September 10, 1993), 63(2):103-200.

**Abstract**: "This paper develops a notion of feasible function of finite type based on the typed lambda calculus. The numerical functions computed by type 1 terms are exactly the polynomial time functions. The notion is employed to provide interpretations of feasibly constructive arithmetic in the style of Buss; versions of modified realizability and Godel's Dialectica interpretation are given. These interpretations are employed to give new versions of Buss's results in intuitionistic feasible arithmetic and to prove independence results for conjectures in computational complexity theory."Review of A.S. Troelstra's

*Lectures on Linear Logic*.*Canadian Philosophical Reviews*(June 1993), 13(3):126-128.Review of Francisco A. Rodriguez-Consuegra's

*The Mathematical Philosophy of Bertrand Russell: Origins and Developement*.*Philosoophia Mathematica*[Series III] (March 1993), 1(1):90-93.Review of Hartry Field's

*Realism, Mathematics and Modality*.*History and Philosophy of Logic*(1993), 14(1):117-119."Russell's Idealist Phase." Review of Nicholas Griffin's

*Russell's Idealist Apprenticeship*.*Russell*(Summer 1993), 13(1):104-108.

Edited (with Albert C. Lewis.) Bertrand Russell's

*Foundations of Logic, 1903-05*. The Collected Papers of Bertrand Russell, 4. London & New York: Routledge, 1994.Review of Grzegorz Malinowski's

*Many-Value Logics*.*Notre Dame Journal of Formal Logic*(Summer 1994), 35(3):469-470.(with Judy Pelham.) "Russellian Propositions." In Dag Prawitz, Brian Skyrms, and Dag Westerstahl, eds.,

*Logic, Methodology, and Philosophy of science IX: Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science, Uppsala, Sweden, August 7-14, 1991*, pp. 307-326. Studies in Logic and the Foundations of Mathematics, 134. Amsterdam & New York: Elsevier, 1994.(with Russell Impagliazzo and Toniann Pitassi.) "Upper and Lower Bounds for Tree-like Cutting-plane Proofs." In IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, INRIA.

*Proceedings: Symposium on Logic in Computer Science : LICS '94, CNAM, Paris, France, July 4-7, 1994*, pp. 220-228. Los Alamitos, Calif.: IEEE Computer Society Press, 1994.(with Nicholas Rescher.) "Zeit und Zeitlogik." In Bertram Kienzle, ed.,

*Zustand und Ereignis*. Frankfurt am Main: Suhrkamp, 1994.

Translation of parts of*Temporal Logic*(1971).

"The Complexity of Propositional Proofs."

*Bulletin of Symbolic Logic*(1995), 4:425-467.(with Toniann Pitassi.) "The Complexity of the Hajós Calculus."

*SIAM Journal on Discrete Mathematics: A Publication of the Society for Industrial and Applied Mathematics*(August 1995), 8(3):464-483."Decision-problems for Distributive Lattice-ordered Semigroups."

*Algebra Universalis*(1995), 33(3):399-418."G. F. Stout and the Theory of Descriptions."

*Russell*(Winter 1994-95), 14(2):163-171.

**Abstract**: "This paper discusses the circumstances under which Russell wrote his 1905 paper "On Denoting" and its reception by Russell's contemporaries, particularly Moore and Stout. It includes a long letter from Stout to Russell written in 1903 in which Stout argues against Russell's ideas on meaning and denotation."Review of Peter Schroeder-Heister and Kosta Dosen, eds.,

*Substructural Logics*.*History and Philosophy of Logic*(1995), 16(1):138-139.

"Duality for Algebras of Relevant Logics."

*Studia Logica*(January-March 1996), 56(1-2):263-276.

**Abstract**: "This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theory piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics."

Special issue on Priestley Duality.Review of A. D. Irvine, G. A. Wedeking's

*Russell and Analytic Philosophy*.*Journal of Symbolic Logic*(December 1996), 61(4):1391-1392.

Text of review.(with Xudong Fu.) "Simplified Lower Bounds for Propositional Proofs."

*Notre Dame Journal of Formal Logic*(1996), 37(4):523-544.

**Abstract**: "This article presents a simplified proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving switching lemmas developed by Razborov and Beame. A similar result is also proved for some examples based on graphs."

"The Graph Constructions of Hajós and Ore."

*Journal of Graph Theory*(December 1997), 26(4):211-215."The Number of Lines in Frege Proofs with Substitution."

*Archive for Mathematical Logic*(December 1997), 37(1):15-19.Review of Jagdish Mehra's

*The Beat of a Different Drum: The Life and Science of Richard Feynman*.*International Studies in the Philosophy of Science*(1997), 11:311-313.Review of Peter Clote and Jeffrey Remmel, eds.,

*Feasible Mathematics II*.*Logic Journal of the IGPL*(March 1997), 5(2):301-302.

"Complexity, Computational." In Edward Craig, ed.,

*Routledge Encyclopedia of Philosophy*, Vol. 2, pp. 471-476. London & New York: Routledge, 1998."The Complexity of Propositional Proofs."

*Canadian Journal of Artificial Intelligence*(Autumn 1998), 42:8-18."The Complexity of Propositional Proofs."

*Bulletin of the European Association for Theoretical Computer Science: EATCS*(1998), 64:128-138.(with James R. Brown.) Review of Adam Morton and Stephen P. Stich, eds.,

*Benacerraf and His Critics*.*Dialogue*(Summer 1998), 37(3):633-637.

"Beth's Definability Theorem in Relevant Logics." In Ewa Orlowska, ed.,

*Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa*, pp. 229-234. Studies in Fuzziness and Soft Computing, 24. Heidelberg & New York: Physica, 1999."The Complexity of Decision Procedures in Relevance Logic." (II)

*Journal of Symbolic Logic*(December 1999), 64(4):1774-1802."From Berkeley to Bourbaki."

*Dialogue*(Summer 1999), 38(3): 587-592.

**Abstract**: "This is a critical notice of the excellent collection of articles,*From Kant to Hilbert: A Source Book in the Foundations of Mathematics*, edited by William Ewald (Oxford 1996)."Review of George Boolos'

*Logic, Logic, and Logic*.*Philosophy in Review*(August 1999), 19(4):244-246.Review of Gregory Landini's

*Russell's Hidden Substitutional Theory*.*Journal of Symbolic Logic*(September 1999), 64(3):1370-1371."The Symmetry Rule in Propositional Logic: The Satisfiability Problem."

*Discrete Applied Mathematics*(October 15, 1999), 97:177-193.

"The Complexity of Linear Logic with Weakening." In Samuel R. Buss, Petr Hajek, and Pavel Pudlak, eds.,

*Logic Colloquium '98: Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic, held in Prague, Czech Republic, August 9-15, 1998*, pp. 500-515. Lecture Notes in Logic, 13. Natick, Mass.: Association for Symbolic Logic, 2000.(with Noriko Arai.) "Local Symmetries in Propositional Logic." In Roy Dyckhoff, ed.,

*Automated Reasoning with Analytic Tableaux and Related Methods: International Conference, TABLEAUX 2000, St Andrews, Scotland, UK, July 3-7, 2000: Proceedings*, pp. 40-51. Lecture Notes in Computer Science, 1847. Lecture Notes in Artificial Intelligence. Berlin & New York: Springer, 2000.Review of Bernard Linsky's

*Russell's Metaphysical Logic*.*University of Toronto Quarterly*(Winter 2000), 70(1):455-457.Review of Michael Astroh and Stephen Read, eds.,

*Proceedings of the Conference: Hugh MacColl and the Tradition of Logic: Ernst-Moritz-Arndt-Universität Greifswald, March 20 - April 1, 1998*.*History and Philosophy of Logic*(December 2000), 21(4):308-309.

Review of Greg Restall's

*An Introduction to Substructural Logics*.*International Studies in the Philosophy of Science*(2001), 15:108-110.

"Metatheory." In Dale Jacquette, ed.,

*A Companion to Philosophical Logic*Blackwell Companions to Philosophy, 22. Malden, MA: Blackwell, 2002.

(with Noriko Arai and Toniann Pitassi.) "The Complexity of Analytic Tableaux." In

*STOC 2001 Proceedings*.Contributions to Ross Brady, ed.,

*Relevant Logics and their Rivals II*."Resolution Proofs and Matching Principles."

*Annals of Mathematics and Artificial Intelligence*."The Theory of Types." In Nicholas Griffin, ed.,

*The Cambridge Companion to Bertrand Russell*.(with Peter Apostoli.) "Weakly Additive Operators in Distributive Lattices."

*Proceedings of the SEP '99 Conference*.

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