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An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Carnap's Tolerance, Language Change and Logical Pluralism
, 2000
"... In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap's Principle of Tolerance from a pluralism which maintains that there are different, equally "good" logical consequence relati ..."
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In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap's Principle of Tolerance from a pluralism which maintains that there are different, equally "good" logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap's more moderate pluralism.
Boolean Conservative Extension Results for some Modal Relevant Logics
, 2010
"... Abstract: This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation. 1 Dedication This paper is dedicated to the memory of Bob Meyer. Bob was our friend and colleague. We miss him greatly. 2 ..."
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Abstract: This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation. 1 Dedication This paper is dedicated to the memory of Bob Meyer. Bob was our friend and colleague. We miss him greatly. 2
Review of Heinrich Wansing, Displaying Modal Logic, Kluwer Academic Publishers, 1998 Trends in Logic: Studia Logica Library
"... disjunction of the material on the right. Sequents feature a single kind of punctuation, here the comma, which is interpreted as conjunction on the left and disjunction on the right. Belnap saw (with others, such as Dunn [2] and Mints [4]) that to model interesting intensional logics such as the lo ..."
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disjunction of the material on the right. Sequents feature a single kind of punctuation, here the comma, which is interpreted as conjunction on the left and disjunction on the right. Belnap saw (with others, such as Dunn [2] and Mints [4]) that to model interesting intensional logics such as the logic R of relevant implication, more punctuation was necessary. In particular, it seems necessary to have an intensional form of conjunction, along with the extensional conjunction of standard Gentzen systems. The addition of more `punctuation' in a Gentzen system brings with it a corresponding complexity in proving that the Gentzen system has desirable properties, such as the admissibility of the Cut rule. If sequents can contain a combination of intensional and extensional punctuation on both the left and the right, then it seems like a cut rule will have to reason from premises: X(A)<F17.28
www.elsevier.com/locate/apal Ternary relations and relevant semantics
"... Modus ponens provides the central theme. There are laws, of the form A → C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theories L; T and U, where U consists of all of the ou ..."
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Modus ponens provides the central theme. There are laws, of the form A → C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theories L; T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product (or fusion) operation ◦ on theories (or other collections of formulas) L and T, whence RLTU i L◦T ⊆ U. These ideas have been expressed, especially with Routley, as (Kripke style) worlds semantics for relevant and other substructural logics. Worlds are best demythologized as theories, subject to truthfunctional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians ’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates. Each logic L gives rise to a Calculus of Ltheories, on which particular candidate logical
How to Universally Close the Existential Rule
"... Abstract This paper introduces a nested sequent system for predicate logic. The system features a structural universal quantifier and a universally closed existential rule. One nice consequence of this is that proofs of sentences cannot contain free variables. Another nice consequence is that the as ..."
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Abstract This paper introduces a nested sequent system for predicate logic. The system features a structural universal quantifier and a universally closed existential rule. One nice consequence of this is that proofs of sentences cannot contain free variables. Another nice consequence is that the assumption of a nonempty domain is isolated in a single inference rule. This rule can be removed or added at will, leading to a system for free logic or classical predicate logic, respectively. The system for free logic is interesting because it has no need for an existence predicate. We see syntactic cutelimination and completeness results for these two systems as well as two standard applications: Herbrand’s Theorem and interpolation.