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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
A Generalization of Analytic Deduction via Labelled Deductive Systems I: Basic Substructural Logics
 Journal of Automated Reasoning
, 1995
"... In this series of papers we set out to generalize the notion of classical analytic deduction (i.e. deduction via elimination rules) by combining the methodology of Labelled Deductive Systems [Gab94] with the classical system KE [DM94]. LDS is a unifying framework for the study of logics and of their ..."
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Cited by 52 (8 self)
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In this series of papers we set out to generalize the notion of classical analytic deduction (i.e. deduction via elimination rules) by combining the methodology of Labelled Deductive Systems [Gab94] with the classical system KE [DM94]. LDS is a unifying framework for the study of logics and of their interactions. In the LDS approach the basic units of logical derivation are not just formulae but labelled formulae, where the labels belong to a given "labelling algebra". The derivation rules act on the labels as well as on the formulae, according to certain fixed rules of propagation. By virtue of the extra power of the labelling algebras, standard (classical or intuitionistic) proof systems can be extended to cover a much wider territory without modifying their structure. The system KE is a new tree method for classical analytic deduction based on "analytic cut". It is a refutation system, like analytic tableaux and resolution, but it is essentially more efficient than tableaux and, un...
Substructural Logics on Display
, 1998
"... Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek ca ..."
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Cited by 50 (16 self)
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Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponentialfree linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic BiLambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bilinear, Birelevant, BiBCK and Biintuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and som...
Truthmakers, entailment and necessity
 Australasian Journal of Philosophy
, 1996
"... Australian Realists are fond of talking about truthmakers. Here are three examples from the recent literature •.. suppose a is F... What is needed is something in the world which ensures that a is F, some truthmaker or ontological ground for a's being F. What can this be except the state of af ..."
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Cited by 36 (7 self)
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Australian Realists are fond of talking about truthmakers. Here are three examples from the recent literature •.. suppose a is F... What is needed is something in the world which ensures that a is F, some truthmaker or ontological ground for a's being F. What can this be except the state of affairs of a's being F? [3, p. 190] If • entails lI, what makes ~ true also makes II true (at least when • and I] are contingent). [8, p. 32] The hallowed path from language to universals has been by way of the correspondence theory of truth: the doctrine that whenever something is true, there must be something in the world which makes it true. I will call this the Truthmaker axiom. The desire to find an adequate truthmaker for every truth has been one of the sustaining forces behind traditional theories of universals... Correspondence theories of truth breed legions of recalcitrant philosophical problems • For this reason I have sometimes tried to stop believing in the Truthmaker axiom. Yet, I have never really succeeded. Without some such axiom, I find I have no adequate anchor to hold me
Let's Plan It Deductively
 Artificial Intelligence
, 1997
"... The paper describes a transition logic, TL, and a deductive formalism for it. It shows how various important aspects (such as ramification, qualification, specificity, simultaneity, indeterminism etc.) involved in planning (or in reasoning about action and causality for that matter) can be modell ..."
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The paper describes a transition logic, TL, and a deductive formalism for it. It shows how various important aspects (such as ramification, qualification, specificity, simultaneity, indeterminism etc.) involved in planning (or in reasoning about action and causality for that matter) can be modelled in TL in a rather natural way. (The deductive formalism for) TL extends the linear connection method proposed earlier by the author by embedding the latter into classical logic, so that classical and resourcesensitive reasoning coexist within TL. The attraction of a logical and deductive approach to planning is emphasized and the state of automated deduction briefly described. 1 Introduction Artificial Intelligence (AI, or Intellectics [Bib92a]) aims at creating artificial (or computational [PMG98]) intelligence. Were there no natural intelligence, the sentence would be meaningless to us. Hence understanding natural intelligence by necessity has always been among the goals of Intel...
Negation In Relevant Logics (How I stopped worrying and learned to love the Routley Star)
 BULLETIN OF THE SECTION OF LOGIC
, 1999
"... Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a `point shift' operator . Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference fr ..."
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Negation raises three thorny problems for anyone seeking to interpret relevant logics. The frame semantics for negation in relevant logics involves a `point shift' operator . Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ¸A B to B to be invalid, because the corresponding relevant conditional A (¸ A B) ! B is not a theorem. Yet we often make the inference from A and ¸A B to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x j= \GammaA if and only if x 6j= A. Problem number three involves deciding which is the `real' negation. How can we decide between orthodox negation and the new, `Boolean' negation. In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow u...
On the Insufficiency of Ontologies: Problems in Knowledge Sharing and Alternative Solutions
"... One of the benefits of formally represented knowledge lies in its potential to be shared. Ontologies have been proposed as the ultimate solution to problems in knowledge sharing. However even when an agreed correspondence between ontologies is reached that is not the end of the problems in knowledge ..."
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Cited by 21 (2 self)
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One of the benefits of formally represented knowledge lies in its potential to be shared. Ontologies have been proposed as the ultimate solution to problems in knowledge sharing. However even when an agreed correspondence between ontologies is reached that is not the end of the problems in knowledge sharing. In this paper we explore a number of realistic knowledgesharing situations and their related problems for which ontologies fall short in providing a solution. For each situation we propose and analyse alternative solutions.
Displaying and Deciding Substructural Logics 1  Logics with Contraposition
 Journal of Philosophical Logic
, 1994
"... Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic (Belnap 1982). However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which mor ..."
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Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic (Belnap 1982). However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to provide decidability proofs for a large range of substructural logics. Displaying and Deciding Substructural Logics 1 Logics with Contraposition Greg Restall Greg.Restall@anu.edu.au There is rather a lot of interest these days in what have come to be called `substructural logics.' The term picks out logics in which the standard complement of structural rules (in, say in a Gentzen proof theory or a natural deduction system) are not all present. While much of this interest is rather recent  arising since Girard's landmark "Linear Logic" (Girard 1987), some of it has quite a history; for example the last 35 years hav...
Multiple conclusions
 In 12th International Congress on Logic, Methodology and Philosophy of Science
, 2005
"... Abstract: I argue for the following four theses. (1) Denial is not to be analysed as the assertion of a negation. (2) Given the concepts of assertion and denial, we have the resources to analyse logical consequence as relating arguments with multiple premises and multiple conclusions. Gentzen’s mult ..."
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Abstract: I argue for the following four theses. (1) Denial is not to be analysed as the assertion of a negation. (2) Given the concepts of assertion and denial, we have the resources to analyse logical consequence as relating arguments with multiple premises and multiple conclusions. Gentzen’s multiple conclusion calculus can be understood in a straightforward, motivated, nonquestionbegging way. (3) If a broadly antirealist or inferentialist justification of a logical system works, it works just as well for classical logic as it does for intuitionistic logic. The special case for an antirealist justification of intuitionistic logic over and above a justification of classical logic relies on an unjustified assumption about the shape of proofs. Finally, (4) this picture of logical consequence provides a relatively neutral shared vocabulary which can help us understand and adjudicate debates between proponents of classical and nonclassical logics. Our topic is the notion of logical consequence: the link between premises and conclusions, the glue that holds together deductively valid argument. How can we understand this relation between premises and conclusions? It seems that any account begs questions. Painting with very broad brushtrokes, we can sketch the landscape