A. Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In W. Hodges, M. Hyland, C. Steinhorn, and J. Truss, editors, Logic: Foundations to Applications. Oxford Science Publications, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....systems. An important exception is the third main formalization of fuzzy logic, namely Godel logic G# . Analytic systems for this logic also known as (Godel )Dummett logic, since Dummett [14] presented the first axiomatization matching Godel s matrix characterization can be found, e.g. in [25, 1 3, 16, 4, 5]. The interest in G# is well motivated by the fact that Research supported by EC Marie Curie fellowship HPMF CT 1999 00301. it naturally turns up in a number of di#erent contexts. Already in the 1930s Godel [18] used it in investigations of intuitionistic logic; later, Dunn and Meyer [15] ....

....as associated with the binary semantic predicate F implies G . In the context of a many valued logic with an ordered set of truth value this can, e.g. be understood as (F ) G) for all interpretations I. The concept of hypersequents (as investigated extensively by A. Avron in, e.g. [2, 3], see Section 4.1) extends the range of logics for which analytic Gentzen style systems can be given. Hypersequents are sequences of sequents understood as disjunctively connected (at the external level) If external contraction and external weakening are present and a splitting rule (which is ....

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Avron, A.: The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium. Oxford Science Publications. Clarendon Press. Oxford (1996) 1--32

....Send The messaging connective can be seen as a form of distributed implication. Instead of adding a formula to the premises of its sequent, like normal implication, it adds a formula to another sequent s premises. The use of multiple, parallel, sequents is reminiscent of Avron s hypersequents [7]. The e ects of actions can be captured by making the environment into a distinguished agent. This agent (the Environment Agent, EA) has no goals, it simply performs a cycle of receiving actions from agents, performing them, and sending percepts (based on the new state of the world) to agents. ....

A. Avron. The Method of Hypersequents in the Proof Theory of Propositional Non-Classical Logics In \Logic: Foundations to Applications" (edited by W. Hodges, M. Hyland, C. Steinhorn and J Truss), Oxford Science Publications, 1-32, 1996

....one obtains that the derivability problem for these logics is 3 In fact, Res1 is redundant in presence of Ax1, Ax2, Ax3. decidable and is at most in PSPACE. HC, HC and GLC are based on hypersequents a simple and natural generalization of Gentzen sequents to multisets of sequents (see [2] for an overview) The axioms and rules of the calculus HC are as follows: Axioms: A A A Cut Rule: G j 1 A G 0 j A; 2 B G j G 0 j 1 ; 2 B (cut) Structural Rules: G j C G j ; A C (W) G j G j j 0 0 (EW) G j j G j (EC) G j 1 ; 1 A G 0 j ....

A. Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium, pages 1-32. Oxford Science Publications. Clarendon Press. Oxford, 1996.

....characterized by Kripke models of depth k. These calculi are obtained by adding one more structural rule to the path hypertableau calculus for Intuitionistic Logic. 1 Introduction Hypersequent calculi are a simple and natural generalization of Gentzen sequent calculi to sets of sequents (see [4] for an overview) Hypersequents allow to formalize logics of a di erent nature ranging from modal to many valued logics. In this paper we are concerned with intermediate logics, that is, logics between Intuitionistic and Classical Logic. In [4, 3, 9, 8] cut free hypersequent calculi have been de ....

....of Gentzen sequent calculi to sets of sequents (see [4] for an overview) Hypersequents allow to formalize logics of a di erent nature ranging from modal to many valued logics. In this paper we are concerned with intermediate logics, that is, logics between Intuitionistic and Classical Logic. In [4, 3, 9, 8] cut free hypersequent calculi have been de ned for: 1. the logics Bw k , with k 1, which are semantically characterized by Kripke models of width k; 2. the logics Bc k , with k 1, which are semantically characterized by Kripke models of cardinality k; 3. the logics G k 1 , with k 1, ....

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A. Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium, pp 1-32, Oxford University Press, 1996.

.... sequents of relations . In RG1 all rules are local, have at most two premises, introduce at most one connective at a time and are invertible. These properties render this calculus particularly apt for (human and automated) proof search. Alternative analytic systems for G1 can be found, e.g. in [11, 1, 2, 3, 8, 4]. In particular, the axioms (basic hypersequents) introduced in [4] are closely related to the axioms of RG1 . Partly supported by the Austrian Science Fund under grant P 12652 MAT y Research supported by EC Marie Curie fellowship HPMF CT 1999 00301 Soundness and completeness of RG1 ....

....as sequences of components. However, it is easy to see that it suffices to consider sets instead of sequences (or multi sets) This allows to drop the external rules of permutation and contraction from RG1 . Sequent calculi of relations are closely related to hypersequent calculi (see, e.g. [2, 3]) We denote sequents (of relations) as A 1 1 B 1 j : j An n Bn ; where the sign i (1 i n) is either or and plays a role similar to the sequent arrow in traditional sequent calculi. A sequent is called atomic if all A i , B i are atomic formulas. The separation sign j is ....

A. Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium, pages 1--32. Oxford Science Publications. Clarendon Press. Oxford, 1996.

....hypersequent calculus for L 4 . Moreover it allows to de ne an alternative axiomatization for L 4 making no use of the Lukasiewicz axiom ( A B) B) B A) A) 2 Hypersequent Calculi Hypersequent calculi are a simple and natural generalization of ordinary Gentzen calculi, see e.g. [1, 2, 4, 3, 5]. De nition 1. A hypersequent is an expression of the form 1 1 j j n n , where for all i = 1; n; i i is an ordinary sequent. i i is called a component of the hypersequent. We say that a hypersequent is singleconclusion if for any i = 1; n, i consists of at ....

A. Avron. The Method of Hypersequents in the Proof Theory of Propositional Nonclassical Logics. In: Logic: from Foundations to Applications. W. Hodges, M. Hyland, C. Steinhorn and J. Truss Eds., European Logic Colloquium. Oxford Science Publications. Clarendon Press. Oxford. pp. 1-32. 1996.

....logics considered in this paper. In this work we will provide cut free hypersequent calculi for logics Bw k , Bc k and G k . Hypersequent calculi are a natural generalization of ordinary sequent calculi and turn out to be very suitable for expressing disjunctive axioms in an analytic way (see [5] for an overview) Indeed a common feature of the above mentioned logics is the fact that their properties can be expressed in a disjunctive form. Our calculi follow a standard pattern: all calculi belonging to the same family (respectively, Bw k , Bc k and G k ) are uniform and are simply ....

.... non empty class F of posets, L(F) is an intermediate logic (see, e.g. 8, 12] We say that an intermediate logic L is characterized by the class of posets F if L = L(F) Hypersequent Calculi Hypersequent calculi are a simple and natural generalization of ordinary Gentzen calculi, see e.g. [5] for an overview. De nition 1 A hypersequent is an expression of the form 1 1 j : j n n where, for all i = 1; n; i i is an ordinary sequent. i i is called a component of the hypersequent. We say that a hypersequent is single conclusion if, for any i = 1; n, i ....

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A. Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium, pages 1-32. Oxford Science Publications. Clarendon Press. Oxford, 1996.

....to their proper extensions C new [ fAbsg and C [ fAbsg (i.e. C ) respectively. 3 Sequent and Hypersequent Calculi In this section we define sequent (and hypersequent) calculi for I # and C # . Hypersequent calculi are a simple and natural generalization of Gentzen sequent calculi. See [2] for an overview. Definition 3.1 A hypersequent is an expression of the form 1 1 j j n n , where for all i = 1; n, i i is an ordinary sequent. The intended meaning of the symbol j is disjunctive. Hypersequent calculi are particularly useful to formalize logics containing ....

A. Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium, pages 1-- 32. Oxford Science Publications. Clarendon Press. Oxford, 1996.

....not only do some of our systems break the subformula property, but most do not possess separate rules for introducing modalities into the right and left sides of sequents. Elegant modal sequent systems respecting these ideals of Gentzen have proved elusive although the very recent work of Avron [Avr94], Cerrato [Cer93] Masini [Mas92, Mas91] and Wansing [Wan94] are attempts to redress this dearth. However, some of these methods have their own disadvantages. The systems of Cerrato enjoy the subformula property and separate introduction rules but do not enjoy cut elimination in general (although ....

....logics K and KD. The systems of Wansing enjoy cut elimination and clear introduction rules but do not immediately give decision procedures, and Tableau Methods for Modal and Temporal Logics 69 cannot handle logics like S4:3:1 and S4Dbr [Kra96] The hypersequents of Pottinger [Pot83] and Avron [Avr94] seem to retain most of the desired properties since they give cut free systems with the subformula property for most of the basic modal logics including S5. It would be interesting to see if they can be extended to handle the Diodorean or provability logics. 5 Tableau Systems For Multimodal ....

Arnon Avron. The method of hypersequents in proof theory of propositional non-classical logics. Technical Report 294-94, Institute of Computer Science, Tel Aviv University, Israel, 1994.

....Linear Logic. Polynomial cut elimination. Elementary cut elimination. Light Linear Logic. Elementary Linear Logic. 1 Introduction The rationale for extending Gentzen s format for sequents is not unidimensional. It is often a blend of several issues that inspires the design of a particular system [11, 2, 3, 16, 1]. The 2 sequent approach [12, 13, 14, 15] is not an exception. Its original goal was notational: providing symmetric and local (i.e. context free) rules for the minimal deontic logic KD. However, we discovered soon that 2 sequents could be used as a uniform tool for several logical systems. In ....

Arnon Avron. The method of hypersequents in proof theory of propositional non-classical logics. In W. Hodges, M. Hyland, C. Steinhorn, and J. Truss, editors, Logic: Foundations to Applications, pages 1--32. Oxford Science Publications, 1996.

....basis, and there is no analogue of Belnap s conditions. See [74] for a correction to the cut elimination proof in some cases. 12.5. 3 Hypersequents Hypersequents, invented independently by Pottinger [64] and Avron [4] have been used to obtain cut free formalisations of many non classical logics [5, 6]. Wansing [82] shows that, at least for some logics, display calculi have some advantages over hypersequents. Wansing gives a thorough description of hypersequents, so we omit details here. But I am not aware of a general cut elimination theorem for hypersequents. 12.5.4 Labelled Deductive ....

A Avron. The method of hypersequents in proof theory of propositional non-classical logics. Technical Report 294-94, Institute of Computer Science, Tel Aviv University, Israel, 1994.

....for hypersequents, S for sequents. 6 The claim is true, in fact also for almost all the extensions of RM . The only exceptions are classical logic and the 3 valued extension of RM . This follows from Dunn s characterizations of all these extensions in [Du70] and the proof below. 7 See [Av95] for an introduction to this method and many examples. The system GF : Axioms: A ) A External Structural rules: G GjH (EW ) GjSjSjH GjSjH (EC) GjS 1 jS 2 jH GjS 2 jS 1 jH (EP ) External Weakening, Contraction and Permutation, respectively) Internal Structural rules: Gj Gamma 1 ; A; ....

Avron A., The Method of Hypersequents in Proof Theory of Propositional NonClassical Logics, In: Logic: Foundations to Applications, Ed. by W. Hodges, M. Hyland, C. Steinhorn and J Truss, Oxford Science Publications (1996), pp. 1-32.

....designated. Not surprisingy, the use of multiple conclusion hypersequents corresponds to the use of structures which are both symmetrical and with a single nondesignated element. Keywords: Hypersequents, Consequence relations, Substructural logics, Relevance logics 1 Introduction Hypersequents ([7]) are essentially finite sets of ordinary sequents. As such they are usually taken as a generalization of Gentzen s sequents, and calculi which manipulate hypersequents are usually classified as a generalization of Gentzen type calculi (this indeed is how they were presented in [7] and previous ....

....Hypersequents ( 7] are essentially finite sets of ordinary sequents. As such they are usually taken as a generalization of Gentzen s sequents, and calculi which manipulate hypersequents are usually classified as a generalization of Gentzen type calculi (this indeed is how they were presented in [7] and previous papers) This picture is not absolutely correct, though. Gentzen s multiple conclusion sequents as introduced in [14] were themselves a generalization. They generalize his much more natural singleconclusion sequents. Now hypersequents in which all the components are singleconclusion ....

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Avron A., The Method of Hypersequents in Proof Theory of Propositional Non-Classical Logics, In: Logic: Foundations to Applications, Ed. by W. Hodges, M. Hyland, C. Steinhorn and J Truss, Oxford Science Publications (1996), pp. 1-32.

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A. Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In W. Hodges, M. Hyland, C. Steinhorn, and J. Truss, editors, Logic: Foundations to Applications. Oxford Science Publications, 1996.

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Arnon Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, editors, Logic: from foundations to applications. Proc. Logic Colloquium, Keele, UK, 1993.

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Arnon Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, editors, Logic: from foundations to applications. Proc. Logic Colloquium, Keele, UK, 1993.

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Arnon Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, editors, Logic: from foundations to applications. Proc. Logic Colloquium, Keele, UK, 1993.

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Avron, A.: The method of hypersequents in proof theory of propositional nonclassical logics. In Hodges, W., Hyland, M., Steinhorn, C., Truss, J., eds.: Logic: Foundations to Applications. Oxford Science Publications (1996) 1--32

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Arnon Avron. The method of Hypersequents in Proof Theory of Propositional Non-Classical Logics, pages 1--32. Oxford Science Publications, 1996.

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Avron, A.: The method of hypersequents in proof theory of propositional nonclassical logics. In Hodges, W., Hyland, M., Steinhorn, C., Truss, J., eds.: Logic: Foundations to Applications. Oxford Science Publications (1996) 1--32

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A. Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In W. Hodges et al. (eds.), Logic: From Foundations to Applications, pages 1--32. Oxford University Press, 1996.

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Arnon Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, editors, Logic: from foundations to applications. Proc. Logic Colloquium, Keele, UK, 1993.

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A. Avron. The Method of Hypersequents in the Proof Theory of Propositional Non-classical Logics. In W. Hodges, M. Hyland, C. Steinhorn and J. Truss, editors, Logic: From Foundations to Applications, pages 1--32. Oxford University Press, Oxford, 1996.

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Avron, A.: The Method of Hypersequents in Proof Theory of Propositional Non-Classical Logics. In Logic: From Foundations to Applications. European Logic Colloquium, Keele, UK, July 20--29,

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A. Avron. The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium, pages 1--32. Oxford Science Publications. Clarendon Press. Oxford, 1996.

No context found.

Arnon Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In W. Hodges, M. Hyland, C. Steinhorn, and J. Truss, editors, Logic: Foundations to Applications, pages 1--32. Oxford Scientific Publications, 1996.

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A Avron. The method of hypersequents in proof theory of propositional non-classical logics. In C Steinhorn J Truss W Hodges, M Hyland, editor, Logic: Foundations to Applications, pages 1-- 32. Oxford Science Publications, 1996.

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