A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

...., c range over C , and range over I . In the following we describe the requirements that such a structure must meet in order to qualify as a sequent system. 13 Constraint satisfaction must obey the basic laws for a (classical) consequence relation, i.e. reflexivity, monotonicity, and cut (cf. [1, 9]) We extend satisfaction to a relation between sets of constraints by defining ec j= ec 1 iff (8c 2 ec 1 ) ec j= c) Let us call schematic entities the entities that I acts on producing entities of the same sort. The collection of schematic entities includes sequents and constraints, and it is ....

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

....specifically j= I , and [ must meet certain requirements, which we describe below. Let s range over S , c range over C , and range over I . Recall our convention that, for example ec will range over P (C ) Satisfaction must obey the basic laws for a (classical) consequence relation (cf. [5, 53]) mon) if ec ec 0 and ec j= c, then ec 0 j= c; ax) if c 2 ec , then ec j= c; cut) if ec j= c and fcg [ ec 0 j= c 0 , then ec [ ec 0 j= c 0 . We extend satisfaction to a relation between sets of constraints by defining ec j= ec 1 , 8c 2 ec 1 ) ec j= c) Let us call ....

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

.... such that: O (D; Gamma M ) fh Gamma O ; AO i : h Gamma M [ Gamma O ; AO i 2 Dg M (D; Gamma O ) fh Gamma M ; AM i : h Gamma M [ Gamma O ; AM i 2 Dg D, O (D; Gamma M ) and M (D; Gamma O ) are not consequence relations (taking the standard notion of consequence relation, see for instance [2], example 3.2) as they don t satisfy the consequence relation closure conditions. D cannot be closed as any expression in LO[LM with subformulas in LO and LM is not a formula. O (D; Gamma M ) and M (D; Gamma O ) 26 are not closed as we want them to capture the pairs whose second element is the ....

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

....range over C , and range over I . In the remainder of this subsection we describe the requirements that such a structure must meet in order to qualify as a sequent system, and introduce some auxiliary definitions. Satisfaction must obey the basic laws for a (classical) consequence relation (cf. [2, 43]) mon) if ec ec 0 and ec j= c, then ec 0 j= c; ax) if c 2 ec , then ec j= c; cut) if ec j= c and fcg [ ec 0 j= c 0 then ec [ ec 0 j= c 0 . We extend satisfaction to a relation between sets of constraints by defining ec j= ec 1 , 8c 2 ec 1 ) ec j= c) Let us call schematic ....

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

....logic and different theories, or, even, the same logics and theories but different deductive machineries. Our interest is in the construction of systems which can be easily tailored and effectively used to represent and solve complex reasoning problems. The work on consequence relations by Avron [5] came about as part of the LF effort to answer questions about the nature of logics. The aim here is to characterize the commonalities and distinctions between inference systems. Standard logical connectives are analyzed in terms of an abstract notion of consequence relation. A variety of logics ....

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

....remainder of this section we describe the requirements that such a structure must meet in order to qualify as a sequent system. We also introduce some auxiliary definitions. Satisfaction must obey the basic laws for a (classical) consequence relation, i.e. reflexivity, monotonicity, and cut (cf. (Avron, 1987; Meseguer, 1989) We extend satisfaction to a relation between sets of constraints by defining e c j= e c 1 iff (8c 2 e c 1 ) e c j= c) We call the collection of entities that I acts on, producing entities of the same sort, schematic entities. This collection includes sequents and constraints, ....

A. Avron. Simple consequence relations. LFCS Report, Laboratory for the Foundations of Computer Science, University of Edinburgh, 1987.

...., assertions or judgments for consideration, and a set of constraints, C , that allow us to construct provisional derivations. There is a constraint solving mechanism, j= P (C ) Theta C (where P (C ) is the set of finite subsets of C ) represented abstractly as a consequence relation (cf. [2,25]) Both sequents and constraints can be schematic. A sequent system contains a set of instantiation maps, I , and an application operation, for filling in schemata, that is [ S Theta I S ] and [ C Theta I C ] Let us call schematic entities the entities that I acts on producing ....

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

....of a hypothetical judgement therefore corresponds to either a rule of derivation or a derivable rule of the system, not simply a rule of proof or an admissible rule. The consequence relations which are directly encoded by the LF s judgements are ordinary single conclusioned consequence relations [2]. A proof of a sequent Phi 1 ; Phi n Psi is encoded by a term of type J ( Phi 1 ) J ( Phi 2 ) Delta Delta Delta J ( Phi n ) J ( Psi) where J is a judgement that is induced by : Note, however, that the type structure of the LF makes it possible to formulate and prove also ....

....on its application. Prawitz gives several possible versions of this side condition. In the first one, for example, all assumptions on which Phi depends should be modal (i.e. the main connective is 2) In all versions the side condition makes this rule impure. This impurity is of the second degree [2]. Thus we lack the coherence, which the LF paradigm expects, between the formulation of the rules of a system and the consequence relation represented by it. In the Hilbert style presentation we can ignore the intended consequence relation of truth and still encode all proofs of theorems using ....

Arnon Avron. Simple Consequence Relations. Technical Report, Laboratory for the Foundations of Computer Science, Edinburgh University, 1987. ECS-LFCS87 -30.

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A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

No context found.

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

No context found.

A. Avron. Simple consequence relations. LFCS Report Series, Laboratory for the Foundations of Computer Science, Computer Science Department, University of Edinburgh, 1987.

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