Logic of approximate entailment in quasimetric spaces

The Logic of Approximate Entailment (LAE) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds with a similarity relation. When using this logic in applications, however, a disadvantage must be accepted; namely, in LAE it is not possible to combine conclusions in a conjunctive way. In order to overcome this drawback, we propose in this paper a modification of LAE where, at the semantic level, the underlying set of worlds is moreover endowed with an order structure. The chosen framework is designed in view of possible applications. *

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“…The original account is, however, due to [22]; see also [13]. A further, recent approach to the axiomatisation of LAE is contained in [25].…”

Section: The Logic Of Approximate Entailmentmentioning

confidence: 99%

Logics for Approximate Entailment in ordered universes of discourse

Vetterlein

Esteva

Godo

2016

International Journal of Approximate Reasoning

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“…In Rodríguez ( 2002 ), for instance, the language is assumed to contain a fixed finite set of variables. This restriction was discarded in Vetterlein ( 2015 ); however, the assumption of symmetry of the distance function was dropped in turn.…”

Section: The Logic Of Approximate Entailmentmentioning

confidence: 99%

“…In our previous paper Vetterlein ( 2015 ), we have dealt with a closely related setting. Namely, we have considered a modification of , the logic , which is based on quasimetric spaces instead of metric spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Namely, we have considered a modification of , the logic , which is based on quasimetric spaces instead of metric spaces. We have shown that a certain calculus, denoted by and consisting of only six rules (Vetterlein 2015 ), is sound and complete for . The question has been open if this result could be useful also for the logic .…”

Section: Introductionmentioning

confidence: 99%

“…We apply this result in Sect. 4 to the Logic of Approximate Entailment , showing that the calculus from Vetterlein ( 2015 ) is sound and complete for . An outlook on possible further work is contained in the concluding Sect.…”

Section: Introductionmentioning

confidence: 99%

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Logic of approximate entailment in quasimetric and in metric spaces

Vetterlein

2016

Soft Comput

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It is known that a quasimetric space can be represented by means of a metric space; the points of the former space become closed subsets of the latter one, and the role of the quasimetric is assumed by the Hausdorff quasidistance. In this paper, we show that, in a slightly more special context, a sharpened version of this representation theorem holds. Namely, we assume a quasimetric to fulfil separability in the original sense due to Wilson. Then any quasimetric space can be represented by means of a metric space such that distinct points are assigned disjoint closed subsets.This result is tailored to the solution of an open problem from the area of approximate reasoning. Following the lines of E. Ruspini’s work, the Logic of Approximate Entailment () is based on a graded version of the classical entailment relation. We present a proof calculus for and show its completeness with regard to finite theories.

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