Abstract
We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on many-valued logic in that it is a multi-signature one, in the spirit of institution theory. We consider graded consequence at three different conceptual levels: entailment, semantic, and closure operators, and explore several interpretations between them. We also study logical connectors and quantifiers both at the entailment and semantic level, compactness and soundness properties.
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What is called ‘external’ here and in Mossakowski et al. (2005) corresponds to what is called ‘internal’ in Diaconescu (2003, 2008). ‘Internal’ suggests that the properties characterising the connectives/quantifications are defined as properties of the respective entailment system, while ‘external’ suggests the involvement of the models as an entity which is outside to the entailment system. One may be tempted to use ‘semantic’ instead of ‘external’ and ‘syntactic’ of ‘proof theoretic’ instead of ‘internal’, but this would not be a good choice of terminology because the ‘syntactic’ stuff for the semantic entailment would have a semantic essence.
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Acknowledgments
This work has been supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0439. Thanks to Ionuţ Ţuţu for enlightening discussions around the temporal logic example.
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Communicated by L. Spada.
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Diaconescu, R. Graded consequence: an institution theoretic study. Soft Comput 18, 1247–1267 (2014). https://doi.org/10.1007/s00500-014-1231-y
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DOI: https://doi.org/10.1007/s00500-014-1231-y