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Title: Entailment : the formalisation of inference
Author: Copeland, Brian John
ISNI:       0000 0001 3562 5375
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1978
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Entailment is the relation which holds between the premisses and conclusion of a. valid argument. Various standards of argument validity are prominent in the current literature: the classical, the relevant, and the intuitionistic. Correspondingly there exist various entailment relations. Classical entailment is studied in Chapters II-V, and relevant entailment in Chapters VI - IX. Entailment is treated syntactically as a sentence-predicate, and sentences asserting entailment are axiomatised hierarchically, one axiom system for each level of language. The axiomatisations used are based on Genzen's sequent calculus. o and 9 are defined in terms of entailment, and thus are treated also as sentence-predicates. Accordingly the hierarchy of entailment systems constitutes modal logic at what Quine has termed the first grade of modal involvement. It is proved that the hierarchy of systems is of SS strength. Thereby the influential conjectures of Montague (1963) are rebutted. The semantics of entailment is studied. In particular semantics giving a substitutional interpretation to the quantifiers are investigated. Such semantics answer Quine's charge that quantification into first grade modal contexts is incoherent. The problem of finding a substitutional semantics which invalidates the Barcan Formula is tackled. Finally, with a view to avoiding the standard objections to substitutional quantification whilst retaining the facility of quantifying into metalinguistic contexts, semantics which mix the substitutional and objectual interpretations are introduced. Reasons given in the literature for rejecting classical entailment theory in favour of relevance logic are surveyed and dismissed. However, the question of the relevance of premisses to conclusion is obviously a bona fide part of the whole business of appraising the worth of an argument. It behooves us as formal logicians to attempt to formalise the relevance appraisal. Given a formal theory of premiss-conclusion relevance one may, if one wishes, include the relevance appraisal alongside the truth-preservation appraisal in the assessment of formal validity. This procedure generates a relevant entailment relation. Anderson and Belnap's system E of entailment is investigated and dismissed as being merely ad hoc, answering to no account of premiss-conclusion relevance. A formal semantical analysis of premiss-conclusion relevance is given. This analysis is used in the construction of a semantical characterisation of a system of relevant entailment. A Genzen-style axiomatisation of the system is then given.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Entailment (Logic)