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Circular demonstration and von Wright-Geach entailment

Students of the literature on entailment [7], [1] are familiar with the von Wright-Geach definition of entailment [6], [2]: p entails q if, and only if, by means of logic, 1 it is possible to come to know the truth of •"/> D cp without coming to know the falsehood of p or the truth of q. A competitor of the Lewis account of entailment as strict implication, the von Wright-Geach definition has enjoyed little success. Perhaps one reason is that the definition is essentially epistemic, and entailment is not widely thought to be an epistemic notion.2 The notion of entailment aside, however, the von Wright-Geach notion does intriguingly seem to be applicable to one interesting epistemic concept of obvious logical interest. In this article,* we suggest that the von Wright-Geach definiens, in effect, partially defines the concept of a non-circular demonstration, and we argue that the von Wright-Geach definiens can be extended in a natural way to yield a full definition for non-circular demonstration. Thus we think of the latter notion in a frankly epistemic way. We have argued in [8] and [9] that circularity of argument (petitio principii) is best thought of as an epistemic matter, and for those who agree with us on this point it may not seem too surprising that there is a connection between circularity and von Wright-Geach "entailment". For those who disagree with our thesis that petitio is essentially epistemic, we hope that establishing the connection in question may serve to diminish the disagreement. In either case, we think that this new application of the von Wright-Geach framework is interesting in its own right. In [9] we argued that the history of the subject, which has largely followed in the tradition set by Aristotle in various remarks in the Topics, De Sophisticis Elenchis, and the Rhetoric? suggests the wisdom of