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Syllogistic logic with complements. (English) Zbl 1317.03011
van Benthem, Johan (ed.) et al., Games, norms and reasons. Logic at the crossroads. Dordrecht: Springer (ISBN 978-94-007-0713-9/hbk; 978-94-007-0920-1/set; 978-94-007-0714-6/ebook). Synthese Library 353, 179-197 (2011).
From the text: This paper presents a logic for statements of the form all \(X\) are \(Y\) and some \(X\) are \(Y\), where the \(X\) and \(Y\) are intended as (plural) nouns or other expressions whose natural denotation is as subsets of an underlying universe. Languages like this have been studied previously, and the novelty here is to add an explicit complement operator to the syntax. So we now can say, for example, all \(X^\prime\) are \(Y\), or some non-\(X\) are \(Y\). The point of the paper is to present a sound and complete proof system for the associated entailment relation. In its details, the work is rather different from previous work in the area. Our particular system is new as far as we know (but see just below). In addition, the work here builds models using a representation theorem coming from quantum logic.
For the entire collection see [Zbl 1260.03003].

03A05 Philosophical and critical aspects of logic and foundations
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