# Book Reviews

Book Reviews above. The first clause is always a statement of a filter condition (i.e., a condition such that if an entity satisfies the condition then so too will any entity by which it is implied). The second clause stipulates that the result of applying the logical operator is the weakest entity satisfying the filter condition. One consequence of this is the definition of an unorthodox logical operator, the dual of negation. A notion of dual implication can be defined on any implication structure. Negation relative to the dual implication is a logical operator relative to the original implication. (Inciden­ tally, the negation of an entity is the weakest entity which together with it implies all other entities.) One upshot of this is a neat characterization of classical implication structures as those in which negation and its dual are equivalent (under =>). Quantification is treated by means of extended implication structures which include predicates. Given an implication structure <S, = » we can extend this to a structure / = <E,Pr,S,^>, where E denotes a set of 'objects' and Pr a set of pre­ dicates, predicates which map infinite sequences of elements of E to elements of S. So although the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Philosophical Quarterly Oxford University Press

# Book Reviews

, Volume 46 (183) – Apr 1, 1996
5 pages

/lp/oxford-university-press/book-reviews-eF67tgEFLV

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Publisher
Oxford University Press
ISSN
0031-8094
eISSN
1467-9213
DOI
10.2307/2956396
Publisher site
See Article on Publisher Site

### Abstract

above. The first clause is always a statement of a filter condition (i.e., a condition such that if an entity satisfies the condition then so too will any entity by which it is implied). The second clause stipulates that the result of applying the logical operator is the weakest entity satisfying the filter condition. One consequence of this is the definition of an unorthodox logical operator, the dual of negation. A notion of dual implication can be defined on any implication structure. Negation relative to the dual implication is a logical operator relative to the original implication. (Inciden­ tally, the negation of an entity is the weakest entity which together with it implies all other entities.) One upshot of this is a neat characterization of classical implication structures as those in which negation and its dual are equivalent (under =>). Quantification is treated by means of extended implication structures which include predicates. Given an implication structure <S, = » we can extend this to a structure / = <E,Pr,S,^>, where E denotes a set of 'objects' and Pr a set of pre­ dicates, predicates which map infinite sequences of elements of E to elements of S. So although the

### Journal

The Philosophical QuarterlyOxford University Press

Published: Apr 1, 1996