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Reasoning with Logical Proportions

Last modified: 2010-04-27

#### Abstract

By logical proportion, we mean a statement that expresses a semantical equivalence between two pairs of propositions. In these pairs, each element is compared to the other in terms of similarities and/or dissimilarities. An example of such a proportion is the well known analogical proportion:

*a*is to*b*as*c*is to*d*. Analogical proportions have been recently characterized in logical terms, but there are many other proportions that are worth of interest. Some of them can be related to the analogical pattern, others are related to semantical equivalence between conditional objects and express statements such as a ressembles to*b*and differs from*b*in the same way as*c*with respect to*d.*We show that there are 5 direct proportions, including the analogical one and 4 others having a conditional object ﬂavor, where the change (if any) from*a*to*b*goes in the same direction as the change from*c*to*d*(if any), together with 5 reverse proportions obtained by switching*c*and*d.*Moreover, there exists only one auto-reverse proportion called paralogy and stating that what*a*and*b*have in common,*c*and*d*have it as well. It is then established that there is none other proportion than these ones (with the exception of 4 degenerated ones) that satisﬁes a natural “full identity” requirement. The paper proposes a structured and uniﬁed view of these logical proportions and discusses their characteristic properties. It extends previous works where only proportions related to analogy were considered. It also explores the use of these logical proportions in transduction-like inference, where new items are classiﬁed on the basis of already classiﬁed items without trying to induce a generic model, considering similarities and differences between items only. Taking advantage of different proportions, a transduction procedure is proposed.
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