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From Analogical Proportion to Logical Proportions: A Survey

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Computational Approaches to Analogical Reasoning: Current Trends

Part of the book series: Studies in Computational Intelligence ((SCI,volume 548))

Abstract

Analogies play an important role in many reasoning tasks. This chapter surveys a series of recent works developing a logical view of the notion of analogical proportion, and its applications. Analogical proportions are statements of the form “\(A\) is to \(B\) as \(C\) is to \(D\)”. The logical representation used for encoding such proportions takes both into account what the four situations \(A, B, C, D\) have in common and how they differ. Thanks to the use of a Boolean modeling extended with suitable fuzzy logic connectives, the approach can deal with situations described by features that may be binary or multiple-valued. It is shown that an analogical proportion is a particular case of a more general concept, namely the one of logical proportion. Among the 120 existing logical proportions, we single out two groups of 4 proportions for their remarkable properties: the homogeneous proportions (including the analogical proportion) which are symmetrical, and the heterogeneous proportions which are not. These eight proportions are the only logical proportions to satisfy a remarkable code-independency property. We emphasize the interest of these two groups of proportions for dealing with a variety of reasoning tasks, ranging from the solving of IQ tests, to transductive reasoning for classification, to interpolative and extrapolative reasoning, and also to the handling of quizzes of the “find the odd one out” type. The approach does not just rely on the exploitation of similarities between pairs of cases (as in case-based reasoning), but rather takes advantage of the parallel made between a situation to be evaluated or to be completed, with triples of other situations.

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Notes

  1. 1.

    The overline denotes Boolean negation.

  2. 2.

    The measure of analogical dissimilarity introduced in [37] is 0 for the valuations corresponding to the characteristic patterns of \(A\), \(P\), and \(I\), maximal for the valuations corresponding to the characteristic patterns of \(R\), and takes the same intermediary value for the 8 valuations characterized by one of the patterns \(xxxy\), \(xxyx\), \(xyxx\), or \(yxxx\).

  3. 3.

    Similarly, the clausal form for paralogy is: \(\{\lnot a \vee c \vee d, \lnot a \vee \lnot b \vee d, a \vee b \vee \lnot c, b \vee \lnot c \vee \lnot d,\) \(a \vee \lnot c \vee \lnot d, a \vee b \vee \lnot d, \lnot a \vee \lnot b \vee c, \lnot b \vee c \vee d \}.\) More generally, analogy, paralogy, reverse analogy, inverse paralogy are each described by a set of 8 clauses which cannot be further reduced by resolution, and these 4 sets do not share any clause.

  4. 4.

    For copyright reasons and to protect the security of the tests, the original Raven test is replaced by specifically designed examples (still isomorphic in terms of logical encoding to the original ones).

  5. 5.

    \((u_1, \dots , u_i, \dots , u_n) >_{lexicographic} (v_1, \dots , v_i, \dots , v_n)\), once the components of each vector have been decreasingly ordered, iff \(\exists j <n \ \forall i =1,j \ u_i=v_i\) and \(u_{j+1}>v_{j+1}\).

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    Henri Prade & Gilles Richard

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    Henri Prade

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    Gilles Richard

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Prade, H., Richard, G. (2014). From Analogical Proportion to Logical Proportions: A Survey. In: Prade, H., Richard, G. (eds) Computational Approaches to Analogical Reasoning: Current Trends. Studies in Computational Intelligence, vol 548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54516-0_9

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