Analogical reasoning has been known as a noticeable form of plausible and creative reasoning since Antiquity. Still it has remained apart from logic, since its conclusions do not offer the guarantees of syllogistic and more generally deductive reasoning. Closely related to analogical reasoning is the notion of analogical proportions. They are statements of the form ‘’a is to b as c is to d’’. For about two decades now, their formalization and use have raised the interest of a number of researchers. Ten years ago, a propositional logic modeling of these proportions has been proposed. This logical view makes clear that analogy is as much a matter of dissimilarity as a matter of similarity.

Moreover, an analogical proportion is a special type of logical proportions, a family of quaternary operators built as a conjunction of two equivalences linking similarity or dissimilarity indicators pertaining to pairs (a,b) and (c,d). Homogeneous logical proportions (which include analogical proportion) and heterogenreous logical proportions are of particular interest. These remarkable proportions play a key role in the solving of various intelligence quizzes. Moreover analogical proportion-based inference has been experimentally shown to be quite good at classification tasks. Recent theoretical results suggest why.

I. The first lecture singles out analogical proportion among logical proportions. Logical proportions, a family of particular quaternary Boolean operators built from similarity or dissimilarity indicators between pairs, are first introduced. Then, different sub-families are identified according to their definitional structure, or some characteristic properties. Analogical proportion appears as one of the four symmetrical logical proportions that are code independent (which means that their truth value does not change when 0 and 1 are exchanged). Analogical proportion is uniquely characterized among these four proportions by satisfying reflexivity (‘’a is to b as a is to b’’) and the central permutation property (if ‘’a is to b as c is to d’’ then ‘’a is to c as b is to d’’). Other noticeable pro perties of analogical proportion and relations with other proportions are presented, as well as a discussion in terms of structures of opposition.

II. The second lecture is devoted to analogical proportion-based inference. Indeed analogical proportions are at the basis of an inference mechanism (which can be related to the basic analogical reasoning pattern) that enables us to complete or create a fourth item (described by means of Boolean attributes) from three other items. The good results of this inference in solving quizzes and in classification problems are then reported. The fact that this inference can never be wrong in case the classification function is an affine Boolean function is emphasized. We also discuss the differences with case-based reasoning and case-based decision.

III. The third lecture is devoted to extensions of analogical proportion beyond the Boolean case on the one hand and to the use of other logical proportions on the other hand. Multiple-valued logic extensions enable us to handle items described with numerical attributes, while the extension of analogical proportion to non distributive lattices make possible to define and identify such a proportion between concepts in a formal context, in the sense of formal concept analysis. Besides, the four non symmetrical code independent logical proportions are also worth of interest since they express that there is an intruder in a 4-tuple that is not in some definite positition in the tuple. Lastly we explain how these proportions can be used as well in classification.

References

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L. Miclet, N.Barbot, and H. Prade. From analogical proportions in lattices to proportional analogies in formal concepts. In T. Schaub, G. Friedrich, and B. O’Sullivan, editors, Proc. 21st Europ. Conf. on Artificial Intelligence (ECAI’14), Prague, Aug. 18-22, volume 263 of Frontiers in Artificial Intelligence and Applications, pages 627–632. IOS Press, 2014.

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H.Prade and G.Richard. Homogenous and heterogeneous logical proportions. IfCoLog J. 
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H.Prade and G.Richard, editors. Computational Approaches to Analogical Reasoning: Current Trends, volume 548 of Studies in Computational Intelligence. Springer, 2014.