Prade, Henri and Richard, Gilles From Analogical Proportion to Logical Proportions. (2013) Logica Universalis, 7 (4). 441505. ISSN 16618297

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Official URL: http://dx.doi.org/10.1007/s1178701300896
Abstract
Given a 4tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators ( a∧b and a¯∧b¯), or dissimilarity indicators ( a∧b¯ and a¯∧b) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which corresponds to a statement of the form “a is to b as c is to d”. The paper inventories the whole set of logical proportions by dividing it into five subfamilies according to what they express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for their negations (a¯,b¯,c¯,d¯)). It appears that only four proportions (including analogical proportion) are homogeneous in the sense that they use only one type of indicator (either similarity or dissimilarity) in their definition. Due to their specific patterns, they have a particular cognitive appeal, and as such are studied in greater details. Finally, the paper provides a discussion of the other existing works on analogical proportions.
Item Type:  Article 

Additional Information:  Thanks to Springer Verlag editor. The definitive version is available athttp://link.springer.com/article/10.1007%2Fs1178701300896 
HAL Id:  hal01138469 
Audience (journal):  International peerreviewed journal 
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Institution:  Université de Toulouse > Institut National Polytechnique de Toulouse  Toulouse INP (FRANCE) French research institutions > Centre National de la Recherche Scientifique  CNRS (FRANCE) Université de Toulouse > Université Toulouse III  Paul Sabatier  UT3 (FRANCE) Université de Toulouse > Université Toulouse  Jean Jaurès  UT2J (FRANCE) Université de Toulouse > Université Toulouse 1 Capitole  UT1 (FRANCE) 
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Deposited On:  02 Apr 2015 07:13 
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