Logical proportions - Further investigations

Prade, H; Richard, G

Logical proportions - Further investigations

Prade, H

Richard, G

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Publication Type:: Conference Proceeding
Citation:: Communications in Computer and Information Science, 2012, 297 CCIS (PART 1), pp. 208 - 218
Issue Date:: 2012-11-02

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Field	Value	Language
dc.contributor.author	Prade, H https://orcid.org/0000-0003-4586-8527	en_US
dc.contributor.author	Richard, G	en_US
dc.date.issued	2012-11-02	en_US
dc.identifier.citation	Communications in Computer and Information Science, 2012, 297 CCIS (PART 1), pp. 208 - 218	en_US
dc.identifier.isbn	9783642317088	en_US
dc.identifier.issn	1865-0929	en_US
dc.identifier.uri	http://hdl.handle.net/10453/32873
dc.description.abstract	Logical proportions may be viewed as a Boolean counterpart of the notion of numerical proportions. They relate two pairs of binary variables, say (a,b) and (c, d), by stating the conjunction of two equivalences between similarity or dissimilarity indicators pertaining to these pairs of variables. Two variables are regarded as similar if they are both true or both false, they are dissimilar if one is true while the other is false. Logical proportions include the logical expression of the analogical proportion as a particular case. Although the phrase 'logical proportion' dates back to an early proposal by Piaget sixty years ago, the general definition of logical proportions has been proposed only very recently. There are 120 distinct logical proportions that can be organized in different subfamilies according to the way similarity or dissimilarity indicators are put in relation. Besides, subsets of logical proportions satisfying noticeable requirements such as permutation properties, or code independency (stability when taking the negation of all literals) have been also identified. The paper pursues the investigation of logical proportions having remarkable properties. In particular, the proportions that are homogeneous in the sense that they involve the same indicators for evaluating the pairs (a,b) and (c, d) are proved to be the 12 symmetric proportions. Some of the results are rather easy to prove, while others may be checked through tedious enumeration procedures. For this reason a program that helps establishing results is presented. A series of remarkable and sometimes surprising results are presented, which help to characterize logical proportions, to understand their potential interest, and to grasp their cognitive appeal. © 2012 Springer-Verlag Berlin Heidelberg.	en_US
dc.relation.ispartof	Communications in Computer and Information Science	en_US
dc.relation.isbasedon	10.1007/978-3-642-31709-5_22	en_US
dc.title	Logical proportions - Further investigations	en_US
dc.type	Conference Proceeding
utslib.citation.volume	PART 1	en_US
utslib.citation.volume	297 CCIS	en_US
utslib.for	0801 Artificial Intelligence and Image Processing	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Software
pubs.organisational-group	/University of Technology Sydney/Strength - CAI - Centre for Artificial Intelligence
utslib.copyright.status	closed_access
pubs.issue	PART 1	en_US
pubs.publication-status	Published	en_US
pubs.volume	297 CCIS	en_US

Abstract:

Logical proportions may be viewed as a Boolean counterpart of the notion of numerical proportions. They relate two pairs of binary variables, say (a,b) and (c, d), by stating the conjunction of two equivalences between similarity or dissimilarity indicators pertaining to these pairs of variables. Two variables are regarded as similar if they are both true or both false, they are dissimilar if one is true while the other is false. Logical proportions include the logical expression of the analogical proportion as a particular case. Although the phrase 'logical proportion' dates back to an early proposal by Piaget sixty years ago, the general definition of logical proportions has been proposed only very recently. There are 120 distinct logical proportions that can be organized in different subfamilies according to the way similarity or dissimilarity indicators are put in relation. Besides, subsets of logical proportions satisfying noticeable requirements such as permutation properties, or code independency (stability when taking the negation of all literals) have been also identified. The paper pursues the investigation of logical proportions having remarkable properties. In particular, the proportions that are homogeneous in the sense that they involve the same indicators for evaluating the pairs (a,b) and (c, d) are proved to be the 12 symmetric proportions. Some of the results are rather easy to prove, while others may be checked through tedious enumeration procedures. For this reason a program that helps establishing results is presented. A series of remarkable and sometimes surprising results are presented, which help to characterize logical proportions, to understand their potential interest, and to grasp their cognitive appeal. © 2012 Springer-Verlag Berlin Heidelberg.

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http://hdl.handle.net/10453/32873