Abstract
Given a 4-tuple 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 distinct logical proportions. One of them models analogical proportions which correspond to statements 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 5 subfamilies according to what their logical proportions 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 4-tuple (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 (¬a,¬b, ¬c, ¬d)). Finally, the paper provides a discussion of the potential interest of logical proportions, which clearly have a cognitive appeal.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bayoudh, S., Miclet, L., Delhay, A.: Learning by analogy: A classification rule for binary and nominal data. In: Proc. IJCAI ’07, pp. 678–683 (2007)
Dubois, D., Fargier, H., Prade, H.: Possibilistic likelihood relations. In: Proc. 7th IPMU’98, Paris, pp. 1196–1202 (1998)
Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. IEEE Trans. on Systems, Man and Cybernetics 24, 1724–1740 (1994)
Klein, S.: Culture, mysticism & social structure and the calculation of behavior. In: Proc. Europ. Conf. in Artificial Intelligence (ECAI), pp. 141–146 (1982)
Lepage, Y.: Analogy and formal languages. In: Proc. FG/MOL 2001, pp. 373–378 (2001) (in French), http://www.slt.atr.cos.jp/lepage/pdf/dhdryl.pdf.gz
Miclet, L., Bayoudh, S., Delhay, A.: Analogical dissimilarity: definition, algorithms and two experiments in machine learning. JAIR 32, 793–824 (2008)
Miclet, L., Prade, H.: Handling analogical proportions in classical logic and fuzzy logics settings. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS, vol. 5590, pp. 638–650. Springer, Heidelberg (2009)
Prade, H., Richard, G.: Analogical proportions: another logical view. In: Nicholson, A., Li, X. (eds.) AI 2009. LNCS, vol. 5866, Springer, Heidelberg (2009)
Prade, H., Richard, G.: Analogy, paralogy and reverse analogy: Postulates and inferences. In: Mertsching, B., Hund, M., Aziz, Z. (eds.) KI 2009. LNCS (LNAI), vol. 5803, pp. 306–314. Springer, Heidelberg (2009)
Prade, H., Richard, G.: Reasoning with logical proportions. In: Proc. Int. Conf. on Principles of Knowledge Representation and Reasoning (KR’10), Toronto (2010)
Stroppa, N., Yvon, F.: Analogical learning and formal proportions: Definitions and methodological issues. ENST Paris report (2005)
Tversky, A.: Features of similarity. Psychological Review 84, 327–352 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Prade, H., Richard, G. (2010). Logical Proportions – Typology and Roadmap. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Computational Intelligence for Knowledge-Based Systems Design. IPMU 2010. Lecture Notes in Computer Science(), vol 6178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14049-5_77
Download citation
DOI: https://doi.org/10.1007/978-3-642-14049-5_77
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14048-8
Online ISBN: 978-3-642-14049-5
eBook Packages: Computer ScienceComputer Science (R0)