Abstract
In this paper, the meaning of a vague concept \(\alpha \) is assumed to be rendered through two (mutually exclusive) finite sets of prototypes and counterexamples. In the remaining set of situations the concept is assumed to be applied only partially. A logical model for this setting can be fit into the three-valued Łukasiewicz’s logic Ł\(_3\) set up by considering, besides the usual notion of logical consequence \(\models \) (based on the truth preservation), the logical consequence \(\models ^\le \) based on the preservation of all truth-degrees as well. Moreover, we go one step further by considering a relaxed notion of consequence to some degree \(a \in [0, 1]\), by allowing the prototypes (counterexamples) of the premise (conclusion) be a-similar to the prototypes (counterexamples) of the conclusion (premise). We present a semantical characterization as well as an axiomatization.
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Notes
- 1.
Actually, one could take \(\rightarrow \) and \(\lnot \) as the only primitive connectives since \(\wedge \) and \(\vee \) can be defined from \(\rightarrow \) and \(\lnot \) as well: \(\varphi \wedge \psi = \varphi \otimes (\varphi \rightarrow \psi )\) and \(\varphi \vee \psi = (\varphi \rightarrow \psi ) \rightarrow \psi \).
- 2.
Although we are using symbols \(\wedge , \vee , \lnot , \rightarrow \) for both formulas of \(\mathcal {L}_{0}\) and \(\mathcal {L}_{1}\), it will be clear from the context when they refer to Ł\(_3\) or when they refer to Boolean connectives.
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Acknowledgments
The authors are thankful to the anonymous reviewers for their helpful comments. Esteva and Godo acknowledge partial support of the project TIN2015-71799-C2-1-P (MINECO/FEDER).
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Dutta, S., Esteva, F., Godo, L. (2016). On a Three-Valued Logic to Reason with Prototypes and Counterexamples and a Similarity-Based Generalization. In: Luaces , O., et al. Advances in Artificial Intelligence. CAEPIA 2016. Lecture Notes in Computer Science(), vol 9868. Springer, Cham. https://doi.org/10.1007/978-3-319-44636-3_47
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