Relevant generalization starts here (and here = 2)

Dmitry Zaitsev, Oleg Grigoriev

DOI: http://dx.doi.org/10.12775/LLP.2010.012

Abstract


There is a productive and suggestive approach in philosophical logic based on the idea of generalized truth values. This idea, which stems essentially from the pioneering works by J.M. Dunn, N. Belnap, and which has recently been developed further by Y. Shramko and H. Wansing, is closely connected to the power-setting formation on the base of some initial truth values. Having a set of generalized truth values, one can introduce fundamental logical notions, more specifically, the ones of logical operations and logical entailment. This can be done in two different ways. According to the first one, advanced by M. Dunn, N. Belnap, Y. Shramko and H. Wansing, one defines on the given set of generalized truth values a specific ordering relation (or even several such relations) called the logical order(s), and then interprets logical connectives as well as the entailment relation(s) via this ordering(s). In particular, the negation connective is determined then by the inversion of the logical order. But there is also another method grounded on the notion of a quasi-field of sets, considered by Białynicki-Birula and Rasiowa. The key point of this approach consists in defining an operation of quasi-complement via the very specific function g and then interpreting entailment just through the relation of set-inclusion between generalized truth values.

In this paper, we will give a constructive proof of the claim that, for any finite set V with cardinality greater or equal 2, there exists a representation of a quasi-field of sets <P(V ), ∪, ∩, −> isomorphic to de Morgan lattice. In particular, it means that we offer a special procedure, which allows to make our negation de Morgan and our logic relevant.


Keywords


relevant logic; first-degree entailment; generalization; information

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References


Anderson, A.R., and N.D. Belnap, Entailment: The Logic of Relevance and Necessity, Vol. I, Princeton University Press, Princeton, NJ, 1975.

Belnap, N.D., “A useful four-valued logic”, pp. 8–37 in: J.M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel Publishing Company, Dordrecht, 1977.

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Białynicki-Birula, A., and H. Rasiowa, “On the representation of quasi-Boolean algebras”, Bulletin de l’Academie Polonaise des Sciences 5 (1957): 259–261.

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Shramko, Y., and H. Wansing, “Hyper-contradictions, generalized truth values and logics of truth and falsehood”, Journal of Logic, Language and Information 15 (2006): 403–424.








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