David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 92 (2):265 - 280 (2009)
In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN 3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data is asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT 4 . Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a –order (truth order) and d –order (falsity order) coincide with first-degree entailment. Finally logic with two kinds of operations ( a –connectives and d –connectives) and consequence relation defined via a –ordering is considered. An adequate axiomatization for this logic is proposed.
|Keywords||Generalized truth values Dunn–Belnap logic Shramko–Wansing logic bilattice trilattice tetralattice first-degree entailment|
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References found in this work BETA
Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton University Press.
Ofer Arieli & Arnon Avron (1996). Reasoning with Logical Bilattices. Journal of Logic, Language and Information 5 (1):25--63.
N. D. Belnap (1977). A Useful Four-Valued Logic. In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel
Nuel Belnap (1977). How a Computer Should Think. In G. Ryle (ed.), Contemporary Aspects of Philosophy. Oriel Press Ltd.
J. Michael Dunn (1976). Intuitive Semantics for First-Degree Entailments and 'Coupled Trees'. Philosophical Studies 29 (3):149-168.
Citations of this work BETA
Dmitry Zaitsev & Yaroslav Shramko (2013). Bi-Facial Truth: A Case for Generalized Truth Values. Studia Logica 101 (6):1299-1318.
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