Journal of Logic, Language and Information 15 (4):403-424 (2006)
In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox.
|Keywords||Hyper-contradiction multilattice Belnap-trilattice first-degree entailment|
|Categories||categorize this paper)|
References found in this work BETA
Entailment: The Logic of Relevance and Neccessity, Vol. I.Alan R. Anderson & Nuel D. Belnap - 1975 - Princeton University Press.
Entailment: The Logic of Relevance and Necessity, Vol. II.Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn - 1992 - Princeton University Press.
A Useful Four-Valued Logic.N. D. Belnap - 1977 - In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel.
Citations of this work BETA
The Logic of Generalized Truth Values and the Logic of Bilattices.Sergei P. Odintsov & Heinrich Wansing - 2015 - Studia Logica 103 (1):91-112.
Suszko's Thesis, Inferential Many-Valuedness, and the Notion of a Logical System.Heinrich Wansing & Yaroslav Shramko - 2008 - Studia Logica 88 (3):405 - 429.
Jaina Logic: A Contemporary Perspective.Graham Priest - 2008 - History and Philosophy of Logic 29 (3):263-278.
Sequent Calculi for Some Trilattice Logics.Norihiro Kamide & Heinrich Wansing - 2009 - Review of Symbolic Logic 2 (2):374-395.
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