David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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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|
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References found in this work BETA
Alan R. Anderson & Nuel D. Belnap (1975). Entailment: The Logic of Relevance and Neccessity, Vol. I. Princeton University Press.
Graham Priest (2001). Introduction to Non-Classical Logic. Cambridge University Press.
Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton University Press.
J. Michael Dunn (2001). Algebraic Methods in Philosophical Logic. Oxford University Press.
Nuel Belnap (1977). How a Computer Should Think. In G. Ryle (ed.), Contemporary Aspects of Philosophy. Oriel Press Ltd.
Citations of this work BETA
Sergei P. Odintsov (2009). On Axiomatizing Shramko-Wansing's Logic. Studia Logica 91 (3):407 - 428.
Sergei P. Odintsov & Heinrich Wansing (2015). The Logic of Generalized Truth Values and the Logic of Bilattices. Studia Logica 103 (1):91-112.
Heinrich Wansing & Yaroslav Shramko (2008). Suszko's Thesis, Inferential Many-Valuedness, and the Notion of a Logical System. Studia Logica 88 (3):405 - 429.
Graham Priest (2008). Jaina Logic: A Contemporary Perspective. History and Philosophy of Logic 29 (3):263-278.
Norihiro Kamide & Heinrich Wansing (2009). Sequent Calculi for Some Trilattice Logics. Review of Symbolic Logic 2 (2):374-395.
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