Journal of Philosophical Logic 34 (2):121-153 (2005)
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Abstract |
In Belnap's useful 4-valued logic, the set 2 = {T, F} of classical truth values is generalized to the set 4 = í ”íČ«(2) = {Ă, {T}, {F}, {T, F}}. In the present paper, we argue in favor of extending this process to the set 16 = á” (4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOURâ with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEENâ with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEENâ coincide with the logic of FOURâ, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli's and Avron's notion of a logical bilattice and state a number of open problems for future research
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Keywords | bi-consequence logic first degree entailment generalized truth values (logical) bilattices trilattices multilattices |
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DOI | 10.1007/s10992-005-0556-5 |
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References found in this work BETA
Entailment: The Logic of Relevance and Neccessity, Vol. I.Alan Ross Anderson & Nuel D. Belnap - 1975 - 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.
How a Computer Should Think.Nuel Belnap - 1977 - In G. Ryle (ed.), Contemporary Aspects of Philosophy. Oriel Press.
Intuitive Semantics for First-Degree Entailments and 'Coupled Trees'.J. Michael Dunn - 1976 - Philosophical Studies 29 (3):149-168.
Entailment: The Logic of Relevance and Necessity, Vol. II.Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn - 1992 - Princeton University Press.
View all 28 references / Add more references
Citations of this work BETA
Truth and Falsehood: An Inquiry Into Generalized Logical Values.Yaroslav Shramko & Heinrich Wansing - 2011 - Dordrecht, Netherland: Springer.
Faulty Belnap Computers and Subsystems of FDE.Thomas Macaulay Ferguson - 2016 - Journal of Logic and Computation 26 (5):1617â1636.
Shifting Priorities: Simple Representations for Twenty-Seven Iterated Theory Change Operators.Hans Rott - 2006 - In David Makinson, Jacek Malinowski & Heinrich Wansing (eds.), Towards Mathematical Philosophy. Dordrecht: Springer. pp. 269â296.
Constructive Negation, Implication, and Co-Implication.Heinrich Wansing - 2008 - Journal of Applied Non-Classical Logics 18 (2-3):341-364.
Suszkoâs Thesis, Inferential Many-Valuedness, and the Notion of a Logical System.Heinrich Wansing & Yaroslav Shramko - 2008 - Studia Logica 88 (3):405-429.
View all 50 citations / Add more citations
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