# Some Useful 16-Valued Logics: How a Computer Network Should Think

Journal of Philosophical Logic 34 (2):121-153 (2005)

 Authors Yaroslav Shramko Kryvyi Rih State Pedagogical University, Ukraine 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 Keywords bi-consequence logic  first degree entailment  generalized truth values  (logical) bilattices  trilattices  multilattices Categories Logic and Philosophy of Logic (categorize this paper) DOI 10.1007/s10992-005-0556-5 Options Mark as duplicate Export citation Request removal from index

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 69,089

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)

## References found in this work BETA

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

Faulty Belnap Computers and Subsystems of FDE.Thomas Macaulay Ferguson - 2016 - Journal of Logic and Computation 26 (5):1617â1636.
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. 269â296.
Constructive Negation, Implication, and Co-Implication.Heinrich Wansing - 2008 - Journal of Applied Non-Classical Logics 18 (2-3):341-364.

## Similar books and articles

Many-Valued Logics.Grzegorz Malinowski - 1993 - Oxford, England: Oxford University Press.
How to Avoid Deviance (in Logic).Walter Sinnott-Armstrong & Amit Malhotra - 2002 - History and Philosophy of Logic 23 (3):215--36.
Some Notes Concerning Fuzzy Logics.Charles Grady Morgan & Francis Jeffry Pelletier - 1977 - Linguistics and Philosophy 1 (1):79 - 97.
Ideal Paraconsistent Logics.O. Arieli, A. Avron & A. Zamansky - 2011 - Studia Logica 99 (1-3):31-60.
A Characteristic Model For Some Tabular Many-Valued Logics.Marion Mircheva - 1986 - Bulletin of the Section of Logic 15 (4):159-161.
Factor Semantics Forn-Valued Logics.A. S. Karpenko - 1983 - Studia Logica 42 (2-3):179 - 185.
Lewis Dichotomies in Many-Valued Logics.Simone Bova - 2012 - Studia Logica 100 (6):1271-1290.