Generalized truth values, ordering relations defined on them, and the resulting lattice structures that give rise to various non-classical logics
Final Report Abstract
In this project considerable progress has been made in the investigation of generalized truth valued viewed as sets of already established truth values. The notion of a truth value has been introduced into logic and philosophy by Gottlob Frege, who considered exactly two truth values, the classical values the True and the False. The powerset 4 of the set of classical values is the basis of the famous bilattice of truth values FOUR2. In addition to an information order, FOUR2 contains a so-called truth order that gives rise to a basic system of non-classical logic known as first-degree entailment or Belnap's useful four-valued logic. A core object of research in this project has been the trilattice SIXTEEN3 defined on the powerset 16 of 4. In SIXTEEN3 in addition to an information order, two independent logical orders naturally emerge, a truth order and a falsity order. Whereas the information order orders the truth values according the amount information they provide concerning a given proposition, the truth (falsity) order arranges the truth values according to the amount of truth (falsity) they assign to a given proposition. Each logical order not only induces its own logical vocabulary, a truth vocabulary and a falsity vocabulary, but also its own semantical entailment relation. The notions of truth and falsity are thereby treated as concepts of equal importance that are independent of each other. In the project, the logic of the truth order and the logic of the falsity order have been thoroughly investigated. Most importantly, in co-operation with researchers from Russia and Japan, axiomatic and sequent-style presentations of these logics and certain fragments of these systems have been obtained. Moreover, by using the conception of multilattices of truth values, the relation between the notion of a truth value and the notion of a consequence relation could be further clarified. In the matrix-based approach to many-valued logic, it is a distinctive feature of truth values understood as sets of algebraic values that truth values are used to canonically define entailment relations. In addition, in the project new insights have been obtained into the structure of so-called slingshot arguments, which, according to some philosophers, show that all true sentences and all false sentences describe the same situation, so that the development of a fact ontology might seem to be hopeless. The results of the project provide a better understanding of how such a conclusion can be avoided.
Publications
- Hyper-contradictions, Generalized Truth Values, and Logics of Truth and Falsehood, Journal of Logic, Language and Information 15 (2006), 403-424
Y. Shramko and H. Wansing
- A note on two ways of defining a many-valued logic, in: M. Pelis (ed.) Logica Yearbook 2007, 2008, 255-266
H. Wansing and Y. Shramko
- Entailment relations and/as truth values, Bulletin of the Section of Logic, 36 (2007), 131-143
Y. Shramko and H. Wansing
- Combining linear-time temporal logic with constructiveness and paraconsistency, 2008, Journal of Applied Logic
N. Kamide and H. Wansing
(See online at https://doi.org/10.1016/j.jal.2009.06.001) - Constructive negation, implication, and co-implication, Journal of Applied Non-Classical Logics 18 (2008), 341-364
H. Wansing
- Harmonious many-valued propositional logics and the logic of computer networks, in: C. Dégremont, L. Keiff and H. Rückert (eds.), Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman, College Publications, London, 2008, 491-516
H. Wansing and Y. Shramko
- Suszko's Thesis, inferential many-valuedness, and the notion of a logical system, Studia Logica 88 (2008), 405-429, 89 (2008), 147
H. Wansing and Y. Shramko
- Sequent calculi for some trilattice logics, Review of Symbolic Logic 2 (2009), 374-395
N. Kamide and H. Wansing
- The Slingshot-Argument and sentential identity, Studia Logica 91 (2009), 429-455
Y. Shramko and H. Wansing
- Truth and Falsehood: what are truth values and what they are needed for (in Russian), Logos 70 (2009), 96-121
Y. Shramko
- Truth values. Part I, Studia Logica 91-3 (2009), Springer-Verlag, Dordrecht/Berlin
Y. Shramko and H. Wansing
- Truth values. Part II, Studia Logica 92-2 (2009), Springer-Verlag, Dordrecht/Berlin
Y. Shramko and H. Wansing
