## Truth Values

Por • 2 nov, 2022 • Sección: Leyes

Yaroslav Shramko, Heinrich Wansing

This tutorial  provides a detailed introduction  into the conception of truth values, an important notion of modern logical semantics and philosophy of logic, explicitly introduced by Gottlob Frege. Frege conceived this notion as a natural component  of his language analysis where sentences, being saturated expressions, are interpreted as a special kind of names referring to a special kind of objects: the True (das Wahre) and the False (das Falsche). These are essentially the truth  values of classical logic, which obey the principle of bivalence saying that there may exist only two distinct logical values.

Truth  values have been put to quite different uses in philosophy and logic and have been characterized,  for example, as:

• primitive  abstract objects denoted by sentences  in natural  and formal languages,
• abstract entities hypostatized as the equivalence  classes of sentences,
• what is aimed at in judgements,
• values indicating the degree of truth  of sentences,
• entities that can be used to explain the vagueness of concepts,
• values that are preserved in valid inferences,
• values that convey information concerning a given proposition.

Depending on their particular use, truth values can be treated as unanalyzed, as defined, as unstructured, or as structured entities.  Moreover, the classical conception of truth values can be developed further and generalized in various ways.   One way is to give up the principle of bivalence, and to proceed to many-valued logics dealing with more than two truth  values. Another way is to generalize the very notion of a truth  value by reconstructing them as complex units with an elaborate nature of their own.

In fact, the idea of truth values as compound entities nicely conforms with the modelling of truth values in some many-valued  systems, such as three-valued (Kleene, Priest) and four-valued (Belnap) logics, as certain subsets of the set of classical truth  values. The latter approach is essentially due to Michael Dunn, who proposed to generalize the notion of a classical truth-value function in order to represent the so-called “underdetermined” and “overdetermined” valuations. Namely, Dunn considers a valuation to be a function not from sentences  to elements of the set {the True, the False } but from sentences to subsets of this set. By developing this idea, one arrives at the concept of a generalized truth value function, which is a function from sentences into the subsets of some basic set of truth values. The values of generalized truth value functions can be called generalized truth values.

The tutorial consists of three sessions, in the course of which we unfold step by step the idea of generalized truth  values and demonstrate its fruitfulness for an analysis of many logical and philosophical problems.

Session 1.

The  notion of a truth value and the ways of its generalization

In the first lecture we explain how Gottlob Frege’s notion of a truth  value has become part of the standard philosophical and logical terminology. This notion is an indispensable instrument of realistic, model-theoretic approaches to logical semantics. Moreover, there  exist well-motivated theories of generalized truth values that lead far beyond Frege’s classical the True and the False. We discuss the possibility of generalizing the notion of a truth  value by conceiving them as complex units which possess a ramified inner structure.  We explicate some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value understood as a subset of some basic set of initial truth values of a lower degree. It turns out that this generalization is well- motivated and leads to the notion of a truth value multilattice.  In particular, one can proceed from the bilattice  F OU R2  with  both an information  and a truth-and-falsity  ordering to another algebraic structure, namely the trilattice SI X T EEN3   with an information ordering together with a truth  ordering and a (distinct)  falsity ordering.

Session 2.

Logics of generalized truth values

In this lecture we present various references

[1] Anderson, A.R. and Belnap N.D., 1975, Entailment:  The Logic of Relevance and Necessity, Vol. I, Princeton University Press, Princeton, NJ.

[2] Anderson, A.R., Belnap, N.D. and Dunn, J. Michael, 1992, Entailment: The Logic of Relevance and Necessity, Vol. II, Princeton University Press, Princeton, NJ.

[3] Arieli,  O. and Avron, A., 1996, Reasoning with logical bilattices, Journal of Logic, Language and Information, 5: 25-63.

[4] Beaney, M. (ed. and transl.),  1997, The Frege  Reader,  Wiley-Blackwell, Oxford.

[5] Belnap, N.D., 1977a, How a computer should think, in G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press Ltd., Stocksfield, 30-55.

[6] Belnap, N.D., 1977b, A useful four-valued logic, in: J.M. Dunn and G. Ep- stein (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel Publishing Co., Dordrecht, 8-37.

[7] Dunn, J.M., 1976, Intuitive semantics for first-degree entailment and ‘coupled trees’, Philosophical Studies, 29: 149-168.

[8] Fitting,  M., 2006, Bilattices are nice things, in: T. Bolander, V. Hendricks, and S.A. Pedersen (eds.), Self-Reference, CSLI-Publications, Stanford, 53-77.

[9] Geach, P. and Black, M. (eds.), 1952, Translations from the Philosophical Writings of Gottlob Frege, Philosophical Library, New York.

[10] Ginsberg, M., 1988, Multivalued logics: a uniform approach to reasoning in AI, Computer Intelligence, 4: 256-316.

[11] Gottwald,  S., 2001, A Treatise  on Many-valued Logic, Research Studies Press, Baldock.

[12] Lukasiewicz, J., 1970, Selected Works, L. Borkowski (ed.), North-Holland, Amsterdam and PWN, Warsaw.

[13] Neale, S., 2001, Facing Facts, Oxford University Press, Oxford.

[14] Odintsov, S., 2009, On axiomatizing Shramko-Wansing’s logic, Studia Log- ica, 93: 407-428.

[15] Priest, G., 1979, Logic of Paradox, Journal of Philosophical Logic, 8: 219-241.

[16] Ryan, M. and Sadler, M., 1992, Valuation systems and consequence relations, in:  S. Abramsky, D. Gabbay, and T. Maibaum (eds.), Handbook of Logic in Computer Science, Vol. 1., Oxford University Press, Oxford, 1-78.

[17] Shramko, Y., Dunn, J. M., and Takenaka, T., 2001, The trilaticce of constructive truth  values, Journal of Logic and Computation, 11: 761-788.

[18] Shramko, Y. and Wansing, H., 2005, Some useful 16-valued logics: how a computer network should think,  Journal of Philosophical Logic, 34: 121-153.

[19] Shramko, Y.  and Wansing, H.,  2006, Hypercontradictions, generalized truth  values, and logics of truth  and falsehood, Journal of Logic, Language and Information, 15: 403-424.

[20] Shramko, Y. and Wansing, H., 2009, The Slingshot-Argument and sentential idendity, Studia Logica, 91: 429-455.

[21] Shramko, Y. and Wansing, H. (eds.), 2009, Truth  Values. Part I, Special Issue of Studia Logica, Vol. 91, No. 3.

[22] Shramko, Y. and Wansing, H. (eds.), 2009, Truth Values. Part II, Special Issue of Studia Logica, Vol. 92, No. 2.

[23] Shramko, Y. and Wansing, H., 2010, Truth  values, The Stanford Encyclopedia of Philosophy  (Summer 2010 Edition), Edward N. Zalta (ed.), URL

[24] Shramko, Y. and Wansing, H., 2011, Truth and Falsehood. An Inquiry into Generalized Logical Values, Springer, 2011, 250 p.

[25] Suszko, R., 1977, The Fregean axiom and Polish mathematical logic in the 1920’s, Studia Logica, 36: 373-380.

[26] Urquhart,  A., 1986, Many-valued logic, in:  D. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, Vol. III., D. Reidel Publishing Co., Dordrecht, 71-116.

[27] Wansing, H.,  2001, Short dialogue between M  (Mathematician)  and P (Philosopher) on multi-lattices, Journal of Logic and Computation, 11: 759-760.

[28] Wansing, H., https://www.uni-log.org/t4-truthvalue.html 2010, The power of Belnap. Sequent systems for S I X T EEN 3 , Journal of Philosophical Logic, 39: 69-393.

[29] Wansing, H. and Belnap, N.D., 2010, Generalized truth  values. A reply to Dubois, Logic Journal of the Interest Group in Pure and Applied Logics 18: 921-935.

[30] Wansing, H. and Shramko, Y.,  2008, Harmonious many-valued propositional  logics and the logic of computer networks, in:  C. D´egremont,  L. Keiff and H. Ruckert (eds.), Dialogues, Logics and Other Strange Things. Essays in Honor of Shahid Rahman, College Publications, 491-516.

[31] Wansing, H. and Shramko, Y.,  2008, Suszko’s Thesis, inferential many- valuedness, and the notion of a logical system, Studia Logica, 88: 405-429, 89: 147.

[32] Wansing, H. and Shramko, Y., 2008, A note on two ways of defining a many- valued logic, in:  M. Pelis (ed.), Logica Yearbook 2007, Filosofia, Prague, 255-266.s approaches to the construction of logical systems related to truth  value multilattices.   More concretely, we investigate the logics generated by the algebraic operations under the truth  order and under the falsity order in bilattices and trilattices,  as well as various interrelations between them. It is also rather natural to formulate the logical systems in the language obtained by combining the vocabulary of the logic of the truth  order and the falsity order. We consider the corresponding  first-degree  consequence systems, Hilbert-style axiomatizations and Gentzen-style sequent calculi for the multilattice-logics.

Session 3.

Generalized  truth-values: logical and philosophical applications

Besides its purely logical impact, the idea of truth  values has induced a radical rethinking of some central issues in ontology, epistemology and the philosophy of logic, including:  the categorial status of truth  and falsehood, the theory of abstract objects, the subject-matter of logic and its ontological foundations, and the concept of a logical system. In the third lecture we demonstrate the wealth of philosophical problems, which can be analyzed by means of the apparatus of trhttps://www.uni-log.org/t4-truthvalue.htmluth  values.  Among these problems are the liar paradox and the notion of hyper-contradiction, the famous slingshot-argument, Suszko thesis, harmonious many-valued logics, and some others.

Yaroslav Shramko: Department of Philosophy, Kryvyi Rih National University, Ukraine

Heinrich Wansing: Department of Philosophy II, Logic and Epistemology, University of Bochum,

Germany

https://www.uni-log.org/t4-truthvalue.html