Geodesic
Natural deduction systems for some modifications of Kleene's and Dunn~--- Belnap's logics
Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 4, pp. 438-452.

Voir la notice de l'article provenant de la source Math-Net.Ru

Kleene introduced the notions of regular logical connective and regular logic as well as considered three-valued examples of such logics. Finn and Komendantskaya studied functional properties of regular three-valued logics. Base ourselves upon their results, we present four-valued analogues of Kleene's three-valued logics. The first four-valued generalization of Kleene's three-valued logics (more exactly, of strong Kleene's logic) is Dunn–Belnap's logic. Two different orders (truth and information ones) can be defined on the set of truth values of this logic (we follow Belnap's semantics). Using them, one can define two sets of logical connectives. Only one of them (which is based on truth order) is presented in Dunn–Belnap's logic itself. Fitting considers two sets at the same time. We study logic (we call it Belnap–Fitting's logic) which have the connectives based on information order. Using these connectives (more exactly, we substitute them into Finn and Komendantskaya's equations instead of the connectives of strong Kleene's logic), we obtain a new class of four-valued logics which are analogues of regular three-valued ones. All the elements of this class are formalized via natural deduction systems.
Keywords: natural deduction system, four-valued logic, Kleene's logics, Dunn–Belnap's logic, regular logic. 
@article{CHFMJ_2018_3_4_a4,
     author = {Ya. I. Petrukhin},
     title = {Natural deduction systems for some modifications of {Kleene's} and {Dunn~---} {Belnap's} logics},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {438--452},
     publisher = {mathdoc},
     volume = {3},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2018_3_4_a4/}
}
TY  - JOUR
AU  - Ya. I. Petrukhin
TI  - Natural deduction systems for some modifications of Kleene's and Dunn~--- Belnap's logics
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2018
SP  - 438
EP  - 452
VL  - 3
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2018_3_4_a4/
LA  - ru
ID  - CHFMJ_2018_3_4_a4
ER  - 
%0 Journal Article
%A Ya. I. Petrukhin
%T Natural deduction systems for some modifications of Kleene's and Dunn~--- Belnap's logics
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2018
%P 438-452
%V 3
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2018_3_4_a4/
%G ru
%F CHFMJ_2018_3_4_a4
Ya. I. Petrukhin. Natural deduction systems for some modifications of Kleene's and Dunn~--- Belnap's logics. Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 4, pp. 438-452. http://geodesic.mathdoc.fr/item/CHFMJ_2018_3_4_a4/

[1] S. C. Kleene, “On a notation for ordinal numbers”, Journal of Symbolic Logic, 3:4 (1938), 150–155 | DOI | MR

[2] F. G. Asenjo, “A calculus of antinomies”, Notre Dame Journal of Formal Logic, 7:1 (1966), 103–105 | DOI | MR | Zbl

[3] G. Priest, “The logic of paradox”, Journal of Philosophical Logic, 8:1 (1979), 219–241 | DOI | MR | Zbl

[4] G. Priest, “Paraconsistent logic”, Handbook of Philosophical Logic, v. 6, 2, eds. M. Gabbay, F. Guenthner, Kluwer, Dordrecht, 2002, 287–393 | DOI | MR

[5] Finn V.K., “Axiomatization of some three-valued propositional calculi and their algebras”, Philosophy in the contemporary world. Philosophy and logic, 1974, 398–438 (In Russ.)

[6] S. Halldén, The logic of nonsense, Uppsala Universitets Arskrift, Uppsala, 1949, 132 pp. | Zbl

[7] S. Bonzio, J. Gil-Férez, F. Paoli, L. Peruzzi, “On paraconsistent weak Kleene logic: axiomatisation and algebraic analysis”, Studia Logica, 105:2 (2017), 253–297 | DOI | MR | Zbl

[8] Kleene S.C., Introduction to Metamathematics, D. Van Nostrand Company, Inc., New York, Toronto, 1952, X + 550 pp. | MR | MR

[9] Komendantskaya E.Y., “Functional interdependence of regular Kleene logics”, Logical investigations, 15 (2009), 116–128 (In Russ.)

[10] M. Fitting, “Kleene's three valued logics and their children”, Fundamenta informaticae, 20:1–3 (1994), 113–131 | MR | Zbl

[11] Y. Petrukhin, “Natural deduction for three-valued regular logics”, Logic and Logical Philosophy, 26:2 (2017), 197–206 | MR | Zbl

[12] Petrukhin Y.I., Shangin V.O., “Correspondence analysis for paraconsistent weak Kleene logic”, Moscow University Philosophical Bulletin, 2017, no. 6, 52–62 (In Russ.)

[13] B. Kooi, A. Tamminga, “Completeness via correspondence for extensions of the logic of paradox”, The Review of Symbolic Logic, 5:4 (2012), 720–730 | DOI | MR | Zbl

[14] A. Tamminga, “Correspondence analysis for strong three-valued logic”, Logical Investigations, 20 (2014), 255–268 | MR | Zbl

[15] N. D. Belnap, “Tautological entailments”, Journal of Symbolic Logic, 24:4 (1959), 316

[16] A. R. Anderson, N. D. Belnap, “Tautological entailments”, Philosophical Studies, 13:1–2 (1962), 9–24 | DOI | MR

[17] J. M. Dunn, The Algebra of Intensional Logics, Doctoral Dissertation, University of Pittsburgh, Pittsburgh, 1966

[18] J. M. Dunn, “Intuitive semantics for first-degree entailment and coupled trees”, Philosophical Studies, 29:3 (1976), 149–168 | DOI | MR | Zbl

[19] N. D. Belnap, “A useful four-valued logic”, Modern Uses of Multiple-Valued Logic, eds. J. M. Dunn, G. Epstein, Reidel Publishing Company, Boston, 1977, 8–37 | MR

[20] N. D. Belnap, “How a computer should think”, Contemporary Aspects of Philosophy, ed. G. Rule, Oriel Press, Stocksfield, 1977, 30–55

[21] J. M. Font, “Belnap's four-valued logic and de Morgan lattices”, Logic Journal of the IGPL, 5:3 (1997), 1–29 | DOI | MR

[22] Zaitsev D.V., Shramko Y.V., “Logical entailment and designated values”, Logical Investigations, 2004, no. 11, 126–137 (In Russ.)

[23] M. L. Ginsberg, “Multivalued logics”, Proceedings of the Fifth National Conference on Artificial Intelligence, Morgan Kaufmann Publ., 1986, 243–247

[24] M. L. Ginsberg, “Multivalued logics: a uniform approach to reasoning in artificial intelligence”, Computational Intelligence, 4:3 (1988), 265–316 | DOI | MR

[25] M. Fitting, “Negation as refutation”, Proceedings of the Fourth Annual Symposium on Logic in Computer Science, IEEE Press Piscataway, New Jersey, 1989, 63–70 | DOI

[26] M. Fitting, “Kleene's logic, generalized”, Journal of Logic and Computation, 1:6 (1992), 797–810 | DOI | MR

[27] Tomova N.E., “On four-valued regular logics”, Logical investigations, 15 (2009), 223–228 (In Russ.) | Zbl

[28] Y. Petrukhin, “Natural deduction for four-valued both regular and monotonic logics”, Logic and Logical Philosophy, 27:1 (2018), 53–66 | MR | Zbl

[29] Y. I. Petrukhin, “Natural deduction for Fitting’s four-valued generalizations of Kleene’s logics”, Logica Universalis, 11:4 (2017), 525–532 | DOI | MR | Zbl

[30] Petrukhin Y.I., “Natural deduction for Yuriev's logic”, Chelyabinsk Physical and Mathematical Journal, 2:1 (2017), 46–52 (In Russ.) | MR

[31] Y. I. Petrukhin, “Sorrespondence analysis for logic of rational agent”, Chelyab. fiz.-mat. zhurn, 2:3 (2017), 329–337 | MR

[32] V. O. Shangin, “A precise definition of an inference (by the example of natural deduction systems for logics $I_{\langle\alpha,\beta\rangle}$)”, Logical Investigations, 23:1 (2017), 83–104 | DOI | MR | Zbl

[33] L. Henkin, “The completeness of the first-order functional calculus”, Journal of Symbolic Logic, 14:3 (1949), 159–166 | DOI | MR | Zbl