Basic Four-Valued Systems of Cyclic Negations

Oleg Grigoriev; Dmitry Zaitsev

doi:10.18778/0138-0680.2022.21

Authors

Oleg Grigoriev Lomonosov Moscow State University, Faculty of Philosophy, Department of Logic, 119234, Lomonosovskiy prospect 24, Building 4, Moscow, Russia https://orcid.org/0000-0002-3984-3289
Dmitry Zaitsev Lomonosov Moscow State University, Faculty of Philosophy, Department of Logic, 119234, Lomonosovskiy prospect 24, Building 4, Moscow, Russia

DOI:

https://doi.org/10.18778/0138-0680.2022.21

Keywords:

Generalized truth values, consequence relation, first degree entailment

Abstract

We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four valued semantics as an operation which acts like a cycling negation. We define two variants of logical matrices performing different orders over the set of truth values. Purely formal logical result of our study consists in axiomatizing the logics of defined matrices as the systems of binary consequence relation and proving correctness and completeness theorems for these deductive systems.

References

N. D. Belnap, A Useful Four-Valued Logic, [in:] J. M. Dunn, G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, Springer Netherlands, Dordrecht (1977), pp. 5–37, DOI: https://doi.org/10.1007/978-94-010-1161-7_2
Google Scholar DOI: https://doi.org/10.1007/978-94-010-1161-7_2

N. D. Belnap, How a Computer Should Think, [in:] H. Omori, H. Wansing (eds.), New Essays on Belnap-Dunn Logic, Springer International Publishing, Cham (2019), pp. 35–53, DOI: https://doi.org/10.1007/978-3-030-31136-0_4
Google Scholar DOI: https://doi.org/10.1007/978-3-030-31136-0_4

J. V. Benthem, Logical dynamics meets logical pluralism?, The Australasian Journal of Logic, vol. 6 (2008), pp. 182–209, DOI: https://doi.org/10.26686/ajl.v6i0.1801
Google Scholar DOI: https://doi.org/10.26686/ajl.v6i0.1801

M. L. D. Chiara, R. Giuntini, R. Greechie, Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics, vol. 22 of Trends in Logic, Springer Science & Business Media (2013), DOI: https://doi.org/https://doi.org/10.1007/978-94-017-0526-4
Google Scholar DOI: https://doi.org/10.1007/978-94-017-0526-4

M. L. D. Chiara, R. Giuntini, R. Leporini, Logics from quantum computation, International Journal of Quantum Information, vol. 03(02) (2005), pp. 293–337, DOI: https://doi.org/10.1142/s0219749905000943
Google Scholar DOI: https://doi.org/10.1142/S0219749905000943

D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic, vol. 6(3) (2000), pp. 265–283, DOI: https://doi.org/10.2307/421056
Google Scholar DOI: https://doi.org/10.2307/421056

Í. M. L. D'Ottaviano, H. de Araujo Feitosa, Many-valued logics and translations, Journal of Applied Non-Classical Logics, vol. 9(1) (1999), pp. 121–140, DOI: https://doi.org/10.1080/11663081.1999.10510960
Google Scholar DOI: https://doi.org/10.1080/11663081.1999.10510960

J. M. Dunn, Star and perp: Two treatments of negation, Philosophical Perspectives, vol. 7 (1993), pp. 331–357, DOI: https://doi.org/10.2307/2214128
Google Scholar DOI: https://doi.org/10.2307/2214128

J. M. Dunn, Partiality and its dual, Studia Logica, vol. 66 (2000), pp. 5–40, DOI: https://doi.org/10.1023/A:1026740726955
Google Scholar DOI: https://doi.org/10.1023/A:1026740726955

J. M. Dunn, G. Hardegree, Algebraic Methods in Philosophical Logic, Oxford University Press (2001).
Google Scholar

H. A. Feitosa, I. M. L. D'Ottaviano, Conservative translations, Annals of Pure and Applied Logic, vol. 108(1–3) (2001), pp. 205–227, DOI: https://doi.org/10.1016/s0168-0072(00)00046-4
Google Scholar DOI: https://doi.org/10.1016/S0168-0072(00)00046-4

R. French, Translational embeddings in modal logic, Ph.D. thesis, Monash University (2010).
Google Scholar

D. M. Gabbay, H. Wansing (eds.), What is a negation?, vol. 13 of Applied Logic Series, Springer Netherlands (1999), DOI: https://doi.org/10.1007/978-94-015-9309-0
Google Scholar DOI: https://doi.org/10.1007/978-94-015-9309-0

L. Humberstone, Negation by iteration, Theoria, vol. 61(1) (1995), pp. 1–24, DOI: https://doi.org/10.1111/j.1755-2567.1995.tb00489.x
Google Scholar DOI: https://doi.org/10.1111/j.1755-2567.1995.tb00489.x

N. Kamide, Paraconsistent double negations as classical and intuitionistic negations, Studia Logica, vol. 105(6) (2017), pp. 1167–1191, DOI: https://doi.org/10.1007/s11225-017-9731-2
Google Scholar DOI: https://doi.org/10.1007/s11225-017-9731-2

S. P. Odintsov, Constructive Negations and Paraconsistency, Springer Netherlands (2008), DOI: https://doi.org/10.1007/978-1-4020-6867-6
Google Scholar DOI: https://doi.org/10.1007/978-1-4020-6867-6

H. Omori, K. Sano, Generalizing functional completeness in Belnap-Dunn logic, Studia Logica, vol. 103(5) (2015), pp. 883–917, DOI: https://doi.org/10.1007/s11225-014-9597-5
Google Scholar DOI: https://doi.org/10.1007/s11225-014-9597-5

H. Omori, H. Wansing, On contra-classical variants of Nelson logic N4 and its classical extension, The Review of Symbolic Logic, vol. 11 (2018), pp. 805–820, DOI: https://doi.org/10.1017/s1755020318000308
Google Scholar DOI: https://doi.org/10.1017/S1755020318000308

F. Paoli, Bilattice Logics and Demi-Negation, [in:] H. Omori, H. Wansing (eds.), New Essays on Belnap–Dunn Logic, Springer International Publishing, Cham (2019), pp. 233–253, DOI: https://doi.org/10.1007/978-3-030-31136-0_14
Google Scholar DOI: https://doi.org/10.1007/978-3-030-31136-0_14

E. Post, Introduction to a general theory of elementary propositions, American Journal of Mathematics, vol. 43 (1921), pp. 163–185, DOI: https://doi.org/10.2307/2370324
Google Scholar DOI: https://doi.org/10.2307/2370324

H. Rasiowa, An Algebraic Approach to Non-Classical Logics, vol. 78 of Studies in Logic and Foundations of Mathematics, North-Holland, Amsterdam (1974).
Google Scholar

P. Ruet, Complete set of connectives and complete sequent calculus for Belnap’s logic, Tech. rep., École Normale Supérieure, Logic Colloquium 96, Document LIENS-96–28 (1996).
Google Scholar

Y. Shramko, J. M. Dunn, T. Takenaka, The trilattice of constructive truth values, Journal of Logic and Computation, vol. 11(6) (2001), pp. 761–788, DOI: https://doi.org/10.1093/logcom/11.6.761
Google Scholar DOI: https://doi.org/10.1093/logcom/11.6.761

Y. Shramko, H. Wansing, Some useful 16-valued logics: How a computer network should think, Journal of Philosophical Logic, vol. 34(2) (2005), pp. 121–153, DOI: https://doi.org/10.1007/s10992-005-0556-5
Google Scholar DOI: https://doi.org/10.1007/s10992-005-0556-5

J. van Benthem, Logical Dynamics of Information and Interaction, Cambridge University Press (2011), DOI: https://doi.org/10.1017/cbo9780511974533
Google Scholar DOI: https://doi.org/10.1017/CBO9780511974533

H. Wansing (ed.), Negation: A notion in focus, vol. 7 of Perspectives in Analytical Philosophy, W. De Gruyter (1996), DOI: https://doi.org/10.1515/9783110876802
Google Scholar DOI: https://doi.org/10.1515/9783110876802

D. Zaitsev, A few more useful 8-valued logics for reasoning with tetralattice EIGHT₄, Studia Logica, vol. 92(2) (2009), pp. 265–280, DOI: https://doi.org/10.1007/s11225-009-9198-x
Google Scholar DOI: https://doi.org/10.1007/s11225-009-9198-x

D. Zaitsev, O. Grigoriev, Two kinds of truth – one logic, Logical Investigations, vol. 17 (2011), pp. 121–139, DOI: https://doi.org/https://doi.org/10.21146/2074-1472-2011-17-0-121-139, (in Russian).
Google Scholar DOI: https://doi.org/10.21146/2074-1472-2011-17-0-121-139

D. Zaitsev, Y. Shramko, Bi-facial truth: A case for generalized truth values, Studia Logica, vol. 101(6) (2013), pp. 1299–1318, DOI: https://doi.org/10.1007/s11225-013-9534-z
Google Scholar DOI: https://doi.org/10.1007/s11225-013-9534-z