Abstract
An involutive Stone algebra (IS-algebra) is simultaneously a De Morgan algebra and a Stone algebra (i.e., a pseudo-complemented distributive lattice satisfying the Stone identity \(\mathop {\sim }x \vee \mathop {\sim }\mathop {\sim }x \approx 1\)). IS-algebras have been studied algebraically and topologically since the 1980s, but a corresponding logic (here denoted \(\mathcal {IS}_{\le }\)) has been introduced only very recently. This logic is the departing point of the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that \(\mathcal {IS}_{\le }\) is a conservative expansion of the Belnap-Dunn four-valued logic (i.e., the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic (i.e., every strengthening of the Belnap-Dunn logic) so as to obtain an extension of \(\mathcal {IS}_{\le }\). We show that every logic thus defined can be axiomatized by adding a fixed finite set of multiple-conclusion rule schemata to the corresponding super-Belnap base logic. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of \(\mathcal {IS}_{\le }\). In fact, as in the super-Belnap case, we establish that the finitary extensions of \(\mathcal {IS}_{\le }\) are already uncountably many. When the base super-Belnap logic possesses a disjunction, we show that we can reduce the multiple-conclusion calculus to a traditional one; some of the multiple-conclusion axiomatizations so introduced are analytic and are thus of independent interest from a proof-theoretic standpoint. We also consider a few extensions of \(\mathcal {IS}_{\le }\) that cannot be obtained in the above-described way, but can nevertheless be axiomatized finitely by other methods.
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Notes
Formally, a Kleene lattice (algebra) is defined as a De Morgan lattice (algebra) that satisfies \(x \wedge \mathop {\sim }x \le y \vee \mathop {\sim }y\). It is well known that the variety of Kleene lattices (algebras) is \({\mathbb {V}}(\mathbf {K_3})\).
Formally, a Stone algebra can be defined as a bounded distributive lattice \(\langle A; \wedge , \vee , \lnot , \bot , \top \rangle \) endowed by an extra unary operation \(\lnot \) that satisfies, for all \(a,b \in A\), the following requirements: (i) \(a \wedge b = 0\) iff \(a \le \lnot b\), and (ii) \(\lnot a \vee \lnot \lnot a = \top \).
This recipe is applicable as long as every multiple-conclusion rule being translated has a finite conclusion-set.
Note that in general \(\vartriangleright ^\varLambda _{\mathsf {R}}\) is not a multiple-conclusion consequence relation. It still satisfies dilution and cut for set properties, but only weaker versions of overlap and substitution invariance.
References
Albuquerque H, Prenosil A, Rivieccio U (2017) An algebraic view of super-Belnap logics. Studia Logica 105(6):1051–1086
Belnap ND Jr. (1977) A useful four-valued logic. In J. M. Dunn and G. Epstein, editors, Modern uses of multiple-valued logic (Fifth Internat. Sympos., Indiana Univ., Bloomington, Ind., 1975), pages 5–37. Episteme, Vol. 2. Reidel, Dordrecht
Burris S, Sankappanavar HP (2000) A course in Universal Algebra. The Millennium edition
Caleiro C, Marcelino S, Marcos J (2019) Combining fragments of classical logic: When are interaction principles needed? Soft Computing 23(7):2213–2231
Caleiro C, Marcelino S (2021) On axioms and rexpansions. In: Arieli O, Zamansky A (ed) Arnon avron on semantics and proof theory of non-classical logics, Outstanding Contributions to Logic, vol 21. Springer, pp 39–69. https://doi.org/10.1007/978-3-030-71258-7_3
Cantú L (2019) Sobre la lógica que preserva grados de verdad asociada a las álgebras de Stone involutivas. Masters dissertation, Universidad Nacional del Sur (Bahía Blanca, Argentina)
Cantú L, Figallo M (2020) On the logic that preserves degrees of truth associated to involutive Stone algebras. Logic J IGPL 28(5):1000–1020
Cignoli R, de Gallego MS (1981) The lattice structure of some Łukasiewicz algebras. Algebra Universalis 13(3):315–328
Cignoli R, de Gallego MS (1983) Dualities for some De Morgan algebras with operators and Lukasiewicz algebras. Journal of the Australian Mathematical Society 34(3):377–393
Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning, vol 7. Trends in Logic-Studia Logica Library. Kluwer Academic Publishers, Dordrecht
Clark DM, Davey BA (1998) Natural dualities for the working algebraist, vol 57. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge
da Costa NCA (1963) Calculs propositionnels pour les systèmes formels inconsistants. Compte Rendu Acad. des Sciences (Paris) 257(3):3790–3793
Davey BA, Priestley HA (1990) Introduction to lattices and order. Cambridge University Press, Cambridge
Font JM (1997) Belnap’s four-valued logic and De Morgan lattices. Logic Journal of the I.G.P.L. 5(3):413–440
Font JM (2016) Abstract algebraic logic: An introductory textbook. College Publications
Font JM, Jansana R (2009) A general algebraic semantics for sentential logics, volume 7 of Lecture Notes in Logic. Springer-Verlag, second edition,
Marcelino S, Caleiro C (2019) Analytic calculi for monadic PNmatrices. International Workshop on Logic. Language, Information, and Computation (WoLLIC 2019). Springer, Berlin, Heidelberg, pp 84–98
Marcelino S, Caleiro C (2021) Axiomatizing non-deterministic many-valued generalized consequence relations. Synthese 198(22):5373–5390
Pietz A, Rivieccio U (2013) Nothing but the truth. Journal of Philosophical Logic 42(1):125–135
Přenosil A (2021) The lattice of super-Belnap logics. The Review of Symbolic Logic, 1–50, https://doi.org/10.1017/S1755020321000204
Rivieccio U (2012) An infinity of super-Belnap logics. Journal of Applied Non-Classical Logics 22(4):319–335
Shoesmith DJ, Smiley TJ (1978) Multiple-conclusion logic. Cambridge University Press, Cambridge
Wójcicki R (1988) Theory of logical calculi. Basic theory of consequence operations, volume 199 of Synthese Library. Reidel, Dordrecht
Acknowledgements
We would like to express our gratitude to Vítor Greati for a number of comments that helped us improve the initial version of the paper.
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Research funded by FCT/MCTES through national funds and when applicable co-funded by EU under the project UIDB/EEA/50008/2020 and by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil), under the grant 313643/2017-2 (Bolsas de Produtividade em Pesquisa - PQ).
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Marcelino, S., Rivieccio, U. Logics of involutive Stone algebras. Soft Comput 26, 3147–3160 (2022). https://doi.org/10.1007/s00500-022-06736-2
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DOI: https://doi.org/10.1007/s00500-022-06736-2