Abstract
Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled How a computer should think. Belnap argued that instead of using a logic with two values, for ‘true’ (\(\varvec{t}\)) and ‘false’ (\(\varvec{f}\)), a computer should use a logic with two further values, for ‘contradiction’ (\(\top \)) and ‘no information’ (\(\bot \)). The resulting structure is equipped with two lattice orders, a knowledge order and a truth order, and hence is called a bilattice.
Prioritised default bilattices include not only values for ‘true’ (\(\varvec{t}_0\)), ‘false’ (\(\varvec{f}_0\)), ‘contradiction’ and ‘no information’, but also indexed families of default values, \(\varvec{t}_1, \dots , \varvec{t}_n\) and \(\varvec{f}_1, \dots , \varvec{f}_n\), for simultaneous modelling of degrees of knowledge and truth.
We focus on a new family of prioritised default bilattices: \(\mathbf {J}_n\), for \(n \in \omega \). The bilattice \(\mathbf {J}_0\) is precisely Belnap’s seminal example. We obtain a multi-sorted duality for the variety generated by \(\mathbf {J}_n\), and separately a single-sorted duality for the quasivariety generated by \(\mathbf {J}_n\). The main tool for both dualities is a unified approach that enables us to identify the meet-irreducible elements of the appropriate subuniverse lattices. Our results provide an interesting example where the multi-sorted duality for the variety has a simpler structure than the single-sorted duality for the quasivariety.
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References
Belnap, N.D.: How a computer should think. In: Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press Ltd. (1977)
Cabrer, L.M., Craig, A.P.K., Priestley, H.A.: Product representation for default bilattices: an application of natural duality theory. J. Pure Appl. Algebra 219, 2962–2988 (2015)
Cabrer, L.M., Priestley, H.A.: Distributive bilattices from the perspective of natural duality. Algebra Universalis 73, 103–141 (2015)
Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)
Craig, A.P.K.: Canonical extensions of bounded lattices and natural duality for default bilattices. DPhil thesis, University of Oxford (2012)
Craig, A.P.K., Davey, B.A., Haviar, M.: Expanding Belnap: dualities for a new class of default bilattices. arXiv:1808.09636v2 [math.LO]
Davey, B.A.: Weak injectivity and congruence extension in congruence-distributive equational classes. Can. J. Math. 24, 449–459 (1977)
Davey, B.A.: The product representation theorem for interlaced pre-bilattices: some historical remarks. Algebra Universalis 70, 403–409 (2013)
Davey, B.A., Haviar, M., Priestley, H.A.: The syntax and semantics of entailment in duality theory. J. Symb. Logic 60, 1087–1114 (1995)
Davey, B.A., Priestley, H.A.: Generalized piggyback dualities and applications to Ockham algebras. Houst. J. Math. 13, 101–117 (1987)
Davey, B.A., Talukder, M.R.: Functor category dualities for varieties of Heyting algebras. J. Pure Appl. Algebra 178, 49–71 (2003)
Fitting, M.: Bilattices in logic programming. In: Proceedings of the 20th International symposium on multiple-valued logic, pp. 238–246. IEEE, New York (1990)
Fitting, M.: Bilattices are nice things. In: Self-reference, pp. 53–77. CSLI Publ., Stanford (2006)
Ginsberg, M.L.: Multi-valued logics. In: Proceedings of the 5th National Conference on Artificial Intelligence, pp. 243–249. Morgan Kaufmann (1986)
Ginsberg, M.L.: Multivalued logics: a uniform approach to reasoning in artificial intelligence. Comput. Intell. 4, 265–316 (1988)
Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)
Jung, A., Rivieccio, U.: Priestley duality for bilattices. Studia Logica 100, 223–252 (2012)
Mobasher, B., Pigozzi, D., Slutzki, G., Voutsadakis, G.: A duality theory for bilattices. Algebra Universalis 43, 109–125 (2000)
Rivieccio, U.: An algebraic study of bilattice-based logics. PhD thesis, Universitat de Barcelona (2010)
Taylor, W.: Residually small varieties. Algebra Universalis 2, 33–53 (1972)
Acknowledgements
The authors would like to thank Jane Pitkethly for carefully preparing their diagrams in TikZ. The first and second author would like to thank Matej Bel University for its hospitality during a research visit in September 2017. The first author would like to thank Hilary Priestley and Leonardo Cabrer for useful discussions and guidance during his DPhil studies at the University of Oxford when the first work on these bilattices took place [5]. The third author acknowledges the hospitality of La Trobe University during his stay there in August 2018.
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Presented by R. Willard.
Dedicated to the memory of Prof. Beloslav Riečan.
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The first author acknowledges the support of the NRF South Africa (grant 127266) and the third author acknowledges the support of Slovak grant VEGA 1/0337/16.
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Craig, A.P.K., Davey, B.A. & Haviar, M. Expanding Belnap: dualities for a new class of default bilattices. Algebra Univers. 81, 50 (2020). https://doi.org/10.1007/s00012-020-00678-2
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DOI: https://doi.org/10.1007/s00012-020-00678-2