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2020 | 49 | 4 | 401-437
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Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations

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Basic results of the paper are that any four-valued expansion L4 of Dunn-Belnap's logic DB4 is de_ned by a unique (up to isomorphism) conjunctive matrix ℳ4 with exactly two distinguished values over an expansion 𝔄4 of a De Morgan non-Boolean four-valued diamond, but by no matrix with either less than four values or a single [non-]distinguished value, and has no proper extension satisfying Variable Sharing Property (VSP). We then characterize L4's having a theorem / inconsistent formula, satisfying VSP and being [inferentially] maximal / subclassical / maximally paraconsistent, in particular, algebraically through ℳ4|𝔄4's (not) having certain submatrices|subalebras. Likewise, [providing 𝔄4 is regular / has no three-element subalgebra] L4 has a proper consistent axiomatic extension if[f] ℳ4 has a proper paraconsistent / two-valued submatrix [in which case the logic of this submatrix is the only proper consistent axiomatic extension of L4 and is relatively axiomatized by the Excluded Middle law axiom]. As a generic tool (applicable, in particular, to both classically-negative and implicative expansions of DB4), we also prove that the lattice of axiomatic extensions of the logic of an implicative matrix ℳ with equality determinant is dual to the distributive lattice of lower cones of the set of all submatrices of ℳ with non-distinguished values.
Rocznik
Tom
49
Numer
4
Strony
401-437
Opis fizyczny
Daty
wydano
2020-12-30
Twórcy
  • National Academy of Sciences of Ukraine V.M. Glushkov Institute of Cybernetics Department of Digital Automata Theory (100) Glushkov prosp. 40 Kiev, 03680, Ukraine
Bibliografia
  • [1] A. R. Anderson, N. D. Belnap, Entailment, vol. 1, Princeton University Press, Princeton (1975).
  • [2] R. Balbes, P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia (Missouri) (1974).
  • [3] N. D. Belnap, A useful four-valued logic, [in:] J. M. Dunn, G. Epstein (eds.), Modern uses of multiple-valued logic, D. Reidel Publishing Company, Dordrecht (1977), pp. 8–37, DOI: http://dx.doi.org/10.1007/978-94-010-1161-7_2
  • [4] J. M. Dunn, Algebraic completeness results for R-mingle and its extensions, Journal of Symbolic Logic, vol. 35 (1970), pp. 1–13, URL: https://projecteuclid.org/euclid.jsl/1183737028
  • [5] J. M. Dunn, Intuitive semantics for first-order-degree entailment and `coupled tree', Philosophical Studies, vol. 29 (1976), pp. 149–168, DOI: http://dx.doi.org/10.1007/978-3-030-31136-0_3
  • [6] J. Łoś, R. Suszko, Remarks on sentential logics, Indagationes Mathematicae, vol. 20 (1958), pp. 177–183, DOI: http://dx.doi.org/10.1016/S1385-7258(58)50024-9
  • [7] A. I. Mal'cev, Algebraic systems, Springer Verlag, New York (1965), DOI: http://dx.doi.org/10.1007/978-3-642-65374-2
  • [8] G. Priest, The logic of paradox, Journal of Philosophical Logic, vol. 8 (1979), pp. 219–241, DOI: http://dx.doi.org/10.1007/BF00258428
  • [9] A. P. Pynko, Characterizing Belnap's logic via De Morgan's laws, Mathematical Logic Quarterly, vol. 41(4) (1995), pp. 442–454, DOI: http://dx.doi.org/10.1002/malq.19950410403
  • [10] A. P. Pynko, On Priest's logic of paradox, Journal of Applied Non-Classical Logics, vol. 5(2) (1995), pp. 219–225, DOI: http://dx.doi.org/10.1080/11663081.1995.10510856
  • [11] A. P. Pynko, Functional completeness and axiomatizability within Belnap's four-valued logic and its expansions, Journal of Applied Non-Classical Logics, vol. 9(1/2) (1999), pp. 61–105, DOI: http://dx.doi.org/10.1080/11663081.1999.10510958 special Issue on Multi-Valued Logics.
  • [12] A. P. Pynko, Subprevarieties versus extensions. Application to the logic of paradox, Journal of Symbolic Logic, vol. 65(2) (2000), pp. 756–766, URL: https://projecteuclid.org/euclid.jsl/1183746075
  • [13] A. P. Pynko, A relative interpolation theorem for infinitary universal Horn logic and its applications, Archive for Mathematical Logic, vol. 45 (2006), pp. 267–305, DOI: http://dx.doi.org/10.1007/s00153-005-0302-2
  • [14] A. P. Pynko, Subquasivarieties of implicative locally-nite quasivarieties, Mathematical Logic Quarterly, vol. 56(6) (2010), pp. 643–658, DOI: http://dx.doi.org/10.1002/malq.200810161
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Bibliografia
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