Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations

Alexej P. Pynko

Bulletin of the Section of Logic (2020)

  • Volume: 49, Issue: 4, page 401-437
  • ISSN: 0138-0680

Abstract

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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.

How to cite

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Alexej P. Pynko. "Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations." Bulletin of the Section of Logic 49.4 (2020): 401-437. <http://eudml.org/doc/296784>.

@article{AlexejP2020,
abstract = {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.},
author = {Alexej P. Pynko},
journal = {Bulletin of the Section of Logic},
keywords = {propositional logic; logical matrix; Dunn-Belnap's logic; expansion; [bounded] distributive/De Morgan lattice; equality determinant},
language = {eng},
number = {4},
pages = {401-437},
title = {Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations},
url = {http://eudml.org/doc/296784},
volume = {49},
year = {2020},
}

TY - JOUR
AU - Alexej P. Pynko
TI - Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations
JO - Bulletin of the Section of Logic
PY - 2020
VL - 49
IS - 4
SP - 401
EP - 437
AB - 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.
LA - eng
KW - propositional logic; logical matrix; Dunn-Belnap's logic; expansion; [bounded] distributive/De Morgan lattice; equality determinant
UR - http://eudml.org/doc/296784
ER -

References

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