Back to Search
Start Over
Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations
- Source :
- Bulletin of the Section of Logic, Vol 49, Iss 4, Pp 401-437 (2020)
- Publication Year :
- 2020
- Publisher :
- Uniwersytet Lodzki (University of Lodz), 2020.
-
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.
- Subjects :
- Pure mathematics
Logic
Distributive lattice
0102 computer and information sciences
propositional logic
01 natural sciences
expansion
[bounded] distributive/de morgan lattice
Computer Science::Logic in Computer Science
dunn-belnap's logic
Logical matrix
lcsh:BC1-199
0101 mathematics
Axiom
Mathematics
logical matrix
010102 general mathematics
Subalgebra
Block matrix
Extension (predicate logic)
lcsh:Logic
Propositional calculus
equality determinant
Philosophy
010201 computation theory & mathematics
Isomorphism
Subjects
Details
- ISSN :
- 2449836X and 01380680
- Volume :
- 49
- Database :
- OpenAIRE
- Journal :
- Bulletin of the Section of Logic
- Accession number :
- edsair.doi.dedup.....0c0bafba6f6b3f4cdccb5710c3bb3a95