The lattice of super-Belnap logics
| Autor: | Adam Přenosil |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Quasivariety Logic Continuum (topology) 010102 general mathematics 06D30 03G27 Graph theory Mathematics - Logic 06 humanities and the arts 0603 philosophy ethics and religion 01 natural sciences Philosophy Lattice (module) Mathematics (miscellaneous) Completeness (logic) Computer Science::Logic in Computer Science 060302 philosophy FOS: Mathematics Finitary Homomorphism Isomorphism 0101 mathematics Logic (math.LO) Mathematics |
| DOI: | 10.48550/arxiv.2111.09818 |
| Popis: | We study the lattice of extensions of four-valued Belnap--Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap--Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap--Dunn logic is iso\-morphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. Comment: 50 pages, 7 figures |
| Databáze: | OpenAIRE |
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