114 research outputs found
An infinity of super-Belnap logics
We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new log- ics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable
Usuzování s nekonzistentními informacemi
Tato dizertační práce studuje extenze čtyřhodnotové Belnapovy-Dunnovy logiky, tzv. superbelnapovské logiky, z pohledu abstraktní algebraické logiky. Popisujeme v ní globální strukturu svazu superbelnapovských logik a ukazu- jeme, že tento svaz lze zcela popsat pomocí tříd konečných grafů splňujících jisté uzávěrové podmínky. Také zde zavádíme teorii tzv. explozivních extenzí a používáme ji k důkazu nových vět o úplnosti pro superbelnapovské logiky. Poté rozvíjeme gentzenovskou teorii důkazů pro tyto logiky a použijeme ji k důkazu věty o interpolaci pro mnoho z těchto logik. Nakonec také studujeme rozšíření Belnapovy-Dunnovy logiky o operátor pravdivosti ∆. Klíčová slova: abstraktní algebraická logika, Belnapova-Dunnova logika, parakonzistentní logika, superbelnapovské logikyThis thesis studies the extensions of the four-valued Belnap-Dunn logic, called super-Belnap logics, from the point of view of abstract algebraic logic. We describe the global structure of the lattice of super-Belnap logics and show that this lattice can be fully described in terms of classes of finite graphs satisfying some closure conditions. We also introduce a theory of so- called explosive extensions and use it to prove new completeness theorems for super-Belnap logics. A Gentzen-style proof theory for these logics is then developed and used to establish interpolation for many of them. Finally, we also study the expansion of the Belnap-Dunn logic by the truth operator ∆. Keywords: abstract algebraic logic, Belnap-Dunn logic, paraconsistent logic, super-Belnap logicsKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult
Adding an Implication to Logics of Perfect Paradefinite Algebras
Perfect paradefinite algebras are De Morgan algebras expanded with a
perfection (or classicality) operation. They form a variety that is
term-equivalent to the variety of involutive Stone algebras. Their associated
multiple-conclusion (Set-Set) and single-conclusion (Set-Fmla) order-preserving
logics are non-algebraizable self-extensional logics of formal inconsistency
and undeterminedness determined by a six-valued matrix, studied in depth by
Gomes et al. (2022) from both the algebraic and the proof-theoretical
perspectives. We continue hereby that study by investigating directions for
conservatively expanding these logics with an implication connective
(essentially, one that admits the deduction-detachment theorem). We first
consider logics given by very simple and manageable non-deterministic semantics
whose implication (in isolation) is classical. These, nevertheless, fail to be
self-extensional. We then consider the implication realized by the relative
pseudo-complement over the six-valued perfect paradefinite algebra. Our
strategy is to expand such algebra with this connective and study the
(self-extensional) Set-Set and Set-Fmla order-preserving logics, as well as the
T-assertional logics of the variety induced by the new algebra. We provide
axiomatizations for such new variety and for such logics, drawing parallels
with the class of symmetric Heyting algebras and with Moisil's `symmetric modal
logic'. For the Set-Set logic, in particular, the axiomatization we obtain is
analytic. We close by studying interpolation properties for these logics and
concluding that the new variety has the Maehara amalgamation property
Semantic inconsistency measures using 3-valued logics
AI systems often need to deal with inconsistencies. One way of getting information about inconsistencies is by measuring the amount of information in the knowledgebase. In the past 20 years numerous inconsistency measures have been proposed. Many of these measures are syntactic measures, that is, they are based in some way on the minimal inconsistent subsets of the knowledgebase. Very little attention has been given to semantic inconsistency measures, that is, ones that are based on the models of the knowledgebase where the notion of a model is generalized to allow an atom to be assigned a truth value that denotes contradiction. In fact, only one nontrivial semantic inconsistency measure, the contension measure, has been in wide use. The purpose of this paper is to define a class of semantic inconsistency measures based on 3-valued logics. First, we show which 3-valued logics are useful for this purpose. Then we show that the class of semantic inconsistency measures can be developed using a graphical framework similar to the way that syntactic inconsistency measures have been studied. We give several examples of semantic inconsistency measures and show how they apply to three useful 3-valued logics. We also investigate the properties of these inconsistency measures and show their computation for several knowledgebases
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