Reasoning with logical bilattices

Arieli, Ofer; Avron, Arnon

doi:10.1007/bf00215626

Cited by 213 publications

(217 citation statements)

References 23 publications

Supporting

Mentioning

215

Contrasting

Unclassified

Order By: Relevance

“…The non-modal fragment of the logics we are going to study is the multi-valued logic of Arieli and Avron [1] (henceforth referred to as bilattice logic or Arieli-Avron logic), which arose from Belnap's famous proposal for a "useful four-valued logic" [3]. Belnap saw that his four truth values can be arranged as lattices in two distinct but equally meaningful ways, ordering them either by information content (the knowledge order ≤ ) or by logical strength (the truth order ≤ ), see Generally, a (bounded) bilattice is an algebra ⟨ , ∧, ∨, ⊗, ⊕, ¬, f, t, ⊥, ⊤⟩ such that ⟨ , ∧, ∨, f, t⟩ and ⟨ , ⊗, ⊕, ⊥, ⊤⟩ are both bounded lattices.…”

Section: Algebraic Preliminariesmentioning

confidence: 99%

“…Just as the notion of (ultra)filter is fundamental in the algebraic study of classical logic, a key notion for the logical study of bilattices is that of bifilter [1,Definition 2.13]. A bifilter of a bilattice B is a non-empty set ⊆ that is a lattice filter with respect to both lattice orders.…”

Section: Algebraic Preliminariesmentioning

confidence: 99%

“…Our logic is four-valued and thus in the tradition of bilattice logic begun by Belnap [2], [3] and subsequently greatly extended by Arieli and Avron [1] and others. The additional values besides the classical t ("true") and f ("false") are ⊥ (for "unknown") and ⊤ (for "contradiction") with obvious interpretations and applications in a computer science setting.…”

Section: Introductionmentioning

confidence: 99%

“…This provides the background for our discussion of four-valued Kripke structures in the following section and the evaluation of the □-operator in particular. In the second part of Section III we present a Hilbert-style axiomatisation of modal bilattice logic, extending the one given by Arieli and Avron in [1] for the non-modal part. Both local and global consequence are covered.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Kripke Semantics for Modal Bilattice Logic

Jung

Rivieccio

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

Abstract-We employ the well-developed and powerful techniques of algebraic semantics and Priestley duality to set up a Kripke semantics for a modal expansion of Arieli and Avron's bilattice logic, itself based on Belnap's four-valued logic. We obtain soundness and completeness of a Hilbert-style derivation system for this logic with respect to four-valued Kripke frames, the standard notion of model in this setting. The proof is via intermediary relational structures which are analysed through a topological reading of one of the axioms of the logic. Both local and global consequence on the models are covered.

show abstract

Section: Algebraic Preliminariesmentioning

confidence: 99%

Section: Algebraic Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Kripke Semantics for Modal Bilattice Logic

Jung

Rivieccio

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

View full text Add to dashboard Cite

show abstract

“…In the literature, these structures are commonly referred to as F OUR (after Belnap's [7,8] original four-valued logic) and N IN E (see e.g. [1,2]), respectively. An example of a square with an infinite amount of elements is ([0, 1], ≤) 2 .…”

Section: Squaresmentioning

confidence: 99%

Preference Modeling by Rectangular Bilattices

Arieli

Cornelis

Deschrijver

2006

Modeling Decisions for Artificial Intelligence

Self Cite

View full text Add to dashboard Cite

Abstract. Many realistic decision aid problems are fraught with facets of ambiguity, uncertainty and conflict, which hamper the effectiveness of conventional and fuzzy preference modeling approaches, and command the use of more expressive representations. In the past, some authors have already identified Ginsberg's/Fitting's theory of bilattices as a naturally attractive candidate framework for representing uncertain and potentially conflicting preferences, yet none of the existing approaches addresses the real expressive power of bilattices, which lies hidden in their associated truth and knowledge orders. As a consequence, these approaches have to incorporate additional conventions and 'tricks' into their modus operandi, making the results unintuitive and/or tedious. By contrast, the aim of this paper is to demonstrate the potential of (rectangular) bilattices in encoding not just the problem statement, but also its generic solution strategy.

show abstract

Bipolarity in bilattice logics

Konieczny

Marquis

Besnard

2008

Int. J. Intell. Syst.

View full text Add to dashboard Cite

This paper is centered on a family of propositional multivalued logics, based on bilattices. The semantics of such logics relies on a set of "truth values," with two orderings that give the set a bilattice structure. Many interesting inference relations can be defined on these grounds, especially paraconsistent ones and/or nonmonotonic ones. The focus is laid on Belnap's fundamental bilattice logic FOUR, with four "epistemic truth values," which proves sufficient for the purpose of inference. We show how the bilattice can be associated with a second biordinal structure, which no longer is bilatticial but bipolar. We show how additional inference relations in the logic FOUR can be obtained by exploiting the two preorders associated with this structure. C 2008 Wiley Periodicals, Inc.

show abstract

Reasoning with logical bilattices

Cited by 213 publications

References 23 publications

Kripke Semantics for Modal Bilattice Logic

Kripke Semantics for Modal Bilattice Logic

Preference Modeling by Rectangular Bilattices

Bipolarity in bilattice logics

Contact Info

Product

Resources

About