Modal Logic

Blackburn, Patrick; Rijke, Maarten de; Venema, Yde

doi:10.1017/cbo9781107050884

Cited by 1,686 publications

(1,284 citation statements)

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“…To prove it, we first establish a bounded model property using a 'selective filtration'-style argument, see e.g. Blackburn et al (2001). The difference to the bounded model property proof given above for concept-circumscribed KBs is that, here, we do not build a quotient model of a given model by identifying nodes using an equivalence relation, but construct a submodel of a given model by selecting relevant nodes.…”

Section: Role-minimizing Ckbs With Empty Tbox In Alcqomentioning

confidence: 99%

The Complexity of Circumscription in DLs

Bonatti¹,

Lutz²,

Wolter³

2009

jair

100

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As fragments of first-order logic, Description logics (DLs) do not provide nonmonotonic features such as defeasible inheritance and default rules. Since many applications would benefit from the availability of such features, several families of nonmonotonic DLs have been developed that are mostly based on default logic and autoepistemic logic. In this paper, we consider circumscription as an interesting alternative approach to nonmonotonic DLs that, in particular, supports defeasible inheritance in a natural way. We study DLs extended with circumscription under different language restrictions and under different constraints on the sets of minimized, fixed, and varying predicates, and pinpoint the exact computational complexity of reasoning for DLs ranging from ALC to ALCIO and ALCQO. When the minimized and fixed predicates include only concept names but no role names, then reasoning is complete for NExp NP . It becomes complete for NP NExp when the number of minimized and fixed predicates is bounded by a constant. If roles can be minimized or fixed, then complexity ranges from NExp NP to undecidability.

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Section: Role-minimizing Ckbs With Empty Tbox In Alcqomentioning

confidence: 99%

The Complexity of Circumscription in DLs

Bonatti¹,

Lutz²,

Wolter³

2009

jair

100

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“…From a semantic point of view, these axioms correspond to geometric frame conditions that must be satisfied by the accessibility relation in Kripke models (see e.g. Fitting & Mendelsohn 1998or Blackburn & al. 2001, Section 3.1 on Frame definability): T corresponds to reflexivity, B corresponds to symmetry, 4 corresponds to transitivity and 5 corresponds to euclidianity.…”

Section: Some Useful Derivable Principlesmentioning

confidence: 99%

Variants of Gödel’s Ontological Proof in a Natural Deduction Calculus

Kanckos

Paleo

2017

Stud Logica

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This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott's manuscript about Gödel's argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic KB instead of S5 and by avoiding the equality relation. The second proof is also technically shorter than the first one, because it eliminates unnecessary detours and uses Axiom 1 for the positivity of properties only once. The third and fourth proofs formalize, respectively, Anderson's and Bjørdal's variants of the ontological argument, which are known to be immune to modal collapse.

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“…Conversely, given a finitary L, we obtain Σ and E as follows [24, 1.5]. For Σ we let each element in Ln be an n-ary operation, 4 that is,…”

Section: Proposition 13 a Functor On Set Has A Finite Presentation Imentioning

confidence: 99%

Presenting Functors by Operations and Equations

Bonsangue

Kurz

2006

Lecture Notes in Computer Science

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Abstract. We take the point of view that, if transition systems are coalgebras for a functor T, then an adequate logic for these transition systems should arise from the 'Stone dual' L of T. We show that such a functor always gives rise to an 'abstract' adequate logic for T-coalgebras and investigate under which circumstances it gives rise to a 'concrete' such logic, that is, a logic with an inductively defined syntax and proof system. We obtain a result that allows us to prove adequateness of logics uniformly for a large number of different types of transition systems and give some examples of its usefulness.

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Modal Logic

Cited by 1,686 publications

References 1 publication

The Complexity of Circumscription in DLs

The Complexity of Circumscription in DLs

Variants of Gödel’s Ontological Proof in a Natural Deduction Calculus

Presenting Functors by Operations and Equations

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