Showing papers on "Normal modal logic published in 1988"

A Resolution Calculus for Modal Logics

Abstract: A syntax transformation is presented that eliminates the modal logic operators from modal logic formulae by shifting the modal context information to the term level. The formulae in the transformed syntax can be brought into conjunctive normal form such that a clause based resolution calculus without any additional inference rule, but with special modal unification algorithms, can be defined. The method works for first-order modal logics with the two operators □ and ♦ and with constant-domain Kripke semantics where the accessibility relation is serial and may have any combination of the following properties: reflexivity, symmetry, transitivity. In particular the quantified versions of the modal systems T, S4, S5, B, D, D4 and DB can be treated. Extensions to non-serial and varying-domain systems are possible, but not presented here.

First-order modal tableaux

Abstract: We describe simple semantic tableau based theorem provers for four standard modal logics, in both propositional and first-order versions. These theorem provers are easy to implement in Prolog, have a behavior that is straightforward to understand, and provide natural places for the incorporation of heuristics.

Necessity and contingency

Abstract: The paper considers the question of when the operator L of necessity in modal logic can be expressed in terms of the operator Δ meaning ‘it is non-contingent that’.

Normal multimodal logics

Abstract: This paper studies what we call normal multimodal logics, which are general modal systems with an arbitrary set of normal modal operators. We emphasize the importance of non-simple systems, for which some interaction axioms are considered. A list of such acceptable axioms is proposed, among which the induction axiom has a special behavior. The class of multimodal logics that can be built with these axioms generalizes many existing modal, temporal, dynamic and epistemic systems, and could also suggest new formalizations using modal logics. The main result is a general determination theorem for these multimodal systems, which establishes a correspondence between our axioms and conditions over Kripke frames; this should avoid the need for showing determination each time a new system is considered.

Graded modalities, II (canonical models)

Abstract: This work intends to be a generalization and a simplification of the techniques employed in [2], by the proposal of a general strategy to prove satisfiability theorems for NLGM-s (= normal logics with graded modalities), analogously to the well known technique of the canonical models by Lemmon and Scott for classical modal logics.

Linear Modal Deductions

Abstract: We present a deduction method for propositional modal logics. It is based on a resolution principle for formulas written in a very simple normal form, close to the clausal form for classical logic. It allows us to extend naturally resolution and refinements of resolution to modal logics.

Logic in Law: Remarks on Logic and Rationality in Normative Reasoning, Especially in Law

Abstract: I Logic.- II Normative Judgements.- III the Possibility of Deontic Logic.- IV Prolegomena for a Deontic Logic.- V A Standard System of Deontic Logic.- VI The Norm-Content of the Standard System.- VII The Negation of Normative Expressions: Weak and Strong Permission, Particularly in Law.- VIII Conditional Norms.- IX The Meaning of Logic for Normative Reasoning.- Notes.- Index of names.- Index of subjects.- A few of the used concepts.

A general proof method for modal predicate logic without the Barcan formula

Abstract: We present a general proof method for normal systems of modal predicate logic with identical inference rules for each such logic. Different systems are obtained by changing the conditions under which two formulas are considered complementary. The paper extends previous work in that we are no longer confined to models in which the Barcan formula and its converse hold. This allows the domain of individuals to vary from world to world. Modifications to the original inference rules are given, and a semantic justification is provided.

On the Lattice of Extensions of the Modal Logics KAltn

Abstract: Some results concerning the lattice of normal modal logics A(K) (see for instance [B1, B2]) show the extreme complexity of this structure and the consequent impossibility of a complete description of it. On the other hand, many other results which appear in the literature give descriptions of significative parts of A(K) (see [B 3, Ma, Ra]). In our paper we concentrate the attention on the lattice of the extensions of the logics KAltn, originally introduced in [-Se]. We recall that KAIt. is the normal modal propositional logic characterized by the axiom

Circumscription in a Modal Logic

Abstract: In this paper, we extend circumscription [McCarthy, 80] to a propositional modal logic of knowledge of one agent. Instead of circumscribing a predicate, we circumscribe the knowledge operator "K" in a formula. In order to have a nontrivial circumscription schema, we extend S5 modal logic of knowledge by adding another modality "V al" and a universal quantifier over base sentences (sentences which do not contain modality). Intuitively, "V al(P)" means that P is a valid formula. It turns out that by circumscribing the knowledge operator in a formula, we completely characterize the maximally ignorant models of the formula (models of the formula where agents have minimal knowledge).

Exploring the Epistemic Labyrinth: New Directions in the Formal Theory of Knowledge Representation

Abstract: Recently enormous interest has been generated in what researchers in AI and Computer Science have termed the formal theory of knowledge representation or what is known to logicians as epistemic logic. New life has been breathed into this field as the means and motivation for the computational investigation of these formal theories is now at hand. Much of this work has arisen from the epistemic interpretation of normal modal logics. However as vehicles for knowledge representation it now seems clear that these systems are completely inadequate. In this paper we investigate the alternative posed by the non-normal modal logics and discuss techniques for constructing efficient automated theorem provers for these systems.

Intuitionist Logic-Subsystem of, Extension of, or Rival to, Classical Logic?

Abstract: Strictly speaking, intuitionistic logic is not a modal logic. There are, after all, no modal operators in the language. It is a subsystem of classical logic, not [like modal logic] an extension of it. But... (thus Fitting, p. 437, trying to justify inclusion of a large chapter on intuitionist logic ‘in a book that is largely about modal logics’).

Thresholds for Certainty and the Modal Logic S3

Abstract: This presents logics which use thresholds for certainty. These logics are related to the modal logic S3. The logic U is a weak logic of this kind. The logic UC is a strong logic of this kind. The logic UC has linear matrix, and hence is of interest in many applications, including fuzzy systems.