Showing papers on "Normal modal logic published in 1990"

Action versus State based Logics for Transition Systems

Abstract: A temporal logic based on actions rather than on states is presented and interpreted over labelled transition systems. It is proved that it has essentially the same power as CTL*, a temporal logic interpreted over Kripke structures. The relationship between the two logics is established by introducing two mappings from Kripke structures to labelled transition systems and viceversa and two transformation functions between the two logics which preserve truth. A branching time version of the action based logic is also introduced. This new logic for transition systems can play an important role as an intermediate between Hennessy-Milner Logic and the modal μ-calculus. It is sufficiently expressive to describe safety and liveness properties but permits model checking in linear time.

A Note on Boolean Modal Logic

Abstract: We present a proof of a theorem mentioned in an earlier paper “Modal environment for Boolean speculations”, devoted to the study of extended modal languages containing the so-called “window” or “sufficiency” modal operator m. The theorem states that a particular axiom system for the poly-modal logic encompassing union, intersection and complement of relations (a Boolean analog of the propositional dynamic logic of Pratt, Fischer, Ladner and Segerberg) is complete for the standard Kripke semantics. Moreover this system modally defines the standard semantics — so in the terminology of the present paper the axiomatics is adequate. On the other hand our logic has the finite model property. Thus a fragment of second order logic, rather powerful with respect to expressiveness, turns out to be decidable.

Half-order modal logic: how to prove real-time properties

Abstract: We introduce a novel extension of propositional modal logic that is interpreted over Kripke structures in which a value is associated with every possible woxld. These values are, however, not treated as full first-order objects; they can be accessed only by a very restricted form of quantification: the “freeze” quantifier binds a variable to the value of the current world. We present a complete proof system for this (“hulf_o4er”) modal logic. As a special case, we obtain the real-time temporal logic TPTL of [AH89]: the models are restricted to infinite sequences of states, whose values are monotonically increasing natural numbers. The ordering relation between states is interpreted as temporal precedence, while the value associated with a state is interpreted as its “real” time. We extend our proof system to be complete for TPTL, and demonstrate how it can be used to derive real-time properties.

Models for relevant modal logics

Abstract: Semantics are given for modal extensions of relevant logics based on the kind of frames introduced in [7]. By means of a simple recipe we may obtain from a class FRM (L) of unreduced frames characterising a (non-modal) logic L, frame-classes FRM□ (L.M) characterising conjunctively regular modal extensions L.M of L. By displaying an incompleteness phenomenon, it is shown how the recipe fails when reduced frames are under consideration.

Destructive Modal Resolution

Abstract: We present non-clausal resolution systems for propositional modal logics whose Kripke models do not involve symmetry, and for first order versions whose Kripke models do not involve constant domains. We give systems for K, T , K4 and S4; other logics are also possible. Our systems do not require preliminary reduction to a normal form and, in the first order case, intermingle resolution steps with Skolemization steps.

The Logic of Objective Knowledge and Rational Belief

Abstract: In this paper, we study the logical relations between different notions of knowledge and belief by means of generalizations of the usual Kripke models for epistemic logic. We argue that the obtained generalized Kripke models might be useful for carefully distinguishing the many different notions of knowledge and belief. We identify the (highly idealized) notions of objective knowledge and rational (introspective) belief, which correspond with fairly standard notions of knowledge and belief, and propose a system OKRIB for combining both notions which differs essentially from some other such proposals found in the literature. We also consider some other notions of knowledge and belief, and study how they relate to objective knowledge and rational belief.

Logics of Modal Terms for Systems Specification

Abstract: The use of modal qualification on terms is advocated for a more intuitive account of the description of the effects of events on objects such as program variables or database attributes, and also for an easier verification of the intended temporal integrity constraints. We develop two logics of modal terms focusing on positional and temporal qualification, and show by means of an example how they can be used to support the description and prescription of actions, as well as to reason about the properties of the specified systems.

Deontic Logic and Legal Knowledge Representation

Abstract: . The current literature in the Artificial Intelligence and Law field reveals uncertainty concerning the potential role of deontic logic in legal knowledge representation. For instance, the Logic Programming Group at Imperial College has shown that a good deal can be achieved in this area in the absence of explicit representation of the deontic notions. This paper argues that some rather ordinary parts of the law contain structures which, if they are to be represented in logic, will call for use of a reasonably sophisticated deontic logic.

Modal logics with n‐ary connectives

Abstract: The aim of this paper is to provide a general framework in order to present from a unitary point of view considerable experience in the field of non classical logics. Usual modal logic deals with a binary relation and a unary operator (generalizations to many binary relations and many unary operators are straightforward at least in the initial stage); although n-ary operators and n + 1-ary relations have been only partially investigated (see [9] where there is also a representation theorem) the experience of the semantics of relevant logics [25] shows how to manage the case of ternary relations.

The modal logic of pure provability.

Abstract: We introduce a propositional modal logic PP of «pure» provability in arbitrary theories (propositional or first-order) where the □ operator means «provable in all extensions». This modal logic has been considered in another guise by Kripke. An axiomatization and a decision procedure are given and the □= subtheory is characterized

Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics

Abstract: Shehtman and Skvortsov introduced Kripke bundles as semantics of non-classical first-order predicate logics. We show the structural equivalence between Kripke bundles for intermediate predicate logics and Kripke-type frames for intuitionistic modal prepositional logics. This equivalence enables us to develop the semantical study of relations between intermediate predicate logics and intuitionistic modal propositional logics. New examples of modal counterparts of intermediate predicate logics are given.

Natural deduction in normal modal logic.

Abstract: A natural deduction system for a wide range of normal modal logics is presented, which is based on Segerberg's idea that classical validity should be preserved «in any modal context». The resulting system has greater flexibility than the common Fitch-style systems

A four-valued modal logic arising from Monteiro's last algebras

Abstract: The class of abstract logics projectively generated by the class of logics defined on tetravalent modal algebras by the family of their filters is studied. These logics are four-valued in the sense that they can be characterized by a generalized matrix on the four-element tetravalent modal algebra which generates this variety together with a family of homomorphisms. They can be called modal since this four-element algebra can be given a nice epistemic interpretation as an extension of Belnap's four-valued logic. The authors also characterize them by their abstract properties and prove a completeness theorem with respect to a sequent calculus suggested by the abstract version. >

Modal logics of higher-order probability

Abstract: This paper discusses the relationship between probability and modal logic. We show that it is both natural and useful to think of probability as a modal operator. Contrary to popular belief in AI, a probability ranging between 0 and 1 represents a range between impossibility and necessity, not between simple falsity and truth. We examine two classes of probability models: flat and staged. The flat models are straightforward generalizations of models for alethic logic. We show that one of the more interesting constraints relating higher- and lower-order probabilities forces all higher-order probabilities in flat models to be either zero or one. We introduce staged models as a means of avoiding this problem. Constraints on the two types of models define various classes of probability logics. We relate some of these probability logics to alethic logic.

An Application of Many-Valued Logic to Decide Propositional S5 Formulae: A Strategy Designed for a Parameterized Tableaux-Based Theorem Prover.

Abstract: It is well known that for deciding on the validity of a S5 formula it is possible to bound the maximum number of worlds (say N) on which the formula must be tested to decide In doing modal logic with many-valued logic the number of truth-values to consider depends on N The idea of using a theorem prover parameterized by the set of truth-values is imperative We profit of possibilities offered by a parameterized theorem prover in order to use the information of a failure for a n-valued logic in the validity test for a 2xn-valued logic (corresponding to a change from m to m+1 worlds) This feature allows us to establish the main result of the paper: a strategy for doing modal logic with many-valued logics To do so, we simulate by a many-valued logic the truth values set of a modal formula in the different worlds Though the idea of doing modal logic with many-valued logic exists in the bibliography (see [HuC 68], [Res 69]) it has been used only in obtaining theoretical results We propose in this work a strategy allowing to mechanize effectively modal logic with many-valued logic Moreover this strategy furnish a decision procedure for S5 not needing transformation into a normal form (as in [HuC 68]) Our aim is not to compete with other ways of doing modal logic (ie with resolution or connexion method - work of Farinas del Cerro, Ohlbach, Wallen and others) but to use a parameterized many-valued tableaux based system to do also modal logic A m-valued logic theorem prover (parameterized by m) based on these ideas and result has been implemented on a SUN-3 workstation Some running examples are shown

Modal counterparts of Medvedev logic of finite problems are not finitely axiomatizable

Abstract: We consider modal logics whose intermediate fragments lie between the logic of infinite problems [20] and the Medvedev logic of finite problems [15]. There is continuum of such logics [19]. We prove that none of them is finitely axiomatizable. The proof is based on methods from [12] and makes use of some graph-theoretic constructions (operations on coverings, and colourings).

Conversion principles and the basis of aristotle's modal logic

Abstract: Aristotle founds his modal syllogistic, like his plain syllogistic, on a small set of ‘perfect’ or obviously valid sylligisms. The rest he reduces to those, usually by means of modal conversion principles. These principles are open to more than one reading, however, and they are in fact invalid on one traditional reading (de re), valid on the other (de dicto). It is argued here that this way of framing the contrast is not Aristotelian, and that an interpretation involving modal copulae allows us to see how these principles, and the modal system as a whole, are to be understood in light of close and precise connections to Aristotle's essentialist metaphysics.

Expressive power and semantic completeness: Boolean connectives in modal logic

Abstract: We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of ‘dense’ Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective).

The ontological foundation of Russell's theory of modality

Abstract: Prominent thinkers such as Kripke and Rescher hold that Russell has no modal logic, even that Russell was indisposed toward modal logic. In Part I, I show that Russell had a modal logic which he repeatedly described and that Russell repeatedly endorsed Leibniz's multiplicity of possible worlds. In Part II, I describe Russell's theory as having three ontological levels. In Part III, I describe six Parmenidean theories of being Russell held, including: literal in 1903; universal in 1912; timeless in 1914; transcendental in 1918–1948. The transcendental theory underlies the primary level of Russell's modal logic. In Part IV, I examine Rescher's view that Russell and modal logic did not mix.

On Halldén-completness of intermediate and modal logics

Abstract: A logic L is said to be Hallden-complete (or Hallden-reasonable) if for any formula A ∨B provable in L, where A and B have no variables in common, L ` A or L ` B. A. Wronski [4] has obtained the algebraic equivalents of Halldencompleteness for intermediate and modal logics. J. van Benthem and I. Humberstone [2] have given in semantic terms a sufficient condition for Hallden-completeness in normal modal logics; it is unknown whether the condition is necessary. In this paper we deal with the problem of deciding, given an axiomatization of intermediate logic or normal modal logic containing S4, whether the logic is Hallden-complete. Recently we have shown [1] that the disjunction property of intermediate logics is undecidable. (Recall that an intermediate logic L is said to have disjunction property if L ` A or L ` B whenever L ` A∨B; the definition of disjunction property for modal logics is as follows: L ` A ∨B ⇒ L ` A or L ` B.) It is obvious that for intermediate logics the disjunction property implies Hallden-completeness (compare with Theorem 6 below). With the help of this fact we have proved in [1] the following