Showing papers on "Normal modal logic published in 1987"

Epistemic logics, probability, and the calculus of evidence

Abstract: This paper presents results of the application to epistemic logic structures of the method proposed by Carnap for the development of logical foundations of probability theory. These results, which provide firm conceptual bases for the Dempster-Shafer calculus of evidence, are derived by exclusively using basic concepts from probability and modal logic theories, without resorting to any other theoretical notions or structures. A form of epistemic logic (equivalent in power to the modal system S5), is used to define a space of possible worlds or states of affairs. This space, called the epistemic universe, consists of all possible combined descriptions of the state of the real world and of the state of knowledge that certain rational agents have about it. These representations generalize those derived by Carnap, which were confined exclusively to descriptions of possible states of the real world. Probabilities defined on certain classes of sets of this universe, representing different states of knowledge about the world, have the properties of the major functions of the Dempster-Shafer calculus of evidence: belief functions and mass assignments. The importance of these epistemic probabilities lies in their ability to represent the effect of uncertain evidence in the states of knowledge of rational agents. Furthermore, if an epistemic probability is extended to a probability function defined over subsets of the epistemic universe that represent true states of the real world, then any such extension must satisfy the well-known interval bounds derived from the Dempster-Shafer theory. Application of this logic-based approach to problems of knowledge integration results in a general expression, called the additive combination formula, which can be applied to a wide variety of problems of integration of dependent and independent knowledge. Under assumptions of probabilistic independence this formula is equivalent to Dempster's rule of combination.

Matrix proof methods for modal logics

Abstract: We present matrix proof systems for both constant- and varying-domain versions of the first-order modal logics K, K4, D, D4, T, 84 and 86 based on modal versions of Herbrand's Theorem specifically formulated to support efficient automated proof search. The systems treat the mil modal language (no normal-forming) and admit straightforward structure sharing implementations. A key fsature of our approach is the use of a specialised unification algorithm to reflect the conditions on the accessibility relation for a given logic. The matrix system for one logic differs from the matrix eystem for another only in the nature of this unification algorithm. In addition, proof search may be interpreted as constructing generalised proof trees in an appropriate tableau- or sequent-based proof system. This facilitates the use of the matrix systems within interactive environments.

A note on the complexity of the satisfiability of modal Horn clauses

Abstract: A bottom-up algorithm is given for testing the satisfiability of sets of propositional modal Horn clauses. For some modal logics, such as S5, the algorithm works in polynomial time, while for other modal logics, like K, Q, T, and S4, the worst case complexity can be considerably higher.

A Modal Logic for a Subclass of Event Structures

Abstract: This paper introduces a non-interleaved model for the behaviour of distributed computing systems, and an accompanying temporal logic with an explicit treatment of concurrency (based on a notion of local rather than global states). A subclass of event structures (called n-agent event structures) is used as the underlying model -- intended to describe the computational behaviour of n communicating, sequential (and possibly non-deterministic) agents. The logic is centered around indexed modalities to describe the states of knowledge of the individual agents during such a computation. An axiom system for the logic is presented, and a full proof of its soundness and completeness (Henkin style proof) is given.

A general proof method for first-order modal logic

Abstract: We present a general sequent-based proof method for first-order modal logics in which the Barcan formula holds. The most important feature of our system is the fact that it has identical inference rules for every modal logic; different modal logics can be obtained by changing the conditions under which two formulas are allowed to resolve against each other It is argued that the proof method is very natural because these conditions correspond to the conditions on the accessibility relation in Kripke semantics.

Modal Sequents and Definability

Abstract: The language of propositional modal logic is extended by the introduction of sequents. Validity of a modal sequent on a frame is defined, and modal sequent-axiomatic classes of frames are introduced. Through the use of modal algebras and general frames, a study of the properties of such classes is begun.

BASES OF ADMISSIBLE RULES OF THE MODAL SYSTEM $ \mathrm{Grz}$ AND OF INTUITIONISTIC LOGIC

Abstract: It is proved that the free pseudoboolean algebra and the free topoboolean algebra do not have bases of quasi-identities in a finite number of variables. A corollary is that the intuitionistic propositional logic and the modal system do not have finite bases of admissible rules. Infinite recursive bases of quasi-identities are found for and . This implies that the problem of admissibility of rules in the logics and is algorithmically decidable.Bibliography: 14 titles.

The recursive resolution method for modal logic

Abstract: In this paper a syntactic proof procedure, Recursive Resolution (RR), will be introduced for a Modal Logic system S4. The RR is analogous to Robinson’s resolution method on predicate calculus. In Predicate Calculus, a well-formed-formula (wff) can be transformed to a set of normal forms (clauses) after a skolemization process where skolem functions are introduced. For any two normal forms, the resolution method tells how to find their resolvent if they are resolvable at all. Similarly, for modal logic S4, there is a preliminary step in RR to transform a wff into a set of normal forms (m-clauses) after a skolemization step of the possibility operator □. For any two normal forms (m-clauses), the RR method determines whether they are modal-resolvable. If so, their modal-resolvent(s) (MR) can be computed. Traditionally, we use the semantic tableau or related method to reason on modal logic which is a direct application of the semantic analysis of the concept of possible world. Computationally, these semantic proof procedures involve unnecessary backtracking and so cannot be efficiently implemented in a computer. RR is also justified by and devised according to the semantic analysis of possible worlds. But it avoids explicit construction of possible worlds or reducing modalities by quantified variables and can be implemented by some recursive functions.

A modal logic for the representation of knowledge

Abstract: A modal logic which axiomatizes the modal concept of logical truth is presented. This modal logic is stronger than S5 modal logic and contains axioms allowing individual facts to be proven to be logically possible with respect to a body of knowledge. An extentional semantics is described and the soundness of the logic is proven.

Infinitary propositional normal modal logic

Abstract: A logic with normal modal operators and countable infinite conjunctions and disjunctions is introduced. A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories. It is also shown that the logic has the interpolation property.

Generic generalized Rosser fixed points

Abstract: To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR.

Individual-actualism and three-valued modal logics, part 2: Natural-deduction formalizations

Abstract: 5. FORMALIZING POSSIBILISTIC LOGICS BASED ON K A sequent calculus X may be viewed as a class-function assigning each appropriate language L = L, to a simultaneous inductive definition of two sets of sequents in L: Th s(L) = the theorems of z(L); WkTh z(L) = the weak theorems of z(L). The base-clauses of this definition shall be called axioms; the inductive clauses shall be rules. 5 will be sound relative to a given logic X iff for any appropriate L: all members of Th Ii(L) are X-valid; all members of WkTh J&L) are weakly X-valid. & will be complete relative to X iff for any appropriate L: all X-valid sequents of L belong to Th g(L); all weakly X-valid sequents of L belong to Wk Th z(L). Where g(L) is fixed, use these abbreviations: l-, A t 4 : (I-, A, 4) E Th s(L); I-, A t” 4 : (I-, A, 4) E Wk Th X(L). For I E A E fml(L), (I, A) is X(L)-inconsistent iff I, A b I; otherwise (I, A) is X(L)-consistent. Where x(L) is fixed, we’ll just write “consistent” or “inconsistent”. Notation: where @ E fml(L), let: 00 = {m#J:c$EcP);O-‘@ = {&of$Eq; define ~0 and 0-l CD similarly.

Using modal structures to represent extensions to epistemic logics

Abstract: Kripke structures have been proposed as a semantic basis for modal logics of necessity and possibility. They consist of a set of states, informally interpreted as "possible worlds", and a binary accessibility relation between states. The primitive notion of a possible world in this context seems highly intuitive, since necessity can be interpreted as "truth in all possible worlds", and possibility as "truth in some possible world. However, modal logics have also been used to model the epistemic notions of knowledge and belief, where an agent at a particular world is said to "know" or "believe" a proposition if that proposition is true in all possible worlds compatible with its beliefs. In this context, it is not as obvious how to interpret a "possible world". Modal structures have recently been introduced as a formally equivalent alternative to Kripke structures for modeling particular states of knowledge and belief. Modal structures consist of an infinite number of recursively defined levels, where each level contains the possible worlds that model an agent's meta-beliefs of a certain depth. For example, beliefs about the world are modeled at level 1 of a modal structure, and beliefs about beliefs about the world are modeled at level 2. Each modal structure corresponds to a single world of a Kripke structure and contains all the worlds that are accessible from that world in its levels. Modal structures are defined for the classical propositional epistemic logics S4 and S5. Recently, the traditional possible worlds approach has been extended to model "explicit", or limited, belief with partial worlds, called situations, in an appropriately modified Kripke structure. In this thesis, I demonstrate how modal structures can replace Kripke structures to interpret three recent logics of explicit and implicit belief. I also extend modal structures to model a first-order predicate logic which includes quantifiers, equality, and standard names. For each logic, I demonstrate the equivalence of the extended modal structure and the Kripke structure that originally provided the semantics for the logic. I discuss the advantages and disadvantages of using modal structures to model logics of knowledge and belief.

The Craig Interpolation Lemma for Certain Modal Logics

Abstract: We prove the Craig, Robinson and Beth theorems for a wide class or K-normal propositional modal logics. The method of the proof is based on an interpretation of modal theories in classical ones and on a representation of modal algebras. The method does not depend on any deduction lemma. Results can be generalized to other Segerberg—complete logics and also to some first order modal logics which do not contain the Barcan formula. Moreover the paper contains a remark on a generalization of Sahlqvist's theorem on correspondence and completeness.

Specification of Distributed Systems Using Modal Logics

Abstract: We demonstrate that variations of modal logics provide a unified framework for expressing important properties of distributed systems, even if these systems are based on completely distinct semantic models. Considerable flexibility is gained if, moreover, the combination of different modal calculi is admitted.

Knowledge Representation and Inference Based on First-Order Modal Logic

Abstract: In this paper, we present a knowledge representation system based on a first-order modal logic and discuss the deductive inference mechanism of this system. A possible-world model, which is used in discussing the semantics of a modal logic, can be regarded as structured knowledge, and modal operators can be used to describe various kinds of properties on a possible-world model. In this paper, we introduce a new concept, "viewpoints of modalities", in order to describe the knowledge structure effectively and compactly. We also show that schema formulas available in this framework are useful for the description of metaknowledge such as property inheritance. Therefore, a modal logic is suitable for representing both structured knowledge and metaknowledge. We construct a knowledge representation system based on a subset of a first-order modal logic, and give a complete deductive inference rule which is as effective as SLD resolution.

The modal status of antinomies.

Abstract: What is the modal status of antinomies? Classical modal logic provides no interesting answer to this question because it lets antinomies turn all wellformed formulas (including all modal formulas) into theorems. In the present note we propose two nonclassical modal systems which do not suffer from this defect. Both systems are obtained by supplementing the semantics of Asenjo's and Tamburino's antinomic propositional logic L (see [1], familiarity with which will be assumed in this article) with a very natural-sounding truth condition for modal formulas. The surprising result is that antinomies are in any case both necessary and impossible: according to the second system we propose, they are both non-necessary and possible as well. It may be doubted whether these results are in accord with our intuitions. However, it should be remembered that our intuitions were formed during centuries of classical slumber; acquiring the right intuitions in antinomic thinking may simply be a matter of time.