Showing papers on "Normal modal logic published in 1985"

Graded modalities. I

Abstract: We study a modal system ¯T, that extends the classical (prepositional) modal system T and whose language is provided with modal operators M inn (neN) to be interpreted, in the usual kripkean semantics, as “there are more than n accessible worlds such that...”. We find reasonable axioms for ¯T and we prove for it completeness, compactness and decidability theorems.

Models for stronger normal intuitionistic modal logics

Abstract: This paper, a sequel to “Models for normal intuitionistic modal logics” by M. Božic and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated.

Sequent-systems for modal logic

Abstract: The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. ?0. Introduction. The aim of this work is to present sequent formulations of the modal logics S5 and S4 based on sequents of higher levels. Sequents of level 1 have collections of formulae of a given formal language on the left and right of the turnstile, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants will involve sequents of level 2, whereas rules for other customary logical constants of first-order logic (with identity) will involve only sequents of level 1. We shall show how a restriction on Thinning of level 2, which when applied to Thinning of level 1 produces intuitionistic out of classical logic, produces in this case S4 out of S5. Both in passing from classical to intuitionistic logic and in passing from S5 to S4, only Thinning is changed-all the other assumptions are unchanged. In particular, this means that S5 and S4 will be formulated with identical assumptions for the necessity operator. We shall also show in what sense our characterization of the necessity operator is Received January 5, 1982; revised December 3, 1983. 1980 Mathematics Subject Classification. Primary 03B45, 03F99. (? 1985, Association for Symbolic Logic 0022-4812/85/5001-001 5/$03.00

An internal semantics for modal logic

Abstract: In Kripke semantics for modal logic, “possible worlds” and the possibility relation are both primitive notions. This has both technical and conceptual shortcomings. From a technical point of view, the mathematics associated with Kripke semantics is often quite complicated. From a conceptual point of view, it is not clear how to use Kripke structures to model knowledge and belief, where one wants a clearer understanding of the notions that are primitive in Kripke semantics. We introduce modal structures as models for modal logic. We use the idea of possible worlds, but by directly describing the “internal semantics” of each possible world. It is much easier to study the standard logical questions, such as completeness, decidability, and compactness, using modal structures. Furthermore, modal structures offer a much more intuitive approach to modelling knowledge and belief.

The Extensions of the Modal Logic $K5$

Abstract: Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K 5. We associate with each logic extending K 5 a finitary index , in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K 5 and an abstract characterization of the lattice of such extensions. This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10. By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A ↔ B infer □ A ↔□ B ) and normal if it is classical and contains □ ⊤ and □ ( P → q ) → (□ p → □ q ). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □ A ).

An incomplete system of modal logic

Abstract: Over the last decade a number of logicians have devoted a considerable amount of attention to a system of propositional modal logic known variously as GL, G, L, and provability logic [1]. In this paper our attention will be devoted to a system of modal logic that is a proper subsystem of the system GL. We shall call this system GH. The axioms of GH are the tautologies, the distribution axioms, and the sentences

An Internal Semantics for Modal Logic: Preliminary Report

Abstract: In Kripke semantics for modal logic, "pos- sible worlds" and the possibility relation are both primitive notions. This has both technical and con- ceptual shortcomings. From a technical point of view, the mathematics associated with Kripke se- mantics is often quite complicated. From a concep- tual point of view, it is not clear how to use Kripke structures to model know!edge and belief, where one wants a clearer understanding of the notions that are primitive in Kripke semantics. We introduce modal structures as models for modal logic. We use the idea of possible worlds, but by directly describing the "internal semantics" of each possible world. It is much easier to study the standard logical questions, such as completeness, decidability, and compactness, using modal structures. Furthermore, modal struc- tures offer a much more intuitive approach to mod- elling knowledge and belief.

Can there be a natural deontic logic

Abstract: Before I can answer the title-question of this piece, I must say what I mean by 'deontic logic' and by 'natural ' . This is in itself a substantial task, not least because there are so many distinct but equally compelling conceptions of deontic logic, and the negative answer erotetically implicit in the title must represent for some of those conceptions a sort of denunciation. Deontic logic is a branch of philosophy, a subject matter that some philosophers spend some of their time writing about. To say that deontic logic thus conceived is unnatural might seem to be putting it (as it were) on a level with bestialism, uranism and door-todoor evangelism. Far be it for me to relegate deontic logic, as I would (cheerfully and in the presented order of rigourousness) the other three activities, to the private indulgence of consenting adults. I do not mean deontic logic as an avocation, preoccupation, passion, vice, or instrument of vexation. I do mean deontic logic as a (possibly finitely axiomatisable) set of sentences in some of which the word 'ought ' or its formal representative occurs as a logical constant. I also mean deontic logic as a codified set of inferences the correctness of some of which turns upon the way in which the word 'ought ' recurs through their component sentences. These are not meant to be definitions but serve merely as indicators of the sort of thing I have in mind. But, only the sort. For there is a newer conception of deontic logic which deserves place here: that of an inference logic which purports to codify the inferential closure of what ought to be the case, by introducing a deontic consequence relation which is required to preserve 'ought ' as an alethic consequence relation is required to preserve truth. 1 There is as well the semantic conception of deontic logic in which various model theoretic representations of axiological notions yield truth conditions for sentences in 'ought ' and are axiomatised by this or that deontic system. ~ In all of these cases the question of naturalness can be raised and its meaning is perhaps clear enough. Does the axiomatic deontic logic reflect the natural inter-

An application of Rieger-Nishimura formulas to the intuitionistic modal logics

Abstract: The main results of the paper are the following: For each monadic prepositional formula φ which is classically true but not intuitionistically so, there is a continuum of intuitionistic monotone modal logics L such that L+φ is inconsistent. There exists a consistent intuitionistic monotone modal logic L such that for any formula φ of the kind mentioned above the logic L+φ is inconsistent. There exist at least countably many maximal intuitionistic monotone modal logics.

Computational Aspects of Proofs in Modal Logic

Abstract: Various modal logics seem well suited for developing models of knowledge, belief, time, change, causality, and other intensional concepts. Most such systems are related to the classical Lewis systems, and thereby have a substantial body of conventional proof theoretical results. However, most of the applied literature examines modal logics from a semantical point of view, rather than through proof theory. It appears arguments for validity are more clearly stated in terms of a semantical explanation, rather than a classical proof-theoretic one. We feel this is due to the inability of classical proof theories to adequately represent intensional aspects of modal semantics. This thesis develops proof theoretical methods which explicitly represent the underlying semantics of the modal formula in the proof. We initially develop a Gentzen style proof system which contains semantic information in the sequents. This system is, in turn, used to develop natural deduction proofs. Another semantic style proof representation, the modal expansion tree is developed. This structure can be used to derive either Gentzen style or Natural Deduction proofs. We then explore ways of automatically generating MET proofs, and prove sound and complete heuristics for that procedure. These results can be extended to most propositional system using a Kripke style semantics and a fist order theory of the possible worlds relation. Examples are presented for standard T, S4, and S5 systems, systems of knowledge and belief, and common knowledge. A computer program which implements the theory is briefly examined in the appendix. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-85-55. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/658 COMPUTATIONAL ASPECTS OF PROOFS IN MODAL LOGIC Gregory Donald Hager MS-CIS-85-55 Department Of Computer and Information Science Moore School University of Pennsylvania Philadelphia, PA 191 04

Modality and Self-Reference

Abstract: Publisher Summary Some self-applied protosyntactical systems related to various modal systems are considered in this chapter . Each of these systems contains one predicate variable P , ranging over properties of the expressions of the system. When an interpretation is provided to the symbol P , each sentence becomes true or false. On selecting an arbitrary set of sentences as axioms and an arbitrary set of inference rules, P can be interpreted to mean provability within the very axiom system. This is essentially referred to self-referential interpretation of the axiom system. The chapter considers only those systems in which this self-referential phenomenon occurs—commonly called as “self-referentially correct.” The chapter also describes δ * , which is an analog of the modal system K4 with certain substitution axioms added, which provides enough fixed-points for the arguments of Godel's second-incompleteness theorem and Lob's theorem to go through the full power of the modal system G. Further, the chapter presents a brief description of some self-referential systems related to modal systems other than K4.

On the Central Principle of Deontic Logic

Abstract: R. M. Chisholm has noted sometime ago that there is a degree of resemblance between empirical confirmation and ethical requirement; that inductive and deontic logic share certain aspects with one another. He has pointed out that, for instance, just as "p confirms q" does not imply either that p occurs or that q occurs, similarly "p requires q" does not imply that p or q has actually taken place either. Also confirmation is defeasible and may be overridden with additional information, and so is requirement.' We shall see that Chisholm has merely touched slightly upon the surface of something far reaching that is of utmost significance as well as usefulness. In fact the relation between two logics is very close, so much so that any theorem-like statement in deontic logic can quickly be determined whether or not it is a valid theorem by examining its counterpart in inductive logic to see whether the latter is or is not a valid theorem. Also it is easy to state the reason why each of these two branches of applied logic should be the replica of the other. The reason is compelling enough not merely to explain retrospectively the various analogies we shall have discovered, but to predict with confidence the concrete manifestations of their close kinship, before having observed them. I shall attempt to distinguish clearly between two types of cases in which a given theorem-like statement in deontic logic has its inductive counterpart: in the first family of cases the validity of the deontic statement is directly dependent on the validity of the parallel inductive statement simply because the presence or absence of a certain moral obligation is determined by the success or failure of the adequate confirmation of the parallel empirical hypothesis. In the second group of cases the confirmation of the empirical counterpart plays no direct role in deter-

The decidable normal modal logics are not recursively enumerable

Abstract: LEMMA 2. S, is decidable if X is recursive. Proof Any wff A will be a theorem of S, iff A is valid on (i) all frames G, (see [ 1, p. 358]), and (ii) all frames & for (Y $ X. Let A be of modal degree n. Test A on Fn+z and Gn+Z. Gn+s is already a frame for Sx, and if A fails on F,,+z, then it will fail on F,, which is a frame for Sx. Otherwise A will be valid on every frame F, and G, , for (Y > n + 1. So test A on all F,,, and G,, where m < n + 1. If A fails on G, then, since G, is a frame for S,, A 4 S,. If A is valid on every G, (m < n + l), then test A on every F, (m < n + 1) such that m $Z X. Since X is recursive and m is bounded, this process is finite. If A fails on some such F, , then A f$ S,. Otherwise A E S,.