Showing papers on "Normal modal logic published in 1979"

The Unprovability of Consistency: An Essay in Modal Logic

Abstract: The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted for at least thirty years between the followers of C. I. Lewis and W. V. O. Quine. The author employs semantic methods developed by Saul Kripke in his analysis of modal logical systems. The book will be of interest to advanced undergraduate and graduate students in logic, mathematics and philosophy, as well as to specialists in those fields.

Interpolation theorems in modal logics and amalgamable varieties of topological Boolean algebras

Abstract: The interpolatlon properties of formal theories are important and interesting. At present there area slgnificant number of works that establish Craig's interpolation theorem [8] or disprove it in various formal theorles. Several investigations have been published that are devoted to the interpolation property in various modal logics. For example» Craig's interpolation theorem for certain well-known predicate modal systems [9, i0], including that

The Modal Logic of Programs

Abstract: We explore the general framework of Modal Logic and its applicability to program reasoning. We relate the basic concepts of Modal Logic to the programming environment: the concept of "world" corresponds to a program state, and the concept of "accessibility relation" corresponds to the relation of derivability between states during execution. Thus we adopt the Temporal interpretation of Modal Logic. The variety of program properties expressible within the modal formalism is demonstrated.

Canonical modal logics and ultrafilter extensions

Abstract: In this paper the canonical modal logics, a kind of complete modal logics introduced in K Fine [4] and R I Goldblatt [5], will be characterized semantically using the concept of an ultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic Theorem 8 of R I Goldblatt and S K Thomason [6] characterizing the modally definable Σ⊿-elementary classes of frames will follow as a corollary A second corollary is Theorem 2 of [4] which states that any complete modal logic defining a Σ⊿-elementary class of frames is canonicalThe main tool in obtaining these results is the duality between modal algebras and general frames developed in R I Goldblatt [5] The relevant notions and results from this theory will be stated in §2 The concept of a canonical modal logic is introduced and motivated in §3, which also contains the above-mentioned theorems In §4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions (and some other relevant operations on frames) is discussedThe modal language to be considered here has an infinite supply of proposition letters (p, q, r, …), a propositional constant ⊥ (the so-called falsum, standing for a fixed contradiction), the usual Boolean operators ¬ (not), ∨ (or), ∨ (and), → (if … then …), and ↔ (if and only if)—with ¬ and ∨ regarded as primitives—and the two unary modal operators ◇ (possibly) and □ (necessarily)— ◇ being regarded as primitive Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals

Of Modal Logics

Abstract: I have distinguished between modal logic and tense logic more than is usual because I hold the distinction to be important. I am speaking of alethic modal logic, and throughout this chapter modal logic otherwise unspecified will mean the alethic variety. The categories of modal logic are not the same as those of tense logic, since the modal categories deal with logical distinctions which are unaffected by time, while the categories of tense logic do not. Modal logic is a branch of formal logic, while tense logic is part of concrete logic, where its treatment in this work will be found.

The modal logic of quantum logic

Abstract: Modal logic is concerned with the concepts of necessity and possibility and a certain class of object propositions. In this paper we develop the basic concepts of a modal logic which is related to propositions about quantum physical objects. Since the object logic of quantum mechanical propositions is given by the calculi of quantum logic, the structure investigated in this paper will be called the modal logic of quantum logic. The object language and logic of quantum physical propositions is developed here within the dialogic approach to quantum logic [1]. On the basis of these object-linguistic structures we investigate the language of meta-propositions which state the material or formal truth of objectpropositions. Applying again the dialogic technique to meta-propositions the important notion of a formally true meta-proposition can be defined. Using this concept it turns out that the formal logic of meta-propositions is equivalent to the corresponding structure in ordinary logic, i.e., to the effective (intuitionistic) logic. The modalities "necessary" and "possible" are introduced here in the framework of a meta-linguistic interpretation, which considers the modalities as statements about the object-propositions under discussion [9]. On this basis it is found that meta-propositions which state the material truth of a special class of object propositions may be considered as quantum logical modalities (Section 2). A detailed investigation then shows that these quantum logical modalities are intimately related to the important quantum mechanical concepts of commensurability and objectivity. An illustration of the quantum logical modalities by relations between projection operators in Hilbert space concludes this part. In the third section we introduce the concept of a formally true modality which leads to the modal logic of quantum logic. It is shown that this logic of quantum logical modalities contains all the quantum logical restrictions which come from the possible incommensurability of quantum physical

Local and global operators and many-valued modal logics.

Abstract: 1 Motivation In formal contexts, we may distinguish between two types of conditional operators.* The truth value of a local conditional is defined in terms of the truth values of its antecedent and consequent. The truth value of a global conditional is defined in terms of the possible truth-values of its antecedent and consequent. Global conditionals are usually formed by applying a modal operator to a local conditional. For example, strict implication is defined by applying the necessity operator to material implication. That is, \"p-$q\" is defined as \"L(p CO # ) \" , where CO is the standard two-valued material implication and the properties of the necessity operator \" L \" are determined by the particular modal logic being employed. There has been some move to formally treat ordinary language counterfactual conditionals as global conditionals of a certain sort. (See for example [1] and [3]). Viewed from this perspective, the local conditional involved is not the standard two-valued material implication, but is rather a three-valued operator. We may use Γ, F, and / for \" t r u e \" , \" fa lse\", and \"indeterminate\", respectively. We may then define the conditional Cl as follows (contrasting it with material implication CO):

Is the semantics of branching structures adequate for chronological modal logics

Abstract: In many important philosophical discussions we need a formal theory of tensed modalities or a combined modal and tense logic. As McArther (1976, Chapter 3), McKim and Davis (1976), Thomason (1970) and the like have argued, the semantics of branching structures is indeed adequate for many non-metric tense logics with modal operators like OT in the sense that semantical completeness can be established. Can we extend this semantical approach to metric tense logics with modal operators or, as I prefer to call them, chronological modal logics? In Nishimura (1979) we have already proved that the semantics of causal structures was indeed adequate for chronological modal logics. Causal structures may be called "parallel histories", "history-time index systems with the likeness relation", etc., if the reader wants to. Thus if we were able to prove the eqiiivalence of branching structures and causal structures, the adequacy of branching structures for chronological modal logics would follow immediately. Which semantics we should adopt be a matter of taste in this case.

An axiomatization of the modal theory of the veiled recession frame

Abstract: The veiled recession frame has served several times in the literature to provide examples of modal logics failing to have certain desirable properties. Makinson [4] was the first to use it in his presentation of a modal logic without the finite model property. Thomason [5] constructed a (rather complicated) logic whose Kripke frames have an accessibility relation which is reflexive and transitive, but which is satisfied by the (non-transitive) veiled recession frame, and hence incomplete. In Van Benthem [2] the frame was an essential tool to find simple examples of incomplete logics, axiomatized by a formula in two proposition letters of degree 2, or by a formula in one proposition letter of degree 4 (the degree of a modal formula ϕ is the maximal number of nested occurrences of the necessity operator in ϕ). In [3] we showed that the modal logic determined by the veiled recession frame is incomplete, and besides that, is an immediate predecessor of classical logic (or, more precisely, the modal logic axiomatized by the formula p↔□p), and hence is a logic, maximal among the incomplete ones. Considering the importance of the modal logic determined by the veiled recession frame, it seems worthwhile to ask for an axiomatization, and in particular, to answer the question if it is finitely axiomatizable. In the present paper we find a finite axiomatization of the logic, and in fact, a rather simple one consisting of formulas in at most two proposition letters and of degree at most three.

Modal logics with no minimal proper extensions

Abstract: We show that neither the descending chain property nor the finite model property is a necessary condition for a model logic having no minimal proper extension. This answers in the negative two questions raised by G. E. Hughes.

On the non-availability of Dawson-modeling into certain relevance alethic modal logics

Abstract: This paper shows that the Dawson technique of modelling deontic logics into alethic modal logics to gain insight into deontic formulas is not available for modelling a normal (in the spirit of Anderson) relevance deontic modal logic into either of the normal relevance alethic modal logics R□ S4or R□ M. The technique is to construct an extension of the well known entailment matrix set M 0and show that the model of the deontic formula P (A v B)→. PA v PB is excluded.