Showing papers on "Normal modal logic published in 1975"

Completeness and Correspondence in the First and Second Order Semantics for Modal Logic

Abstract: Publisher Summary This chapter focuses on normal logics and first order properties of the accessibility relation. There are three main themes, a positive, a negative, and a comparative. The chapter outlines that the first order relational semantics is adequate for all tense logics and that there is an equivalent algebra for each first order relational frame. The same may be done for normal modal logics. And by defining a first order neighborhood semantics, one may—via an algebraic semantics—prove the adequacy for all classical modal logics. This theorem is strong enough because it does not require any connections between the modal and Boolean operations. For tense logic or normal modal logic, stronger representation theorems are needed.

Some Connections Between Elementary and Modal Logic

Abstract: Publisher Summary This chapter outlines that a common way of proving completeness in modal logic is to look at the canonical frame. It discusses that the method is applicable to any complete logic whose axioms express a ΣΔ-elementary condition or to any logic complete for a Δ-elementary class of frames. The chapter also proves two mild converses to this result. The first is that any finitely axiomatized logic has axioms expressing an elementary condition if it is complete for a certain class of natural subframes of the canonical frame. The second result is obtained from the first by dropping finitely axiomatized and weakening elementary to Δ-elementary. Classical logic is used in the formulation and proof of these results. The proofs are not hard, but they show that there may be a fruitful and non-superficial contact between modal and elementary logic. The chapter also outlines some basic notions and results of modal logic. For simplicity, this is taken to be mono-modal. However, the results can be readily extended to multi-modal logics and, in particular, to tense logic.

Axiomatic classes in propositional modal logic

Abstract: In his review (Kaplan [1966]) of the article in which Kripke first proposed his relational semantics for modal logic, David Kaplan posed the question: which properties of a binary relation are expressible by formulas of propositional modal logic? A class of Kripke frames is said to be modal-axiomatic if it comprises exactly the frames on which every one of some set of formulas of propositional modal logic is valid. This work is addressed to the problem, suggested by Kaplan's question, of characterizing the modal-axiomatic classes of Kripke frames. In §i we obtain such a characterization, in terms of closure under certain constructions. In §2 we show that, in the case of classes closed under elementary equivalence, much simpler constructions suffice.

Categories of Frames for Modal Logic

Abstract: §1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = { y ∣ (∃ x )( y x Є X )}; and if is a CAMA then is a frame, where is the set of atoms of and b 1 b 2 ⇔ b 1 ∩ mb 2 ≠∅. Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence. The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames. The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].

First-Order Definability in Modal Logic

Abstract: In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp . Definition 1. A modal frame ℱ = 〈 W, R 〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

The Inadequacy of the Neighbourhood Semantics for Modal Logic

Abstract: We present two finitely axiomatized modal propositional logics, one between T and S 4 and the other an extension of S 4, which are incomplete with respect to the neighbourhood or Scott-Montague semantics. Throughout this paper we are referring to logics which contain all the classical connectives and only one modal connective □ (unary), no propositional constants, all classical tautologies, and which are closed under the rules of modus ponens (MP), substitution, and the rule RE (from A ↔ B infer α A ↔ □ B ). Such logics are called classical by Segerberg [6]. Classical logics which contain the formula □ p ∧ □ q → □( p ∧ q ) (denoted by K ) and its “converse,” □{ p ∧ q )→ □ p ∧ □ q (denoted by R ) are called regular; regular logics which are closed under the rule of necessitation, RN (from A infer □ A ), are called normal . The logics that we are particularly concerned with are all normal, although some of our results will be true for all regular or all classical logics. It is well known that K and R and closure under RN imply closure under RE and also that normal logics are also those logics closed under RN and containing □{ p → q ) → {□ p → □ q ).

Logical atomism, nominalism, and modal logic

Abstract: universals. 15 Nominalism, however, or so it would seem, cannot resort to this criterion since nexuses are supposedly not part of the world and cannot therefore be quantified over for the determination or representation of their number. Instead, as metalinguisticaUy distinguishable syncategorematic ties of a formal ontology nexuses would seem to be limited in number, as would all the syncategorematic or formal elements of the system, so as not to exceed ~loFor this reason in what follows we shall assume that both the number of objects in the world and the number of nexuses as metalinguistically distinguishable syncategorematic ties between these objects do not exceed blo.

Finite-level modal logics

Abstract: We shall say that a logic ~£~ belongs to level ~ where O<~<~, if in ~ there holds i~formula ~ but not formula 6n_i ; M£~ o if ~0 o The logics of ~ ~ shall be called the finite-level logics; 6w-d~Uw6 ~ Since in S# the formulas ~---6n+ , are deducible for each ~ , each logic MeJ~ belongs to exactly one of the levels ~n ' g~w. This proposed classification of modal logics is closely related to be classification of hyperintuitionistic logics proposed by Hosoi [6]. As was shown in [2] there is a homomorphism ~ of the lattice J~ onto the lattice of hyperintuitionistic logics. For g-$4 .... a modal logic ME~ belongs to level ~ if and only if the hyperintuitionistic logic ff(~) is a logic of level g in Hosoi's sense [6]. A. V. Kuznetsov [i] proved that all finitelevel hyperintuitionistic logics are locally tabular (i.e., the sets of pseudo-Boolean algebras corresponding to these logics are locally finite). Our basic result in this paper is in some sense a generalization of Kuznetsov's result. A modal logic M is locally tabular if the set of topological Boolean algebras (TBAs) in which all formulas of M are true is locally finite, i.e., every finitely generated algebra in this set is finite. In other words, M is locally tabular if for every n the number of pairwise inequivalent (in M ) formulas in the variables ~,...,~ is finite. (Formulas ~ and # are considered equivalent in M if