General frames are often used in classical modal logic. Since they are duals of modal algebras, completeness follows automatically as with algebras but the intuitiveness of Kripke frames is also retained. This paper develops basics of general frames for relevant modal logics by showing that they share many important properties with general frames for classical modal logic.

The Logic R4 is obtained by adding the axiom □(A v B) → (◇A v □B) to the modal relevant logic NR. We produce a model theory for this logic and show completeness. We also show that there is a natural embedding of a Kripke model for S4 in each R4 model structure.

We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.

This paper uses an "admissible set semantics" to treat quantification in quantified modal logics. The truth condition for the universal quantifier states that a universally quantified statement (x)A(x) is true at a world w if and only if there is some proposition true at that world that entails every instance of A(x). It is shown that, for any canonical propositional modal logic the corresponding admissible set semantics characterises the quantified version of that modal logic.

In 1988, Kit Fine published a semantic theory for quantified relevant logics. He referred to this theory as stratified semantics. While it has received some attention in the literature, 1–20, 1992; Mares & Goldblatt, Journal of Symbolic Logic 71, 163–187, 2006), stratified semantics has overall received much less attention than it deserves. There are two plausible reasons for this. First, the only two dedicated treatments of stratified semantics available are, 27–59, 1988; Mares, Studia Logica 51, 1–20, 1992), both of which (...) are quite dense and technically challenging. Second, there are a number of prima facie reasons to be worried about stratified semantics. The purpose of this paper is to revitalize research on stratified semantics. I will do so by giving a ‘user friendly’ presentation of the semantics, and by giving reasons to think that the prima facie reasons to be worried about it are too simplistic. (shrink)

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment.

The Logic R4 is obtained by adding the axiom □ → to the modal relevant logic NR. We produce a model theory for this logic and show completeness. We also show that there is a natural embedding of a Kripke model for S4 in each R4 model structure.

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.

The quantified relevant logic RQ is given a new semantics in which a formula for all xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of 'extensional confinement': for all x(A V (...) B) -> (A V for all xB). with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation. (shrink)

Although we believe the results reported below to have direct philosophical import, we shall for the most part confine our remarks to the realm of mathematics. The reader is referred to [4] for a philosophically oriented discussion, comprehensible to mathematicians, of tense logic.The “minimal” tense logicT0is the system having connectives ∼, →,F(“at some future time”), andP(“at some past time”); the following axioms:(whereGandHabbreviate ∼F∼ and ∼P∼ respectively); and the following rules:(8) fromαandα → β, inferβ,(9) fromα, infer any substitution instance ofα,(10) fromα, (...) inferGα,(11) fromα, inferHα.A tense logic is a systemTwhose language is that ofT0and whose axioms and rules include (1)–(11). The axioms and rules ofTother than (1)–(11) are calledproperaxioms and rules.We shall investigate three systems of semantics for tense logics, i.e. three notions ofstructureand three relations ⊧ which stand between structures and formulas. One reads⊧αas “αis valid in.” A structureis amodelof a tense logicTif every formula provable inTis valid in. A semantics isadequateforTif the set of models ofTin the semantics is characteristic forT, i.e. if wheneverT ∀ αthen there is a modelofTin the semantics such that∀ α. Two structuresand, possibly from different semantics, are calledequivalent(∼) if exactly the same formulas are valid inas in. (shrink)

This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation.

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered are (...) semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t . The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras. (shrink)

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley- Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.

This paper extends the argument of Mares, ``Classically Complete Modal Relevant Logics'' Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, to show that the system RK is complete over an Extension of the Routley-Meyer semantics.