Modal Languages and Bounded Fragments of Predicate Logic

Studia Logica - An International Journal for Symbolic Logic, 1995

Abstract.

2006

Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an epistemic reading of this formula: "t is a possible justification of φ". Motivated, in part, by this epistemic reading, Fitting introduced a Kripke style semantics for LP in . In this note, we prove soundness and completeness of some axiom systems which are not covered in [8].

MISCELLANEA LOGICA

Journal of Symbolic Logic, 1986

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Annals of Pure and Applied Logic, 1990

Studia Logica - An International Journal for Symbolic Logic, 2006

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics.

Notre Dame Journal of Formal Logic, 1981

The weakest modal logics have not come in for much attention from logicians or philosophers principally, it seems, because they are supposedly incapable of supporting interpretations of much philosophical succulence. But from the point of view of general semantic theory they deserve more attention than they get, for it is only by a study of weak modal logics that we come to appreciate many of the limitations of the now standard semantical methods. A multitude of examples bears this out. In the theory of firstorder definability in modal logic valuable insights would have been lost had we restricted our attention to extensions of S4. The McKinsey formula M o , DOp-> ODp, is characterized by a first-order condition on transitive binary relations, but KM 0 is not defined by any first-order condition. Similarly, looking to logics weaker than K reveals limitations of first-order definability which would otherwise go unnoticed. Here it is that we see that while D, Dp-> Op, and G, ODp-» DOp, are definable in a first-order language with a single binary predicate, neither is definable in a first-order language with a single ternary or n-ary (n>3) predicate (see [3]). Furthermore, weak modal logics preserve philosophically significant distinctions which are lost in stronger logics. If a proposed interpretation requires even so obvious a distinction as that between Con, ~1D1, and D, Dp-> Op, or that between D\ D(Dp-> Op), and £>*, DDp-> DOp, then a logic weaker even than K is required. These two facts are not unrelated. Taken together they amount to this: formulas like D and G are first-order definable only if we restrict ourselves to a first-order language so crude that it cannot *Partially supported by National Research Council Grants A4085 and A4523.

Studies in Logic and Practical Reasoning, 2007

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1966

Journal of Philosophical Logic, 1981

This paper is concerned with the technical implications of a certain view connecting existence to predication. This is the view that in no possible world is there a genuine relation among the nonexistents of that world or between the nonexistents and the existents! The meaning of the term 'genuine' here may be variously explained. On an extreme interpretation, all relations are 'genuine', so that none of them are to relate non-existents. On a milder interpretation, the genuine relations are those that are simple or primitive in some absolute sense. But even without appeal to an absolute concept of simplicity, we can require that all relations should be analyzable in terms of some suitable set of relations, relating only existents to existents. In order to make our results applicable to the thesis, we shall suppose that the primitive non-logical predicates of our language correspond to the genuine relations, whatever they might be taken to be. Thus, the linguistic formulation of the thesis becomes that the primitive predicates of the language should only be true, in each world, of the existents of that world. Of course, the thesis might have been given a linguistic formulation, without any reference to relations, in the first place. The thesis is an instance of what has been called Actualism. This is the ontological doctrine that ascribes a special status to actual or existent objects. Another form of the doctrine, so-called World Actualism, says that the behaviour of nonexistents is supervenient upon the behaviour of the existents, that two possible worlds which agree in the latter respect cannot differ in the former respect. The present thesis, by contrast, might be called Redicate Acrualism. It should be clear that Predicate Actualism implies World Actualism, at least if the predicates used to describe the world are to express 'genuine' relations; for then there are no relationships involving nonexistents by which two worlds might be distinguished. On the other hand, World Actualism does not, as it stands, imply Predicate Actualism. For our purposes, it will be useful to distinguish two versions of the actualist doctrine for predicates. Under one, the primitive predicates do not