Quantified Modal Logic Research Articles

AXIOMATIZABILITY OF PROPOSITIONALLY QUANTIFIED MODAL LOGICS ON RELATIONAL FRAMES

AbstractPropositional modal logic over relational frames is naturally extended with propositional quantifiers by letting them range over arbitrary sets of worlds of the relevant frame. This is also known as second-order propositional modal logic. The propositionally quantified modal logic of a class of relational frames is often not axiomatizable, although there are known exceptions, most notably the case of frames validating the strong modal logic $\mathrm {S5}$ . Here, we develop new general methods with which many of the open questions in this area can be answered. We illustrate the usefulness of these methods by applying them to a range of examples, which provide a detailed picture of which normal modal logics define classes of relational frames whose propositionally quantified modal logic is axiomatizable. We also apply these methods to establish new results in the multimodal case.

On the translation from quantified modal logic to Counterpart Theory

Lewis (1968) claims that his language of Counterpart Theory (CT) interprets modal discourse and he adverts to a translation scheme from the language of Quantified Modal Logic (QML) to CT. However, everybody now agrees that his original translation scheme does not always work, since it does not always preserve the ‘intuitive’ meaning of the translated QML-formulas. Lewis discusses this problem with regard to the Necessitist Thesis, and I will extend his discourse to the analysis of the Converse Barcan Formula.Everyone also agrees that there are CT-formulas that can express the QML-content that gets lost through the translation. The problem is how we arrive to them. In this paper, I propose new translation rules from QML to CT, based on a suggestion by Kaplan. However, I will claim that we cannot have ‘the’ translation scheme from QML to CT. The reason being that de re modal language is ambiguous. Accordingly, there are different sorts of QML, depending on how we resolve such ambiguity. Therefore, depending on what sort of QML we intend to translate into CT, we need to use the corresponding translation scheme. This suggests that all the translation problems might just disappear if we do what Lewis did not: begin with a fully worked out QML that tells us how to understand de re modal discourse.

The Extension of Bisimulation Quantified Modal Logic Based on Covariant-Contravariant Refinement

The notion of covariant-contravariant refinement (CC-refinement, for short) is a generalization of the notions of bisimulation and refinement. This paper interprets semantically a CC-refinement as bisimulation plus model restriction, that is, a CC-refinement model of a given model may be obtained from one bisimilar duplicate of this model by adding some transitions labelled by covariant actions and removing some transitions labelled by contravariant actions. By using certain proposition letter to witness a contravariant action, the standard bisimulation quantified modal logic is able to capture the characterization of this action, however, this fails for covariant actions. This paper, based on the notion of CC-refinement, introduces an extended bisimulation quantified modal logic with the universal modality (EBQML), describes syntactically CC-refinement quantification as the extended bisimulation quantification plus relativization, and establishes a translation from the language of CC-refinement modal μ-calculus to the language of EBQML such that every CC-refinement modal μ-formula is equivalent to its translation. The language of EBQML may be considered as a specification language for describing the properties of a system referring to reactive and generative actions, which are represented respectively by covariant and contravariant actions, and may be used to formalize some interesting problems in the field of formal methods.

Problemi kvantificirane epistemičke logike

Within the framework of quantified modal logic (QML) the author, in the first part of his paper, on the basis of the actualist approach to the problem of founding logical moalities, attempts to found a system of quantified epistemic logic (QEL) which presupposes a characteristic manner of problem-solving if compared to the rule of existential generalisation (EG). This is Hintikka's approach. The central issue of the second part of the paper tackles the rule of existential, as well as some other problems in connection with their applications in quantified epistemic logic. The conclusion gives counter arguments to two systems QEL (Hintikka' system K and B and Carlson's system C).

Modal Logic and Contingentism: A Comment on Timothy Williamson’sModal Logic as Metaphysics

Logic, as standardly conceived, is the science of consequence: a logic tells us which claims follow from which. Quantified modal logic (QML) is supposed to tell us which claims follow from which in a language which provides the means to express both metaphysical modality and quantification. One might think on this basis that we can determine which is the correct QML independently of substantive questions of science and metaphysics: modal logic, on this view, cannot tell us whether it is possible that there be no numbers; nor can it tell us whether, necessarily, everything is at bottom physical. At most, it might be held, logic can tell us what follows from these claims. We might dub this view neutralism about QML, since it holds that the correct QML must be neutral on substantive disputes in modal metaphysics. It is difficult to find an extended, full-throated defense of neutralism in the literature. But Timothy Williamson’s Modal Logic as Metaphysics provides an extended, full-throated criticism. Williamson aims to show how, in particular, the model theory of QML bears on a substantive, if highly abstract, dispute in modal metaphysics. Consider one of the most obvious things about the Sun: it is such that there is something identical to it. More briefly, it is something. Being something, of course, does not make it unique. It shares this feature with everything. Is 1Of course, there are applications of the study of the formally specified languages of QML in which ‘ ’ is interpreted as something other than metaphysical modality. Those applications of the formalism are irrelevant to our discussion. 2Perhaps [Kaplan, 1989, pp. 42-43] offers a defense.

Actualism and Higher-Order Worlds

It has been argued that actualism – the view that there are no non-actual objects – cannot deal adequately with statementsinvolving iterated modality, because such claims require reference, either explicit or surreptitious, to non-actualobjects. If so, actualists would have to reject the standard semantics for quantified modal logic (QML). In this paper I develop an account of modality which allows the actualist tomake sense of iterated modal claims that are ostensibly aboutnon-actual objects. Every occurrence of a modal operatorinvolves the stipulation of a possible world, and nestedmodal operators require stipulation of nested possible worlds.I provide an actualistically acceptable (AA) semantics for QMLwherein the nesting relation is irreflexive and intransitive and forms a tree. Despite these restrictions, AA models can beshown to be sound and complete for a wide variety of modal logics.

In Defense of the Simplest Quantified Modal Logic

By combining the laws of classical quantification theory with the modal propositional logic K in the most direct manner, one produces the simplest Quantified Modal Logic. The models of this simple QML relativize predication to possible worlds and interpret the quantifiers as ranging over a single, fixed domain. But simple QML has many controversial features, not the least of which are that it validates the Barcan formula and appears to require quantification over possibilia. Whereas possibilists employ distinctions that render these features of simple QML unobjectionable,I actualists find the distinctions and the controversial features difficult to accept. Many thought that Kripke-models, with their varying domains and restricted quantifiers, promised to rid QML of the Barcan formula, quantification over possibilia, and other objectionable features we haven't yet described. Unfortunately, Kripke-models themselves have features at which actualists balk, and so these philosophers have had to modify (our understanding of) Kripke-models to find an acceptable QML. Though we are possibilists, we understand the suspicion with which actualists regard simple QML and Kripke-models. But in their attempt to modify Kripke-models to produce an acceptable QML, actualists employ sophisticated maneuvers and machinery. To us, this seems analogous to introducing epicycles into an already flawed geocentric astronomy. For we have reexamined the simplest QML and have discovered that it has the following, unheralded virtue: in addition to having an interpretation compatible with possibilism, it has an interpretation compatible with actualism! In this paper, we concentrate our energies on motivating and describing the interpretation of simple QML that is compatible with actualism, since the possibilist interpretations of simple QML are well known. We argue that actualists need not use nor reconceive the varying domains of Kripke-models to avoid the objectionable features of simple QML.

QUANTIFIED MODAL LOGIC WITH NEIGHBORHOOD SEMANTICS

AbstractThe paper presents a semantics for quantified modal logic which has a weaker axiomatization than the usual Kripke semantics. In particular, the Barcan Formula (BF) and its converse are not valid with the proposed semantics. Subclasses of models which validate BF and other interesting formulas are presented. A completeness theorem is proved, and the relation between this result and completeness with respect to Kripke models is investigated.