Modern modal logic originated as a branch of philosophical logic in which the concepts of necessity and possibility were investigated by means of a pair of dual operators that are added to a propositional or first-order language. The field owes much of its flavor and success to the introduction in the 1950s of the “possible-worlds” semantics in which the modal operators are interpreted via some “accessibility relation” connecting possible worlds. In subsequent years, modal logic has received attention as an attractive (...) approach towards formalizing such diverse notions as time, knowledge, or action. Nowadays, modal logics are applied in various disciplines, ranging from economics to linguistics and computer science. Consequently, there is by now a large variety of modal languages, with an even greater wealth of interpretations. For instance, many applications require a poly-modal framework consisting of a language with a family of modal operators and a semantics in which the corresponding accessibility relations are connected somehow. (shrink)

Modern modal logic originated as a branch of philosophical logic in which the concepts of necessity and possibility were investigated by means of a pair of dual operators that are added to a propositional or first-order language. The field owes much of its flavor and success to the introduction in the 1950s of the “possible-worlds” semantics in which the modal operators are interpreted via some “accessibility relation” connecting possible worlds. In subsequent years, modal logic has received attention as an attractive (...) approach towards formalizing such diverse notions as time, knowledge, or action. Nowadays, modal logics are applied in various disciplines, ranging from economics to linguistics and computer science. Consequently, there is by now a large variety of modal languages, with an even greater wealth of interpretations. For instance, many applications require a poly-modal framework consisting of a language with a family of modal operators and a semantics in which the corresponding accessibility relations are connected somehow. (shrink)

We show how belief revision can be treated systematically in the format of dynamicepistemic logic, when operators of conditional belief are added. The core engine consists of definable update rules for changing plausibility relations between worlds, which have been proposed independently in the dynamic-epistemic literature on preference change. Our analysis yields two new types of modal result. First, we obtain complete logics for concrete mechanisms of belief revision, based on compositional reduction axioms. Next, we show how various abstract postulates for (...) belief revision can be analyzed by standard modal frame correspondences for model-changing operations. (shrink)

We consider modal logics whose intermediate fragments lie between the logic of infinite problems [20] and the Medvedev logic of finite problems [15]. There is continuum of such logics [19]. We prove that none of them is finitely axiomatizable. The proof is based on methods from [12] and makes use of some graph-theoretic constructions (operations on coverings, and colourings).

Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic logic is the class of Medvedev frames $\langle W,R\rangle$ where $W$ is the set of nonempty subsets of some nonempty finite set $S$, and $xRy$ iff $x\supseteq y$, or more liberally, where $\langle W,R\rangle$ (...) is isomorphic as a directed graph to $\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle$. Prucnal [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those $y$ such that $x\not= y$), or a complement modality (for quantifying over those $y$ such that $x\not\supseteq y$). It follows that either the logic of Medvedev frames in one of these tense languages is finitely axiomatizable---which would answer the open question of whether Medvedev's [31] "logic of finite problems'' is decidable---or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics. (shrink)

We continue the investigations initiated in the recent papers where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among (...) these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited. (shrink)

In the article [2] a hierarchy of modal logics has been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to the modal logics of Medvedev frames it has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. However, the infinite case (...) remained open. In this article we prove that the modal logic of Bayesian belief revision determined by standard Borel spaces is also not finitely axiomatizable. (shrink)

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If (...) a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. (shrink)

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of (...) its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate "partial meet contraction functions", which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gardenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are "relational" and "transitively relational", are studied in detail, and their connections with certain "supplementary postulates" of Gardenfors investigated, with a further representation theorem established. (shrink)