J.F.A.K. van Benthem

Situations, Language and Logic

I. Introduction.- 1. Prom linguistic form to situation schemata.- 2. Interpreting situation schemata.- 3. The logical point of view.- II. From Linguistic Form to Situation Schemata.- 1. Levels of linguistic form determining meaning.- 2. Motivation for the use of constraints.- 3. The modularization of the mapping from form to meaning.- 4. Situation schemata.- 5. The algorithm from linguistic form to situation schemata.- III. Interpreting Situation Schemata.- 1. The art of interpretation.- 2. The inductive definition of the meaning relation.- 3. A remark on the general format of situation schemata.- 4. Generalizing generalized quantifiers.- IV. A Logical Perspective.- 1. The mechanics of interpretation.- 2. A hierarchy of formal languages.- 2.1. Propositional logic.- 2.2. Predicate logic.- 2.3. Tense logic.- 2.4. Temporal predicate logic.- 2.5. Situated temporal predicate logic.- 3. Mathematical study of some formal languages.- 3.1. Definition of structure.- 3.2. The system L3.- 3.3. Modal operators.- 4. On the model theoretic interpretation of situation schemata.- 4.1. The basic correspondence.- 4.2. The correspondence extended.- V. Conclusions.- Appendices.- A. Prepositional Phrases in Situation Schemata.- by Erik Colban.- B. A Lyndon type interpretation theorem for many-sorted first-order logic.- C. Proof of the relative saturation lemma.- References.

The stories of logic and information

Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information.

Multimo dal Logics of Products of Topologies

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies.We prove that both of these logics are complete for the product of rational numbers ℚ × ℚ with the appropriate topologies.

Structures and Norms in Science

Will reading habit influence your life? Many say yes. Reading structures and norms in science is a good habit; you can develop this habit to be such interesting way. Yeah, reading habit will not only make you have any favourite activity. It will be one of guidance of your life. When reading has become a habit, you will not make it as disturbing activities or as boring activity. You can gain many benefits and importances of reading.

Logic games are complete for game logics

Game logics describe general games through powers of players for forcing outcomes. In particular, they encode an algebra of sequential game operations such as choice, dual and composition. Logic games are special games for specific purposes such as proof or semantical evaluation for first-order or modal languages. We show that the general algebra of game operations coincides with that over just logical evaluation games, whence the latter are quite general after all. The main tool in proving this is a representation of arbitrary games as modal or first-order evaluation games. We probe how far our analysis extends to product operations on games. We also discuss some more general consequences of this new perspective for standard logic.Game logics describe general games through powers of players for forcing outcomes. In particular, they encode an algebra of sequential game operations such as choice, dual and composition. Logic games are special games for specific purposes such as proof or semantical evaluation for first-order or modal languages. We show that the general algebra of game operations coincides with that over just logical evaluation games, whence the latter are quite general after all. The main tool in proving this is a representation of arbitrary games as modal or first-order evaluation games. We probe how far our analysis extends to product operations on games. We also discuss some more general consequences of this new perspective for standard logic.

Logical Constants across Varying Types

We investigate the notion of «logicality» for arbitrary categories of linguistic expression, viewed as a phenomenon which they can all possess to a greater or lesser degree. Various semantic aspects of logicality are analyzed in technical detail: in particular, invariance for permutations of individual objects, and respect for Boolean structure. We show how such properties are systematically related across different categories, using the apparatus of the typed lambda calculus

Some kinds of modal completeness

In the modal literature various notions of “completeness” have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and preservation under ultrafilter extensions are introduced and studied as well.

Modal Reduction Principles

Modal reduction principles (MRPs) are modal formulas of the following form: Mp → Np , where M, N are (possibly empty) sequences of modal operators (i.e. □ or ◊). The notation M, N will be used to abbreviate such an MRP. We study a certain semantic correspondence between modal formulas and relational properties and obtain two main results. (1) On transitive semantic structures every MRP corresponds to a first-order relational property. (2) For the general case a syntactic criterion exists for distinguishing modal formulas with corresponding first-order properties from the others.

A Note on Modal Formulae and Relational Properties

Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators □ and ⋄. The semantic structures are frames , i.e., pairs W, R > with R ⊆ W 2 . Let F, V be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of W . Then the relation may be defined, for arbitrary formulae α, following the Kripke truth-definition. From this relation we may further define Now, to every modal formula α there corresponds some property P α of R . A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first-order predicate. For these particular P α we have for all F and w ∈ W . These formulae P α are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when P α may be taken to be some first-order formula. For example, it can be seen that for all F and w ∈ W . It is customary to talk about a related correspondence, namely when for all F we have Note that this correspondence holds whenever the first one above holds.

A New Modal Lindström Theorem

Abstract.We prove new Lindström theorems for the basic modal propositional language, and for some related fragments of first-order logic. We find difficulties with such results for modal languages without a finite-depth property, high-lighting the difference between abstract model theory for fragments and for extensions of first-order logic. In addition we discuss new connections with interpolation properties, and the modal invariance theorem.