Some Observations About Generalized Quantifiers in Logics of Imperfect Information

Abstract

We analyse the two definitions of generalized quantifiers for logics of dependence and independence that have been proposed by F. Engström, comparing them with a more general, higher-order definition of team quantifier. We show that Engström's definitions (and other quantifiers from the literature) can be identified, by means of appropriate lifts, with special classes of team quantifiers. We point out that the new team quantifiers express a quantitative and a qualitative component, while Engström's quantifiers only range over the latter. We further argue that Engström's definitions are just embeddings of the first-order generalized quantifiers into team semantics, and fail to capture an adequate notion of teamtheoretical generalized quantifier, save for the special cases in which the quantifiers are applied to flat formulas. We also raise several doubts concerning the meaningfulness of the monotone/nonmonotone distinction in this context. In the appendix we develop some proof theory for Engström's quantifiers.

Safe Dependency Atoms and Possibility Operators in Team Semantics

“…For simplicity, we will assume that all expressions are in Negation Normal Form: Definition 6 (Team Semantics for First Order Logic) Let M be a first order model with domain M, let φ (x) be a first order formula in negation normal form with free variables contained in x, and let X be a team over M with domain Dom(X ) ⊇ x. 1 Then we say that the team X satisfies φ (x) in M, and we write M |= X φ , if this can be derived via the following rules: It is worth remarking that the above semantics for the language of first order logic involves second order existential quantifications in the rules TS-∨ and TS-∃. This is a crucial fact for understanding the expressive power of logics based on Team Semantics, and it is furthermore the reason why Team Semantics constitutes a viable tool for describing and studying fragments of existential second order logic.…”

“…The richer nature of the satisfaction relation of Team Semantics, however, makes it possible to extend First Order Logic in novel ways, such as by introducing new operators or quantifiers [1,4,6,23] or new types of atomic formulas which specify dependencies between different assignments contained in a team. Examples of important logics obtained in the latter way are Dependence Logic [22], Inclusion Logic [5], and Independence Logic [11].…”

“…Exceptions to this are given for instance by the study of logics which add to Team Semantics generalised quantifiers[1,4], or a contradictory negation[23] 3. This is a special case of the more general -and not necessarily first order -notion of dependency used in[7], which comes from[19].…”

Characterizing Downwards Closed, Strongly First-Order, Relativizable Dependencies

“…Another possible direction in which the present work can be expanded could be to consider not only logics FO(D) obtained by adding dependency atoms to First Order Logic, but more in general logics L(D) where L can be based on arbitrary choices of connectives and operators (interpreted in Team Semantics). Much like studying the more general notion of safety gave us in this work the tools necessary to prove the above mentioned result about strongly first order dependencies, it is possible that studying the notions of safety and strong first orderness in the more general L(D) case may give us the tools for solving the FO(D) one; and, moreover, such an investigation would connect the research program to which this work belongs to the related area of the study of generalized quantifiers in Team Semantics [10,35,11,2].…”

“…The study of strongly first-order dependencies, in particular, can be thought of as an attempt to investigate the border between first order and second order "from below" by seeking to characterize precisely which choices of dependency atoms (or families of dependency atoms) breach or fail to breach it. The conjecture according to which a dependency is strongly first order if and only if it is definable in terms of upwards closed dependencies and constancy dependencies 2 is, however, still unproven. In this work, a proof for a special case of it will be found: if a nontrivially-false dependency is strongly first order, is downwards closed, and is furthermore relativizable (a new, natural property of dependencies which will be introduced in this work) 3 then it is definable in terms of constancy atoms alone.…”