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We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent from the variables z if the variables x are kept constant. We contrast this with dependence logic D based on the atomic formula =(x, y), actually equivalent to y ⊥ x y, saying that the variables y are totally determined by the variables x. We show that y ⊥ x z(More)
We study the expressive power of open formulas of dependence logic introduced in Väänänen [Dependence logic (Vol. 70 of London Mathematical Society Student Texts), 2007]. In particular, we answer a question raised by Wilfrid Hodges: how to characterize the sets of teams definable by means of identity only in dependence logic, or equivalently in independence(More)
We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on(More)
The concept of a generalized quantiier of a given similarity type was deened in Lin66]. Our main result says that on nite structures diierent similarity types give rise to diierent classes of generalized quantiiers. More exactly, for every similarity type t there is a generalized quantiier of type t which is not deenable in the extension of rst order logic(More)