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By reading, you can know the knowledge and things more, not only about what you get from people to people. Book will be more trusted. As this dependence logic a new approach to independence friendly logic, it will really give you the good idea to be successful. It is not only for you to be success in certain life you can be successful in everything. The… (More)
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 introduce a modal language which involves the concept of dependence. We give two game-theoretic definitions for the semantics of the language, and one inductive, and prove the equivalence of all three.
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 bymeans of identity only in dependence logic, or equivalently in independence… (More)
In this paper all quantiiers are assumed to be so called simple unary quantiiers, and all models are assumed to be nite. We give a necessary and suucient condition for a quantiier to be deenable in terms of monotone quantiiers. For a monotone quantiier we give a necessary and suucient condition for being deenable in terms of a given set of bounded monotone… (More)
The semantics of the independence friendly logic of Hintikka and Sandu is usually defined via a game of imperfect information. We give a definition in terms of a game of perfect information. We also give an Ehrenfeucht-Fräıssé game adequate for this logic and use it to define a Distributive Normal Form for independence friendly logic.
Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set… (More)
We discuss the differences between first-order set theory and secondorder 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)