Dependence Logic: A New Approach to Independence Friendly by Jouko Väänänen

By Jouko Väänänen

Dependence is a typical phenomenon, at any place one seems: ecological structures, astronomy, human background, inventory markets - yet what's the good judgment of dependence? This booklet is the 1st to hold out a scientific logical examine of this significant thought, giving at the manner an exact mathematical remedy of Hintikka's independence pleasant common sense. Dependence common sense provides the idea that of dependence to first order common sense. the following the syntax and semantics of dependence common sense are studied, dependence good judgment is given an alternate online game theoretic semantics, and effects approximately its complexity are confirmed. this can be a graduate textbook compatible for a different direction in common sense in arithmetic, philosophy and desktop technological know-how departments, and includes over two hundred workouts, a lot of that have an entire answer on the finish of the booklet. it's also available to readers, with a uncomplicated wisdom of common sense, attracted to new phenomena in common sense.

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4). 3 If M is a set, X is a team with M as its codomain and F : X → M, we let X (F/xn ) denote the supplement team {s(F(s)/xn ) : s ∈ X }. A duplicate team is obtained by duplicating the agents of a team until all possibilities occur as far as a particular feature is concerned. Suppose a strategy officer of a company comes to a director with a plan for a team for a company wide committee. The strategy officer says, “I decided we need a programmer, an analyst, and a sales person. I chose such people from 20 Dependence logic X (M/ xn) Fig.

12) If M |=s ¬∃xn φ for all s ∈ X , then for all a ∈ M we have for all s ∈ X M |=s(a/xn ) ¬φ. Now (φ, X (M/xn ), 0) ∈ T . Thus (∃xn φ, X, 0) ∈ T . Now for the other direction. 31 If an L-formula φ of dependence logic is first order, then: (i) if (φ, X, 1) ∈ T , then M |=s φ for all s ∈ X ; (ii) if (φ, X, 0) ∈ T , then M |=s ¬φ for all s ∈ X . Proof We use induction as follows. If (t1 = t2 , X, 1) ∈ T , then t1M s = t2M s for all s ∈ X by (E1). If (t1 = t2 , X, 0) ∈ T , then t1M s = t2M s for all s ∈ X by (E2).

3) contradicts (D1). 18 A team X is of type ∃x0 (=(x2 , x0 ) ∧ x0 = x1 ) if and only if every s ∈ X can be modified to s(as /x0 ) such that for all s ∈ X (i) as = s(x1 ), (ii) as depends only on s(x2 ) in X , if and only if X is of type =(x2 , x1 ). 19 A team X is of type ∃x0 (= (x2 , x0 ) ∧ ¬(x0 = x1 )) if every s ∈ X can be modified to s(as /x0 ) in such a way that (i) as = s(x1 ), (ii) as is dependent only on s(x2 ) in X . This means that we have to be able to determine what s(x1 ) is, in order to avoid it, only on the basis of what s(x2 ) is.

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