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By Hernandes M. E.

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Example text

We assume that μi ≥ λi > 0 for some i, 1 ≤ i ≤ k as otherwise Q is derivable easily using (EW ) and (M ). For h = 0 there are two possibilities. If Q = q q then Σ1 , μ1 q which is derivable in GL. If Q = ⊥ A then Q = Q1 = G1 |Π1 Q = G1 |Π1 Σ1 , A, (μ1 − 1)⊥ which is derivable by Lemma 1. For h > 0 we have several cases according to the last rule r applied. – r is (W L). There are two subcases. (a) We step to: Q = Γ1 , (λ1 − 1)q Δ1 | . . |Γk , λk q Δk Since μ1 ≥ λ1 ≥ 1 we have by Lemma 2 that Q1 = G1 |Π1 Σ1 , (μ1 − 1)q is derivable in GL.

Q’s state of knowledge regarding x is uniquely determined at each point in time by a “conjunction” of the answers given by A. In general this conjunction fails to obey the rules of classical logic. For example, neither the principle of noncontradiction nor idempotency hold. If A answers “x is 2” and then “x is not 2”, this is not inconsistent, it just means that A has one less lie to use. Similarly, repeated assertions that “x is 2” are more informative than one such assertion; indeed n − 1 such assertions guarantee truth.

B) We step to: Q = Γ1 − {A}, λ1 q Δ1 | . . |Γk , λk q Δk and the claim follows by the induction hypothesis and (W L). 7 Note that an atomic formula is either a propositional variable or ⊥. 42 Agata Ciabattoni and George Metcalfe – r is (EW ), (EC), (⊃, l) or (⊃, r). These involve straightforward applications of the induction hypothesis and the corresponding rule. – r is (M ). We step to: Q1 = Γ11 , λ11 q Δ11 | . . |Γk , λk q Δk and Q2 = Γ12 , λ21 q Δ21 | . . |Γk , λk q Δk By the induction hypothesis applied to Q1 and Q1 , .

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Ae -codimension of germs of analytic curves by Hernandes M. E.


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