Elements of logic and foundations of mathematics in problems by Wiktor Marek

By Wiktor Marek

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Given two valuations v1 , v2 , we write v1 ∼ v2 if size(v1 ) = size(v2 ). Clearly, ∼ is an equivalence relation, and it splits valuations according to the cardinality of the set of atoms evaluated as 1. We now have the following fact. 25 A formula ϕ is symmetric if and only if for every equivalence class C of ∼, either all valuations in C satisfy ϕ, or none of them does. 7 Duality The structure Bool = {0, 1}, ∧, ∨, 0, 1 is isomorphic to the structure {0, 1}, ∨, ∧, 1, 0 . One can visualize this isomorphism as a “flip,” 0 goes to 1, and conversely 1 turns into 0.

This formula ϕM has the property that M is the only model of ϕM . Now let us form ϕ as follows: ϕ= ϕM . M∈M © 2009 by Taylor and Francis Group, LLC 42 Introduction to Mathematics of Satisfiability We claim that M = Mod (ϕ). Indeed, if M ∈ M then, since M |= ϕM , M |= ϕ. , N ∈ M. 9 Exercises 1. Let Var 1 , Var 2 be two sets of propositional variables, Var 1 ⊆ Var 2 . Let ϕ ∈ LVar 1 . Let v be a valuation of Var 2 . Prove that v(ϕ) = v|Var 1 (ϕ) and in particular v |= ϕ if and only if v|Var 1 |= ϕ.

Like above, our goal is to establish several representations of partial valuations (we will have three) so we can use them whenever convenient. To this end we can think about partial valuations as partial functions from Var to {0, 1} or as total functions, but with values in a set consisting of three, not two, values, {0, 1, u}. Specifically, we assign to a partial function v from Var to {0, 1} a total function v ⋆ : Var → {0, 1, u} as follows:   0 if v(p) = 0 v ⋆ (p) = 1 if v(p) = 1   u if v(p) is undefined.

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