By Ryszard Wojcicki

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**Extra resources for Lectures on Propositional Logic**

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Let they be ♦ and . By ♦m we shall denote m repetitions of ♦. The similar convention applies to . e. it has the least closure base but it does not have Lindenbaum property. Of course, L is structural, it is defined by sequential rules. Hence the example we have produced (the same remark applies to the examples given earlier) covers both the case when C is assumed to be a consequence without specifying whether it is structural or not and the case when C is assumed to be a structural consequence.

As known, the maximal theories of J are the same as those of K, and the closure base they form is a closure base for K, not for J. (ii)→(i). In this case Lω is a good example. Let v be a valuation in a ω-valued Lukasiewicz matrix. Then Xv = {α : v(α) = 1} is, as one may prove, a maximal theory. In order to see this apply McNaughton 56 CHAPTER 5. HENKIN’S STYLE COMPLETENESS PROOFS. . [1951] results to show that for each x ∈ Lω there exists a formula α(p) in one variable p such that for all y ∈ L, α(p) = 1 iff y = x (pedantically: v(α(p)) = 1 iff v(p) = x, for all valuations in Lω ).

B. H is an adequate semantics for a consequence C iff {Xv : v ∈ H} is a closure base for C. Proof. Of course a. and b. are equivalent and hence it suffices to prove any of these conditions, say a. Let X be a closure base for C. Suppose that α ∈ C(X). If for some Y ∈ X, XY verifies all β ∈ X then X ⊆ Y and hence, C(X) ⊆ C(Y ) = Y . This yields XY (α) = 1. Now, suppose that α ∈ / C(X). Then for some Y ∈ X, X ⊆ Y and α ∈ / Y , which yields XY (X) ⊆ q and XY (α) = 0. Thus, indeed {XX : X ∈ X} is adequate for C.