# Mathematical Logic by Simpson S.G.

By Simpson S.G.

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**Example text**

The objects are signed sequents. 3 This amounts to saying that at least one of the truth values T and F is an interpolant for A ⇒ B. 4 An unsigned sequent is just what we have previously called a sequent. 49 2. We have M, X → X, N and M, X, X → N and M → X, X, N for all X. 3. For each signed tableau rule of the form .. X .. .. X .. | Y | Y1 Y2 .. X .. Y .. X .. / \ Z / \ Y1 Z1 Y2 Z2 we have a corresponding pair of signed sequent rules M, X, Y → N M, X → N M → X, Y , N M → X, N M, X, Y1 , Y2 → N M, X → N M → X, Y1 , Y2 , N M → X, N M, X, Y → N M, X, Z → N M, X → N M → X, Y , N M → X, Z, N M → X, N M, X, Y1 , Y2 → N M, X, Z1 , Z2 → N M, X → N M → X, Y1 , Y2 , N M → X, Z1 , Z2 , N M → X, N respectively.

8, we have (∃x P x) & (∃x Qx) ≡ (∃x P x) & (∃y Qy) ≡ ≡ ∃x (P x & (∃y Qy)) ∃x ∃y (P x & Qy) and this is in prenex form. 13. Let A and B be quantifier-free formulas. Put the following into prenex form. 1. (∃x A) & (∃x B) 2. (∀x A) ⇔ (∀x B) 3. 14 (universal closure). Let A be a formula. The universal closure of A is the sentence A∗ = ∀x1 · · · ∀xk A, where x1 , . . , xk are the variables which occur freely in A. Note that A∗∗ = A∗ . 15. Let A be a formula. 1. Show that A is logically valid if and only if A∗ , the universal closure of A, is logically valid.

2. Each element of U σ is a term of sort σ. 3. If f is an n-ary operaton of type σ1 × · · · × σn → σn+1 , and if t1 , . . , tn are terms of sort σ1 , . . , σn respectively, then f t1 . . tn is a term of sort σn+1 . An atomic L-U -formula is an expression of the form P t1 . . tn , where P is an n-ary predicate of type σ1 × · · · × σn , and t1 , . . , tn are terms of sort σ1 , . . , σn respectively. 3, with clause 5 modified as follows: 5’. If xσ is a variable of sort σ, and if A is an L-U -formula, then ∀xσ A and ∃xσ A are L-U -formulas.