Lecture 4 Notes

Ai d1 dn does not hold if t aind1 dn for

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Unformatted text preview: terms of L+ }† 2. 3. Distinguished elements of DI are the constant letters: ai is ai, and bi is bi. Relations on DI are defined by: n Ai (d1 , ..., dn ) holds if T Ain(d1, ..., dn) n 4. Ai (d1 , ..., dn ) does not hold if T ∼Ain(d1, ..., dn), for d1, ..., dn ∈ DI. Functions on DI are defined by: n n fi (d1 , ..., dn ) = fi (d1 , ..., dn ) , for d1, ..., dn ∈ DI. Lemma 4: Proof: By Base Step: 1. "⇒". For any closed wf A of L+ , T A iff I A. induction on the number n of connectives/quantifiers in A. n = 0, A is an atomic formula Ai n(d1, ..., dn), where d1, ..., dn are closed terms. Supppose T A. n Then: Ai (d1 , ..., dn ) holds in DI. (definition of I.) So: For every valuation v of I, v satisfies Ai n(d1, ..., dn). Thus I A. 2. "⇐". Suppose T A. Then: T ∼A. (T is complete and A is closed.) n So: Ai (d1 , ..., dn ) doesn't hold in DI. (definition of I.) Thus: For every valuation v of I, v doesn't satisfy Ai n(d1, ..., dn). So I A. Induction Step: Suppose A has n...
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