1 - A4 Logic(25 points Here is one form of the Completeness...

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Unformatted text preview: A4 Logic (25 points) Here is one form of the Completeness Theorem for first order predicate logics, and you may and should use thtsform without proof in this problem.2 Theorem 1 (Completeness) Suppose f is a first order predicate logic language. Suppose (F U {A}) is a set of formulas of 1?. l- is the pmoobiltty relation for a fixed sound and complete system of proof for 2.3 i: is the (semantic) consequence relation for 2.4 Then: P|=AerP|—A. Here is one form of the Compoctness Theorem for (first order) predicate logics, Theorem 2 (Compactness) Suppose E isa first order predicate logic language. Suppose I‘ is a set of formulas of E. IThen: 1" has a model 41> (V finite A Q 1")[A has a model]. ' Prove the above form of the Compactness Theorem from the above form of the Completeness Theorem employing the Hint just below. Hint: Assume the above form of the Completeness Theorem. Of course, do not assume 1" is finite. (V) The (=>) direction of Compactness is easy Without even explicitly employing the Completeness assumption. For the (4:) direction of Compactness, prove the contrapositive, i.e., prove the negation of the left-hand side implies the negation of the right-hand side. First assume I" has no model. Suppose B is a. fixed closed formula. Show that l" |= (B /\ —xB). Employ each direction of the Completeness Theorem and something about the SIZE of proofs (in the fixed system of proof) to get a finite A g I‘ such that A ,5 (B /\ —.B) too. Then show that this A has no model. 2IF you want to use a. different form, you will have to state and prove it. 3This system can be, for example, a. resolution system, one of many tableaux systems, a Hilbert style system, a. Gentzen style system, . 1" I— A means that A is provable from 1" in the fixed system of proof. ‘1" I: A means that every model of I‘ satisfies A. ...
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