# 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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