MIT24_241F09_lec23

# MIT24_241F09_lec23 - Logic I Session 23 Completeness of PD...

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Logic I - Session 23 Completeness of PD

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Soundness recap Last time we sketched a proof that PD is sound. I.e., if Γ P in PD then Γ P . The main part of the proof is proving that for any derivation: If Γ i P i for all i k, then Γ k+1 P k+1. We prove this by showing that it holds for each rule that could justify line k+1.
Soundness recap The strategy for the individual cases goes like this: Given the rule justifying line k+1, try to draw conclusions about the form of P k+1. Then draw conclusions about the structure of the derivation above line k+1 and about the forms of sentences on earlier lines, e.g. Q i and R j. Apply the inductive hypothesis to Q i and R j. Note the relationships among Γ i, Γ j, and Γ k+1. Draw conclusions about relationship between Γ k+1 and Q i and R j. Put this together with semantic deﬁnitions and

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Completeness Next up: prove if Γ P then Γ P in PD Remember the main strategy for completeness of SD. We argued that for sets of SL sentences: Any C-SD set is a subset of a MC-SD set Every MC-SD set is TF-C Every subset of a TF-C set is TF-C. So any C-SD set is TF-C. We then appealed to connections between consistency and entailment and derivability. We’ll have a similar strategy for PD.
Preliminary deﬁnitions MC-PD: Γ * is M aximally C onsistent in PD iff Γ * is consistent in PD and Γ * { P } is inconsistent for any P not already in Γ *. C: Γ is E xistentially C omplete iff for each sent. in Γ of the form ( x)P , there’s a substitution instance of ( x)P in Γ , e.g. P(a/x) . ES sets: Set Γ e is E venly S ubscripted iff it is the result of doubling the subscript of every i.c. in Γ .

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Completeness Γ P ES-variant of Γ { d P } a MC- C-PD set Γ * (11.4.4) ES-variant of Γ { d P } is C-PD Γ { d P } a Q-C set Γ * Γ { d P } is Q-C Γ P If Γ * is MC- C-PD then Γ * is Q-C (11.4.8)
Assume Γ P . Then if Γ { d P } were IC-PD, then we could derive some Q and d Q from Γ { d P }. And in that case, from Γ we could derive P by d E, contradiction the assumption that Γ P . So Γ { d P } is C-PD. Now we want to show that the ES variant of Γ { d P } is C-PD. So lets show that for any Γ , if Γ is C-PD, then Γ e is C-PD. Γ P The ES-variant of Γ { d P } is C-PD

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Suppose Γ is C-PD. Then Γ e is the result of doubling the subscript on each individual constant in each sentence in Γ . To show that Γ e is C-PD: Suppose, for reductio, that Γ e were IC-PD. Then we could derive some Q and d Q from Γ e. And then a certain
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MIT24_241F09_lec23 - Logic I Session 23 Completeness of PD...

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