MIT24_241F09_lec13

MIT24_241F09_lec13 - Logic I Session 13 Plan Damien on...

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Unformatted text preview: Logic I - Session 13 Plan Damien on psets Quick summary of completeness Compactness Limitations of SL Intro to PL Completeness Γ ⊬ P ↓ Γ ∪ { ∼ P } is C-SD ↓ Γ ∪ { ∼ P } ⊆ a MC-SD set Γ * ↓ If Γ * is MC-SD then Γ * is TF-C ↓ Γ ∪ { ∼ P } ⊆ a TF-C set Γ * ↓ Γ ∪ { ∼ P } is TF-C ↓ Γ ⊭ P Compactness A cool result of completeness: Compactness : Γ is TF-C iff every ¡nite subset of Γ is TF-C. So: a set Γ is TF-IC only if a ¡nite subset of Γ is TF-IC. So, intuitively, there’s no TF inconsistency that you need an in¡nite number of SL sentences to get! Let’s prove compactness by proving each direction. Compactness First, left-to-right: If Γ is TF-C, then every ¡nite subset of Γ is TF-C. If there were a subset Γ- such that no TVA m.e.m. Γ- true, then there would be no TVA m.e.m. Γ true. Now, right-to-left: If every ¡nite subset of Γ is TF-C, then Γ is TF-C....
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This note was uploaded on 01/25/2012 for the course PHIL 201H1F taught by Professor Derekallen during the Fall '10 term at University of Toronto.

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MIT24_241F09_lec13 - Logic I Session 13 Plan Damien on...

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