Lecture10 - Lecture 10 Compactness/rules for the missing...

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Today’s lecture is the last in propositional logic. What it will contain: •Some final remarks on the completeness theorem and its proof •A proof of compactness •A discussion of how the natural deduction proof system can be expanded so that it has rules for ˮ , ¬, and  Lecture 10 Compactness/rules for the missing connectives
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Decidability is a property of a logical proof system---i.e. a system where notions of sentence and derivability has been fixed. A logical system is decidable if there is an effective procedure which determines whether or not Γ | ± ϕ for a set of sentences Γ and sentence ϕ in the system. Effective procedure an algorithm something you could program a computer to do. (In Phil 152 this notion is given a mathematically precise meaning.) Question: Is the natural deduction system for propositional logic (the system with rules ˭ I, ˭ E, I, E, , and RAA) decidable? Answer: Yes. We can define an algorithm that computes the truth tables for Γ and ϕ and then checks whether or not Γ | = ϕ . By the completeness theorem, the algorithm is also checking whether or not Γ | ± ϕ . Completeness and decidability
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This algorithm would thus tell us indirectly whether Γ | - ϕ by telling us whether Γ | = ϕ . It does not produce a derivation that confirms Γ | - ϕ directly. This leads to the question: is there an algorithm that produces a derivation of Γ from ϕ given that Γ | = ϕ ? One might hope that the completeness proof would help with this question, as it shows that if Γ | = ϕ then Γ | - ϕ . But the completeness proof we studied is not effective (as van Dalen puts it).
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Lecture10 - Lecture 10 Compactness/rules for the missing...

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