2010_lecture10 - CS 245 Lecture 10 January 2010 Shai...

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CS 245 Lecture 10 January 2010 Shai Ben-David from last class: α denotes is provable ' Equivalently, " α is a formal theorem of propositional calculus" Equivalently, α є I(Axioms, {MP}) Proofs From Assumptions Let Σ be a set of propositional formulas, and α be a propositional formula. We say that α is provable from Σ if α є I(Axioms U Σ ,{MP}) . We denote that by ' Σ α ' example: {A, A →B}├ B 1. A (element of Σ) 2. (A →B) (element of Σ) 3. B (MP on 1 & 2) example: {A, ( A)} B (for every propositions A and B) 1. (( ┐A)→(( ┐B)→( A))) Axiom 1 2. ( A) member of Σ 3. (( ┐B)→( A)) MP on 1 & 2 4. ((( ┐B)→( A)) →(A→B)) Axiom 3 5. (A →B) MP on 3 & 4 6. A member of Σ 7. B MP on 5 & 6 Conclusion: Propositional calculus demonstrates that assuming both a statement and it's negation implies EVERYTHING. An interesting property of our notion of proofs from assumptions is the monotonicity of our logic. Namely,
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2010_lecture10 - CS 245 Lecture 10 January 2010 Shai...

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