19 lines
744 B
Plaintext
19 lines
744 B
Plaintext
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Class 14 : *** Proof system for logic
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** Desirable properties of a proof system
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** Proof systems has a set of axioms (AXIOMS) and rules (R).
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** Set of formulas provable in the system = I(AXIOMS, R).
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Alternatively, provable formulas have legal construction sequences over AXIOMS and R.
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** Hilbert's proof system for Propositional Logic:
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* Axioms :
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* Ax1 : (\alpha --> (\beta --> \alpha))
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* Ax2 : ( (\alpha --> (\beta --> \gamma)) --> ((\alpha --> \beta) --> (\alpha --> \gamma)) )
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* Ax3 : ( (\neg \beta --> \neg \alpha) --> (\alpha --> \beta) )
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* Rules : Modus Ponens
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If we have proofs of \alpha, \(alpha --> \beta)
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then we can derive \beta
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** Example : proved (\alpha --> \alpha) in Hilbert's proof system.
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