CS 245 W15
Lecture 6
A. Lubiw, U. Waterloo
CS 245 / Winter 2015
Propositional Logic
CS 245 W15
Lecture 6
A. Lubiw, U. Waterloo
Summary of Inference Rules for Natural Deduction
Recall
Introduction
- natural deduction — proof system for propositional logic designed to
Elimination
be more like real math proofs
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- notation α ⊢ β means from α we can prove β
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CS 245 / Winter 2015
Propositional Logic
Summary of Inference Rules for Natural Deduction
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Introduction
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CS 245 / Winter 2015
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Summary of Inference Rules for Natural 4Deduction
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Lecture 6
A. Lubiw, U. Waterloo
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CS 245 W15
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CS 245 W15
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Lecture 5
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A. Lubiw, U. Waterloo
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CS 245 / Winter 2015
Propositional Logic
Summary of Inference Rules for Natural Deduction
CS 245 W15
Lecture 5
A. Lubiw, U. Waterloo
CS 245 W15
Introduction
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Elimination
Lecture 5
A. Lubiw, U. Waterloo
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CS 245 W15
Lecture 5
Some very useful “derived” rules:
modus tollens: α → β, ¬β ⊢ ¬α
A. Lubiw, U. Waterloo
CS 245 W15
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Lecture 5
A. Lubiw, U. Waterloo
Notes about rules of natural deduction
- this set is consistent with Huth and Ryan (section 1.2)
- different books present slightly different rules
law of excluded middle:
⊢ α ⋁ ¬α
- the set of rules is not minimal (some can be derived from others)
- ⊥ is not truly necessary (can replace by p ⋀ ¬p )
- these are rules of “classical” logic.
“Intuitionist logic” (which has applications in CS for type theory) eschews
the law of excluded middle (and also
¬¬α ⊢ α ). See [H&R 1.2.5].
CS 245 W15
Proof of modus tollens:
CS 245 W15
examples
Lecture 5
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Lecture 5
Proof of law of excluded middle:
α → β, ¬β ⊢ ¬α
Lecture 5
CS 245 W15
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CS 245 W15
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⊢ α ⋁ ¬α
Lecture 6
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CS 245 W15
Lecture 6
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CS 245 W15
Lecture 6
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CS 245 W15
Lecture 6
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CS 245 W15
Lecture 5
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CS 245 W15
Lecture 6
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CS 245 W15
Lecture 5
Proof of law of excluded middle:
CS 245 W15
Lecture 6
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CS 245 W15
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⊢ α ⋁ ¬α
Lecture 6
A. Lubiw, U. Waterloo
CS 245 W15
Lecture 6
A. Lubiw, U. Waterloo
Suggestions for Natural Deduction proofs
- try to work forward from the premises — can you apply an
elimination rule?
- try to work backwards from the conclusion — what introduction
rule do you need to use at the end?
- think about why the formula is true, and use that to guide your proof
- if a direct proof does not work, try a proof by contradiction
CS 245 W15
Lecture 6
What I expect you to know/do:
- do natural deduction proofs
- know what soundness and completeness mean
A. Lubiw, U. Waterloo