Formal Methods in Computer Science CS1502 Proofs involving Conditionals

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Formal Methods in Computer ScienceCS1502Proofs
involving Conditionals

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To learn proof rules involving conditionals.
• To gain skills in constructing proofs involving
  conditionals.
• (Soundness and Completeness of a deductive system)

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Conditional Proof

• If argument is valid, it means
• P implies Q.
• We can capture this in an FOL sentence
• P ? Q
• Hence, the rule ? Intro

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? Intro (cond intro)
Conditional proof

• 3 P
• 7 Q
• P ? Q ? Intro 3-7

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Conditional Proof
A /\ B implies A \/ B.
(A /\ B) ? (A \/ B) is a logical truth.
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? Elim
(Modus Ponens)

• 3 P ? Q
• 5 P
• Q ? Elim 3,5

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A Simple Proof

• A ? B
• B ? C
• A ? C

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? Intro (bicond intro)

• 2 P
• 5 Q
• 7 Q
• 9 P
• P ? Q ? Intro 2-5,7-9

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? Elim

• 3 P ? Q (or Q ? P)
• 5 P
• Q ? Elim 3,5

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Example Exercise

• C ? (A \/ E)
• E ? B
• C ? (A \/ B)

• C ? (A \/ E)
• E ? C
• C ? (A \/ B)

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Problem 8.32

• Horned(c) ? (Elusive(c) /\ Dangerous(c))
• (Elusive(c) \/ Mythical(c)) ? Rare(c)
• Mammal(c) ? Rare(c)
• Horned(c) ? Mammal(c)

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Proving a set of equivalent sentences

• The following conditions are all equivalent Q1,
  Q2, , Qn

Q1 ? Q2 Q1 ? Q3 Q1 ? Q4 How many equivalent
pairs?
How many pairs do we need to prove?
Is there a better way?
Q1 ? Q2, Q2 ? Q3, , Qn ? Q1
We can use any ordering besides Q1, Q2, Q3, ,Qn