Formal Methods in Computer Science CS1502 Proof by Contradiction

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Formal Methods in Computer ScienceCS1502Proof
by Contradiction

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To understand what a contradiction is.
• 4 new proof rules
• __ Intro, __ Elim, Intro, Elim
• To understand the method of proof by
  contradiction. ( Intro)
• To gain skills in using
• proof by contradiction by itself and
• in combination with proof by cases.
• To develop intuition when to use proof by cases
  and when to use proof by contradiction.
• Summary of proof rules page 557 in text

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Negation Elimination ( Elim)
Double negation

• 3. A
• A Elim 3

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Example

• Let p be an integer.
• Prove that if p is even, then p2 is even.
• Easy
• If p2 is even, then p is even
• Harder

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Even(p2) implies Even(p)

• We know that p2 is even. This is a given fact.
• Suppose that p is odd. We will show that this
  leads to a contradiction. (Therefore, the
  opposite must be true)
• Since p is odd, then p 2 k 1 for some integer
  k.
• Then p2 (2 k 1)(2 k 1) 4 k2 4 k 1.
• Thus, p2 is odd.
• But this contradicts to the fact that p2 is
  even.
• Our earlier assumption that p is odd must be
  false.
• Thus, the opposite must be true, i.e. p is even.

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What is a contradiction?

• What is a logical truth?
• Tet(a) \/ Tet(a)
• a a
• A contradiction is
• a sentence that is false in any circumstance
• A /\ A
• a ! a
• FrontOf(a,b) /\ FrontOf(b,a)
• Tet(a) /\ Cube(a) /\ Dodec(a)
• Symbol __ reads contradiction
• But lets call it bottom.

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Bottom Introduction (__ Intro)

• A
• A
• __ __ Intro 3,7

Now that we have __, why is this useful? Can we
use it somehow?
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• Suppose we have __ in our proof.Does anything
  follow from it?

?
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• Tet(a)
• Cube(a)
• Large(c)
• Is the argument valid?

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• Definition An argument is valid if
• whenever all the premises are true, the
  conclusion is also true.
• An argument is not valid if
• there is at least one circumstance such that all
  the premises are true, but the conclusion is
  false.

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Validity revisited

• Recall validity vs soundness of arguments
• Valid doesnt mean the premises are true.
• Valid In any circumstance, whenever all the
  premises are true, the conclusion is true.
• Sound Valid and all premises are true.
• Consider the followings.
• There is no circumstance in which the premises
  are all true.
• There is no circumstance in which the premises
  are all true and the conclusion is false.
• There is no counterexample.
• Thus, the argument is valid.
• This is called Vacuously valid.

Tet(a) Cube(a) Large(c)
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Also works in the body of proof

• If the premises are inconsistent, the argument
  is always valid.
• This also works in the body of a proof.
• If we have a set of inconsistent sentences in our
  proof, we can assert any sentence P whatsoever.
• We need a proof rule for this.

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__ Intro and __ Elim

• __ Intro __ Elim
• 3. A 3. __
• 7. A
• __ __ Intro 3,7 B __ Elim 3

for any sentence B
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Example

• Tet(a) /\ Cube(b)
• Tet(a)
• Small(d)

• Tet(a) \/ Cube(b)
• Tet(a)
• Dodec(b)
• Small(d)

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Why do we care?

• So, if the premises are inconsistent, the
  argument is always valid.
• But the argument is always unsound (There is no
  circumstance in which all premises are true.)
• There are no real situations where the argument
  is applicable. Why do we care about this?

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Example

• Tet(a) \/ Cube(b)
• Tet(a)
• Dodec(b)
• Small(d)
• from earlier

• Tet(a) \/ Cube(b)
• Tet(a)
• Dodec(b)
• Cube(b)
• modified

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First use of __

• Use __ in subproofs in proof by cases for cases
  that are impossible to happen.
• Anywhere else?
• Yes. __ is used in proof by contradiction.
• How?

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• Suppose we know that
• A /\ B /\ C is true
• and
• A /\ B /\ C /\ D is false
• What can be said about D?

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• We begin with the fact that A /\ B /\ C is true.
• But when we add an assumption that D is true, the
  whole thing becomes inconsistent.
• Thus, A /\ B /\ C being true implies that D must
  be false (D must be true)

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Negation Introduction ( Intro)
Proof by contradiction

• 5. D
• 9. __
• D Intro 5-9

(1) Premises and other sentences derived from the
premises.
(2) Added assumption.
(3) A contradiction is derived from (1) AND (2).
(4) We can conclude the opposite of (2).
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• Enough background
• Lets put it to use!!

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Even(p2) implies Even(p)

• We know that p2 is even. This is a given fact.
• Suppose that p is odd. We will show that this
  leads to a contradiction. (Therefore, the
  opposite must be true)
• Since p is odd, then p 2 k 1 for some integer
  k.
• Then p2 (2 k 1)(2 k 1) 4 k2 4 k 1.
• Thus, p2 is odd.
• But this contradicts to the fact that p2 is
  even.
• Our earlier assumption that p is odd must be
  false.
• Thus, the opposite must be true, i.e. p is even.

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Even(p2) implies Even(p)

• 01. Even(p2)
• 02. ---
• 03. Even(p)
• 04. ---
• 05. p 2k1 for some integer k from 03.
• 06. p2 (2k1)(2k1) 4k2 4k 1 from
  05.
• 07. Even(p2) from 06.
• 08. __ __ Intro 01,07.
• 09. Even(p) Intro 03-08.

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Even(p2) implies Even(p)

• 01. Even(p2)
• 02. ---
• 03. p2 2g for some integer g
• 04. p sqrt(2 g)
• 05. stuck
• 06.
• 07. Even(p)

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• In the subproof of a proof by contradiction, we
  get an additional assumption to use. This allows
  us to make progress.
• The goal of the subproof in proof by
  contradiction is __.

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Example

• A \/ B
• A
• B

• (A /\ B)
• A
• B

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Exercise

• Cube(b) \/ Tet(a)
• Dodec(a)
• (Cube(b) /\ Large(d))
• ???
• What interesting fact can be concluded?
• Prove it

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Exercise

• (B /\ A) \/ C
• B
• ???
• A
• What non-trivial premise makes the argument
  valid?
• Prove it.

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More proofs
(P /\ P)
P \/ P
P \/ Q (P /\ Q)
(P /\ Q) P \/ Q