EE1J2

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EE1J2 Discrete Maths Lecture 5

• Analysis of arguments (continued)
• More example proofs
• Formalisation of arguments in natural language
• Proof by contradiction

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Logical Consequence

• Let ? be a set of formulae and f a formula
• f is a logical consequence of ? if for any
  assignment of truth values to atomic propositions
  for which all of the members of ? true, f is also
  true
• If f is a logical consequence of ?, write ??f
• Note this is consistent with ?f when f is a
  tautology

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Arguments

• An argument consists of
• A set ? of formulae, called the assumptions or
  hypotheses
• A formula f, called the conclusion
• If ??f then the argument is a valid argument
• In other words, an argument is valid if its
  conclusion is a logical consequence of its
  assumptions.

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Notation

• An intuitive way to write an argument with a set
  of hypotheses ? and conclusion f is as follows

hypotheses
? --- ?f
conclusion
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Example proof 4

• Show that
• is a valid argument

(p ?q) ?(p ? r) r ------- ? q
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Proof 4

• (1) (p ? q)
• ?(p ? r)
• (3) r

(4) ?p ? ?r (from (2))
(5) ? ? r ? ?p (from (4)) (6) r ? ?p
(from (5))
(7) ?p ? q (from (1))
(8) r ? q (from (6) and (7))
(9) q (from (8) and (3))
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Alternative proof
(p ? q) ?(p ? r) r ------- ? q
(1) (2) (3)

• Assume that the conclusion is false
• i.e q is False
• Therefore p must be true (from (1))
• But p and r cannot both be true, by (2)
• Therefore r is false
• But this contradicts (3), so assumption must have
  been wrong

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Proof by Contradiction

• This is an example of proof by contradiction
• Basic idea is
• Assume that the conclusion is false
• Use this to deduce a contradiction
• Hence the conclusion must be true

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Proof by Contradiction

• Proof by contradiction is another powerful
  technique to show that an argument is valid
• Proof by contradiction is also known as
  reductio ad absurdum

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Reductio ad Absurdum

• Youve already met proof by contradiction as a
  rule of deduction
• This is also known as Reductio ad Absurdum

?p?(r??r) --------------- ? p
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Analysis of an Argument

• The meeting can take place if all members are
  informed in advance, and it is quorate. It is
  quorate provided that there are at least 15
  members present, and members will have been
  informed in advance if there is not a postal
  strike. Therefore, if the meeting was cancelled,
  there were fewer than 15 members present or there
  was a postal strike

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Identification of atomic propositions

• Atomic propositions are
• m the meeting takes place
• a all members have been informed in advance
• t - there are at least 15 members present
• q the meeting is quorate
• p there is a postal strike

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Formalisation of assumptions

• The meeting can take place if all members are
  informed in advance, and it is quorate
• becomes (a ? q ) ? m
• It is quorate provided that there are at least
  15 members present, and members will have been
  informed in advance if there is not a postal
  strike
• becomes ( t ? q) ? ( ? p ? a)

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Formalisation of assumptions (continued)

• So,
• ? (a ? q ) ? m , ( t ? q) ?( ? p ? a)
• These are the assumptions

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Formalisation of conclusion

• The argument concludes
• Therefore, if the meeting was cancelled, there
  were fewer than 15 members present or there was a
  postal strike
• which becomes ? m ? (?t ? p )
• So f ? m ? (?t ? p )
• Is f a logical consequence of ??

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Formal notation

• In our formal notation, the argument becomes

(a ? q ) ? m ( t ? q) ?( ? p ? a) ----------------
--------- ? ? m ? (?t ? p )
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Is this argument valid?

• 2 assumptions
• (a ? q ) ? m
• ( t ? q) ?( ? p ? a)
• 1 conclusion
• ? m ? (?t ? p )
• 5 atomic propositions implies 25 32 different
  allocations of truth values to atomic
  propositions

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Proof by Contradiction

• Proof by contradiction
• Assume ??f is false
• Then there is an allocation of truth values to
  atomic propositions for which all of the formulae
  in ? are true but f is false called a
  counter-example
• Show that the existence of a counter-example
  leads to a contradiction (e.g. that one of the
  formulae in ? must be false)

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Proof by contradiction is NOT

• where you prove that something is true by
  proving that it is false
• Anon., EE2F1 exam 2002

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Example

• Proof that is not a rational number

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Example Proof by Contradiction

• Suppose there exists an assignment of truth
  values to m, a, t, q and p such that
• (a ? q ) ? m, and ( t ? q) ?( ? p ? a)
• are both true, but
• ?m ? (?t ? p ) is false
• If ?m ? (?t ? p ) is false, then
• ?m must be true, and (?t ? p ) must be false

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Proof continued

• 3. It follows that
• m is false, t is true and p is false
• 4. Now consider the first formula in ?, namely
  ( t ? q) ? ( ? p ? a)
• 5. Since this is true, t ? q and ?p ? a must both
  be true
• 6. Hence a and q are true, because t and ?p are
  true (from above)

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Proof continued

• 7. Finally consider the second formula in ?,
  namely (a ? q ) ? m
• 8. Since q is true and a is true (from 6 on the
  previous slide), a ? q is true,
• 9. Hence m must be true
• 10. But this contradicts the assertion that m is
  false in part 3 on the previous slide

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Summary

• In summary, we have shown that the existence of
  an assignment of truth values for which ? is true
  and f is false leads to a contradiction.
• Hence such an assignment cannot exist.
• Hence ??f

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Example 2

• If the Big Bang theory is correct, then either
  there was a time before anything existed, or the
  world will come to an end. The world will not
  come to an end. Therefore, if there was no time
  before anything existed, the Big Bang theory is
  incorrect.

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Identification of atomic propositions

• Atomic propositions
• b the big bang theory is correct
• t there was a time before anything existed
• w the world will come to an end
• Formal statement of premises
• b ? (t ? w)
• ?w
• Formal statement of conclusion
• ?t ? ?b

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Proof by contradiction

• Formally, if
• ? b ? (t ? w), ?w,
• f is ?t ? ?b
• Is it the case that ? ? f ?
• Assume that f is not a logical consequence of ?
• Then there is an assignment of T and F to the
  atomic propositions such that each formula in ?
  is true and f is false

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Proof (continued)

• If ?t ? ?b is false, then
• ?t is true and ?b is false
• Hence t is false and b is true
• Now use the fact that, by assumption,
• b ? (t ? w) is true
• Since b is true, (t ? w) must be true
• But t is false. Hence w must be true. This
  contradicts assertion that ?w is true
• Hence ? ? f

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Summary

• In summary, we have shown that the existence of
  an assignment of truth values for which ? is true
  and f is false leads to a contradiction.
• Hence such an assignment cannot exist.
• Hence ??f

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Adequacy

• A set of propositional connectives is adequate if
  
• For any set of atomic propositions p1,,pN and
• For any truth table for these propositions,
• There is a formula involving only the given
  connectives, which has the given truth table.

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Adequacy

• The goal of the next lecture will be to show that
  the set ?, ?, ?, ? is adequate and contains
  redundancy, in the sense that it contains subsets
  which are themselves adequate
• We shall also introduce other sets of adequate
  connectives

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Summary

• More analysis of arguments
• Proof by contradiction