Methods of Proof for Boolean Logic

1
Methods of Proof for Boolean Logic
Language, Proof and Logic
Chapter 5
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Beyond truth tables
5.0

• Why truth tables are not sufficient
• Exponential sizes
• Inapplicability beyond Boolean connectives
• Need proofs, whether formal or informal.
• For informal proofs, it is relevant who your
  listener is.
• This section talks about some informal proof
  methods.

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Valid inference steps in informal proofs
5.1

• In giving an informal proof from some premises,
  if Q is already
• known to be a logical consequence of some already
  proven sentences,
• then you may assert Q in your proof.
• 2. Each step should be significant and easily
  understood (this is where
• your audiences level becomes relevant).

• Valid patterns of inference that generally go
  unmentioned
• From P?Q, infer P (conjunction
  elimination)
• From P and Q, infer P?Q (conjunction
  introduction)
• From P, infer P?Q (disjunction
  introduction)

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Indirect proof (proof by contradiction)
5.2
Contradiction --- any claim that cannot possibly
be true. Proof of ?Q by contradiction assume Q
and derive a contradiction. Proving that ?2
is irrational Suppose ?2 is rational. So, ?2
a/b for some integers a,b. We may assume at
least one of a,b is odd, for otherwise divide
both a and b by their greatest common divisor.
From ?2a/b we find 2a2/b2. Hence a22b2. So,
a is even. So, a2 is divisible by 4. So, b2 is
even. So, b is even. Contradiction.
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Proof by cases (disjunction elimination)
5.3

• To prove Q from a disjunction, prove it from each
  disjunct separately.
• There are irrational numbers b,c such that bc is
  rational.
• ?2?2 is either rational or irrational.
• If rational, then take bc ?2, already known to
  be irrational.
• If irrational, take b?2?2 and c ?2.

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Arguments with inconsistent premises
5.4
Premises from which a contradiction follows are
said to be inconsistent. You can prove anything
from such premises! An argument with
inconsistent premises is always valid yet never
sound!