Formal%20Proofs%20and%20Boolean%20Logic

1
Formal Proofs and Boolean Logic
Language, Proof and Logic
Chapter 6
2
Conjunction rules
6.1
? Intro P1 ? Pn P1??Pn
? Elim P1??Pi??Pn Pi

1. A?B?C
2. B ? Elim 1
3. C ? Elim 1
4. C?B ? Intro 3,2

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Disjunction rules
6.2a
? Elim P1??Pn ? S
? Intro Pi P1??Pi??Pn
P1 S
Pn S
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Example
6.2b
1. (A?B) ? (C?D) 2. A?B
3. B ? Elim 2
4. B?D ? Intro 3 5. C?D
6. D ? Elim 5 7.
B?D ? Intro 6 8. B?D
? Elim 1, 2-4, 5-7

You try it, page 152
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Contradiction and negation rules
6.3
? Intro P ?P ?
? Elim ? P
? Intro P ? ?P
? Elim ??P P
You try it, p.163
6
The proper use of subproofs
6.4
A subproof may use any of its own assumptions and
derived sentences, as well as those of its
parent (or grandparent, etc.) proof. However,
once a subproof ends, its statements are
discharged. That is, nothing outside that
subproof (say, in its parent or sibling proof)
can cite anything from within that subproof.
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Strategy and tactics
6.5

• When looking for a proof, the following would
  help
• Understand what the sentences are saying.
• Decide whether you think the conclusion follows
  from the premises.
• If you think it does not follow, look for a
  counterexample.
• If you think it does follow, try to give an
  informal proof first, and
• then turn it into a formal one.
• 5. Working backwards is always a good idea.
• 6. When working backwards, though, always check
  that your
• intermediate goals are consequences of the
  available information.
• 
  You try it, page 170.

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Proofs without premises
6.6
The conclusion of such a proof is always
logically valid!
1. P??P 2. P ? Elim 1
3. ?P ? Elim 1 4. ?
? Intro 2,3 5. ?(P??P) ? Intro
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