Propositional Logic

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Propositional Logic
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Sentence Restrictions

• Precise use of natural language is difficult.
• Want a notation that is suited to precision.
• Restrict discussion to sentences that are
• declarative
• either true or false but not both.
• Such sentences are called propositions.

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Examples of propositions

• Which of the sentences below are propositions?
• Mastercharge, dig me into a hole!
• This class is fascinating.
• Do I exist yet?
• This sentence is false.

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5 Basic Connectives

• Not () p is true exactly when p is false.
• Denote by p This class is the greatest
  entertainment since the Rockford files.
• p denotes It is not the case that this class is
  the greatest entertainment since the Rockford
  files.

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Or operator (disjunction)

• Or (? ) proposition p ?q is true exactly when
  either p is true or q is true

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And operator (conjunction)

• And (? ) proposition p ? q is true exactly when
  p is true and q is true

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If and only if operator (iff)

• If and only if (?) proposition p ? q is true
  exactly when (p ? q) or ( p ? q)

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Implies operator (if then)

• Implies (?) proposition p ? q is true exactly
  when p is false or q is true

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If then ...

• Example If pigs had wings they could fly.
• In English, use of implies normally connotes a
  causal relation
• p implies q means that p causes q to be true.
• Not so with the mathematical definition!
• If 1 ? 1 then this class is fun.

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p ? q may be expressed as

• p implies q
• if p then q
• p only if q (if q then p)
• q if p
• q follows from p
• q provided p
• q is a consequence of p
• q whenever p
• q is a necessary condition for p (if q then p)
• p is a sufficient condition for q

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Converse inverse

• The converse of p ? q is q ? p.
• The inverse of p ? q is p ? q.
• The contrapositive of p ? q is q ? p.
• If p ? q then which, if any, is always true
• Its converse?
• Its inverse?
• Its contrapositive?
• Use a truth table to find the answer.
• Describe the contrapositive of p ? q in terms of
  converse inverse.

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Operator Precedence

• ?
• ?
• ?
• ?
• ?
• Thus, p ? q ? p ? q means
• (p ? q) ? ((p) ? (q)).

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Capturing the form of a Proposition in English

• Let g, h, and b be the propositions
• g Grizzly bears have been seen in the area.
• h Hiking is safe on the trail.
• b Berries are ripe along the trail.
• Translate the following sentence using g, h, and
  b, and logical operators
• If berries are ripe along the trail, hiking is
  safe on the trail if and only if grizzly bears
  have not been seen in the area.

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• If berries are ripe along the trail, hiking is
  safe on the trail if and only if grizzly bears
  have not been seen in the area.
• If b, (h if and only if ? g).
• b ? ( h ? ? g).

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Characters

• ? ? ?
• ? ? ? ? ? ? ?
• ? ? ?
• ? ?
• ? ? ? ? ?
• ? ? ? ? ? ? ? ?