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Propositional Logic. 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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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.
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.”
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.”
Or operator (disjunction) • Or ( ): proposition p q is true exactly when either p is true or q is true:
And operator (conjunction) • And ( ): proposition p q is true exactly when p is true and q is true:
If and only if operator (iff) • If and only if (): proposition p q is true exactly when (p q) or (~ p ~ q):
Implies operator (if … then) • Implies (): proposition p q is true exactly when p is false or q is true:
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.
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
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.
Operator Precedence • • • • • Thus, p q ~p ~q means (p q) ((~p) (~q)).
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.
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).
Characters • • • • • •
