Propositional logic
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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
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
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
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 operator (disjunction)

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


And operator conjunction
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 operator (iff)

  • If and only if (): proposition p  q is true exactly when (p  q) or (~ p  ~ q):


Implies operator if then
Implies operator (if … then)

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


If then
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  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
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
Operator Precedence

  • Thus, p  q  ~p  ~q means

    (p  q)  ((~p)  (~q)).


Capturing the form of a proposition in english
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.


Propositional logic


Characters
Characters trail if and only if grizzly bears have not been seen in the area.

  •   

  •       

  •   

  •  

  •     

  •        