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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Propositional Logic

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Propositional logic

Propositional Logic


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

  • 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

Characters

  •   

  •       

  •   

  •  

  •     

  •        


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