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# Propositional Logic PowerPoint PPT Presentation

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.

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.

### 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

•   

•       

•   

•  

•     

•        