Propositional LogicPropositional Logic

Sentence RestrictionsSentence Restrictions

• Precise use of natural language is Precise use of natural language is difficultdifficult..

• Want a notation that is suited to Want a notation that is suited to precisionprecision..

• Restrict discussion to sentences that are: Restrict discussion to sentences that are:

• declarative

• either true or false but not both.

• Such sentences are called Such sentences are called propositionspropositions..

Examples of propositionsExamples of propositions

Which of the sentences below are 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 Connectives5 Basic Connectives• Not (~)Not (~): p is true exactly when ~p is false.: p is true exactly when ~p is false.

• Denote by p “This class is the greatest Denote by p “This class is the greatest entertainment since the Rockford files.” entertainment since the Rockford files.”

• ~~p denotes “p denotes “It is not the case thatIt is not the case that this class is the this class is the greatest entertainment since the Rockford files.” greatest entertainment since the Rockford files.”

p ~p

T F

F T

Or operator (disjunction)Or operator (disjunction)

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

p q p q

T T T

T F T

F T TF F F

And operator (conjunction)And operator (conjunction)

• And (And ( )): proposition p : proposition p q is true exactly q is true exactly when p is true when p is true andand q is true: q is true:

p q p q

T T T

T F F

F T FF F F

If and only if operator (iff)If and only if operator (iff)

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

p q pq

T T TT F FF T FF F T

Implies operator (if … then)Implies operator (if … then)

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

p q p q

T T T T F F F T T F F T

If … then ...If … then ...

• Example: “If pigs had wings they could fly.”Example: “If pigs had wings they could fly.”

• In English, use of In English, use of impliesimplies normally connotes normally connotes

a a causalcausal relation: relation:

p implies q means that p causes q to be true.

• Not so with the mathematical definition!Not so with the mathematical definition!

If 1 1 then this class is fun.

p p q may be expressed as 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 & inverseConverse & inverse• The The converseconverse of p of p q is q q is q p. p.• The The inverseinverse of p of p q is ~p q is ~p ~q. ~q.• The The contrapositivecontrapositive of p of p q is ~q q is ~q ~p. ~p.• If p If p q then which, if any, is always true: 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 Describe the contrapositive of p q in terms of q in terms of converse & inverse.converse & inverse.

Operator PrecedenceOperator Precedence

1.1. 2.2. 3.3. 4.4. 5.5. Thus, p Thus, p q q ~p ~p ~q means ~q means

(p (p q) q) ((~p) ((~p) (~q)). (~q)).

Capturing the Capturing the formform of a of a Proposition in EnglishProposition in English

• Let Let gg, , hh, and , and bb be the propositions 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 Translate the following sentence using gg, , hh, , and and bb, and logical operators:, 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.

1.1. If berries are ripe along the trail, hiking is If berries are ripe along the trail, hiking is safe on the trail if and only if grizzly bears safe on the trail if and only if grizzly bears have not been seen in the area.have not been seen in the area.

2.2. If If bb, (, (hh if and only if if and only if g). g).

3.3. bb ( ( hh gg).).

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