Chapter 1

The Logic of Compound Statements

Section 1.2 – 1.3 (Modus Tollens)

Conditional and Valid & Invalid Arguments

Conditional Statements

• A conditional statement is a sentence of the form “if p then q” or p -> q (p implies q).– p is the hypothesis– q is the conclusion

Example

• “If you show up for work Monday morning, then you will get the job.”– p = You show up for work Monday Morning.– q = You will get the job.– p -> q

• When is this statement false?

Example ->

• p v ~q -> ~p• Order of precedence: 1. ~, 2. ^,v, 3. ->, <->• p v ~q -> ~p (pv~q) -> (~p)

Logical Equivalence ->

• pq -> r (p ->r) ^ (q ->r)

Equivalence -> & or

• p -> q ~p v q

• Example– ~p v q = “Either you get to work on time or you are

fired.”– ~p = You get to work on time.– q = You are fired.– p = You do not get to work on time.– p -> q = “If you do not get to work on time, then you are

fired.”

Negation of Conditional

• Negation of if p then q “p and not q”• ~(p -> q) p^ ~q• Derivation from Theorem 1.1.1– ~(p -> q) ~(~p v q)– ~(~p) ^ (~q) by DeMorgan’s– p ^ ~q by the double neg law

• Example– If Karl lives in Wilmington, then he lives in NC.– Karl lives in Wilmington and he does not live in NC.

Contrapositive of a Conditional

• The contrapositive of p -> q is ~q -> ~p.• Conditional is logically equivalent to its

contrapositive: p -> q ~q -> ~p

p q ~p ~q p->q ~q -> ~p

T T F F T T

T F F T F F

F T T F T T

F F T T T T

Example

• Conditional p->q– If Howard can swim across the lake, then Howard

can swim to the island.– p = “Howard can swim across the lake.”– q = “Howard can swim to the island.”

• Contrapositive ~q -> ~p– If Howard cannot swim to the island, then Howard

cannot swim across the lake.

Converse of Conditional

• Converse of conditional “if p then q” (p->q) is “if q then p” (q->p)

• Converse is not logically equivalent to the conditional.

• Example– (conditional) If today is Easter, then tomorrow is

Monday.– (converse) If tomorrow is Monday, then today is

Easter.

Inverse of Conditional• Inverse of conditional “if p then q” (p->q) is “if ~p then ~q”

(~p->q)• Inverse is not logically equivalent to the conditional.• Example

– (conditional) If today is Easter, then tomorrow is Monday.– (inverse) If today is not Easter, then tomorrow is not Monday.

• However, the converse and inverse are logically equivalent.

•

p q ~p ~q p->q q->p ~p->~q

T T F F T T T

T F F T F T T

F T T F T F F

F F T T T T T

Biconditional

• Biconditional is “p if, and only if q”.• Biconditional is T when both p and q have the

same logic value and F otherwise. • Symbolically – p <-> q

Biconditional Truth Table

Necessary & Sufficient Conditions

• For statements r and s,– r is a sufficient condition for s (if r then s) means

“the occurrence of r is sufficient to guarantee the occurrence of s”.

– r is a necessary condition for s (if not r then not s) means “if r does not occur, then s cannot occur”.

Valid & Invalid Arguments

• An argument is a sequence of statements.• All statements in an argument, except for the

final one, is the premises (hypotheses).• The final statement is the conclusion.

• Valid argument occurs when the premises are TRUE, which results in a TRUE conclusion.

Testing Argument Form

• Identify the premises and conclusion of the argument form.

• Construct a truth table showing the truth values of all the premises and the conclusion.

• If the truth table reveals all TRUE premises and a FALSE conclusion, then the argument from is invalid. Otherwise, when all premises are TRUE and the conclusion is TRUE, then the argument is valid.

Example

• If Socrates is a man, then Socrates is mortal.• Socrates is a man.• :. Socrates is mortal.

• Syllogism is an argument form with two premises and a conclusion. Example Modus Ponens form:– If p then q.– p– :. q

Example Valid Form

• p v (q v r)• ~r• :. p v q

Example Invalid Form

• p -> q v ~r• q -> p ^ r• :. p -> r

Modus Tollens

– If p then q.– ~q– :. ~p

– Proves it case with “proof by contradiction”– Example:– if Zeus is human, then Zeus is mortal.– Zeus is not mortal.– :. Zeus is not human.