chapter 1 the logic of compound statements. section 1.2 – 1.3 (modus tollens) conditional and...
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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)
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
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
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