Chapter 1

1. Chapter 1 The Logic of Compound Statements

2. Section 1.2 – 1.3 (Modus Tollens) Conditional and Valid & Invalid Arguments

3. 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

4. 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?

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

6. Logical Equivalence -> • pq -> r (p ->r) ^ (q ->r)

7. Equivalence -> & or • p -> q ~pvq • Example • ~pvq = “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.”

8. 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)  ~(~pvq) •  ~(~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.

9. Contrapositive of a Conditional • The contrapositive of p -> q is ~q -> ~p. • Conditional is logically equivalent to its contrapositive: p -> q ~q -> ~p

10. 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.

11. 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.

12. 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.

13. 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

14. Biconditional Truth Table

15. 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 scannot occur”.

16. 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.

17. 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.

18. 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

19. Example Valid Form • pv (qvr) • ~r • :. pvq

20. Example Invalid Form • p -> qv ~r • q -> p ^ r • :. p -> r

21. 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.