Valid and Invalid Arguments - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

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Chapter 1

The Logic of Compound Statements

Conditional and Valid & Invalid Arguments

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.
  • ~(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.

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.

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

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 Valid Form

• p v (q v r)
• ~r
• :. p v q

Example Invalid Form

• p -> q v ~r
• q -> p ^ r
• :. p -> r