Chapter 1

The Logic of Compound Statements

Section 1.3

Valid & Invalid Arguments

Review

• Review of last lecture– Conditional Statement

• if-then, ->• p -> q ~p v q

– Negation of Conditional• ~(p -> q) p^ ~q

– Contrapositive of Conditional• p -> q ~q -> ~p

• Review– Converse of Conditional

• (p->q) is (q->p)

– Inverse of Conditional• (p->q) is (~p->q)

– Converse Inverse – Biconditional

• “p if, and only if q”, p <-> q, TRUE when both p and q have same logic value

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 form 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.

Examples

• Modus Ponens– “If you have a current password, then you can log

on to the network”– “You have a current password”– :. ???

• Modus Tollens– Construct the valid argument using modus tollens.

• p->q, ~q, :. ~p• What is p and q?• What is ~q?

Rules of Inference

• Rule of inference is a form of argument that is valid.– Modus Ponens, Modus Tollens– Generalization, Specialization, Elimination,

Transitivity, Proof by Division, etc.

Rules of Inference

• Generalization– p :. p v q– q :. p v q

• Specialization– p ^ q :. p– p ^ q :. q– Example:

• Karl knows how to build a computer and Karl knows how to program a computer

• :. Karl knows how to program a computer

Rules of Inference

• Elimination– p v q, ~q, :. p– p v q, ~q, :. p– Example• Karl is tall or Karl is smart.• Karl is not tall.• :. Karl is smart.• x-3=0 or x+2=0• x ~< 0 • :. x = 3 (x-3=0)

Rules of Inference

• Transitivity (Chain Rule)– p -> q, q -> r, :. p -> r– Example• If 18,486 is divisible by 18, then 18486 is divisible by 9.• If 18,486 is divisible by 9, then the sum of the digits of

18,486 is divisible by 9.• :. 18,486 is divisible by 18, then the sum of the digits

18,486 is divisible by 9.

Rules of Inference

• Proof by Division– p v q, p->r, q->r, :.r– Example• x is positive or x is negative.• If x is positive, then x2 > 0.• If x is negative, then x2 > 0.• :. x2 > 0

Fallacies

• A fallacy is an error in reasoning that results in an invalid argument.

• Converse Error– If Zeke is a cheater, then Zeke sits in the back row.– Zeke sits in the back row.– :. Zeke is a cheater.

• Inverse Error– If interest rates are going up, then stock market prices will go

down.– Interest rates are not going up.– :. Stock market prices will not go down.

Contradictions and Valid Arguments

• Contradiction Rule – If you can show that the supposition that statement p is false leads logically to a contradiction, then you can conclude that p is true.– ~p -> c, :. p