Chapter 1 The Logic of Compound Statements Section

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 – 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 x 2 > 0. If x is negative, then x 2 > 0. : . x 2 > 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 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