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 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 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