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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
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.
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