Transcript Chapter 1 The Logic of Compound Statements Section 1.3 Valid & Invalid Arguments.

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