Transcript Slide 1

Discrete Structures
Chapter 3: The Logic of Quantified Statements
3.4 Arguments with Quantified Statements
The only complete safeguard against reasoning ill, is the habit of
reasoning well; familiarity with the principles of correct
reasoning; and practice in applying those principles.
– John Stuart Mill, 1806 – 1873
3.4 Arguments with Quantified Statements
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Universal Instantiation
(in-stan-she-AY-shun)
• The rule of universal instantiation is that if
some property is true of everything in a set,
then it is true of any particular thing in the set.
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Universal Instantiation
• One of the most famous examples of universal
instantiation is as follows:
– All men are mortal.
– Socrates is a man.
– Socrates is mortal.
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Universal Modus Ponens
• Universal modus ponens is a combination of universal
instantiation and modus ponens. It is Latin for "mode that
affirms".
Formal Version
x, if P  x  then Q  x  .
P  a  for a particular a.
 Q  a .
Informal Version
If x makes P( x) true, then x makes Q( x) true.
a makes P( x) true.
 a makes Q( x) true.
• The first line is called the major premise and the second
line is the minor premise.
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Example – pg. 142 # 2
• Use universal instantiation or universal modus
ponens to fill in valid conclusions.
– If an integer n equals 2k and k is an integer, then n
is even.
– 0 equals 20 and 0 is an integer.
–
.
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Example – pg. 142 # 3
• Use universal instantiation or universal modus
ponens to fill in valid conclusions.
– For all real numbers a, b, c, and d, if b  0 and
d  0, then a/b + c/d = (ad + bc)/(bd).
– a = 2, b = 3, c = 4, and d = 5 are particular real
numbers such that b  0 and d  0.
–
.
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Definition
• Valid
– To say that an argument form is valid means the
following: No matter what particular predicates are
substituted for the predicate symbols in its
premises, if the resulting premise statements are all
true, then the conclusion is also true. An argument
is called valid iff its form is valid.
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Using Diagrams to Show Validity
• Example
– (informal) All integers are rational numbers.
– (formal) ∀integers n, n is a rational number.
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Using Diagrams to Show Invalidity
• Show invalidity of the following argument
– All human beings are mortal.
– Felix is mortal.
– ∴ Felix is a human being.
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Modus Ponens in Diagrams
x, if P  x  then Q  x  .
P  a  for a particular a.
 Q  a .
x | Q( x)
x | P( x)
a
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Example – pg. 143 # 21
• Indicated whether the arguments are valid or
invalid. Support your answers by drawing
diagrams.
– All people are mice.
– All mice are mortal.
–  All people are mortal.
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Universal Modus Tollens
• Universal modus ponens is a combination of universal
instantiation and modus tollens. It is Latin for "mode that
denies".
Formal Version
Informal Version
x, if P  x  then Q  x  .
~ Q  a  for a particular a.
 ~P  a  .
If x makes P( x) true, then x makes Q( x) true.
a does not make Q( x) true.
 a does not make P( x) true.
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Example – pg. 142 # 5
• Use universal modus tollens to fill in valid
conclusions for the arguments.
– All irrational numbers are real numbers.
– 1/0 is not a real number.
–
.
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Converse Error
• This claim is most simply put as
– If A, then B.
– B.
–  A.
• It's a fallacy because at no point is it shown
that A is the only possible cause of B;
therefore, even if B is true, A can still be false.
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Converse Error
• In other words, a universal conditional statement is not
logically equivalent to its converse.
Formal Version
Informal Version
x, if P  x  then Q  x  .
If x makes P( x) true, then x makes Q( x) true.
a makes Q( x) true.
 a makes P( x) true.
Q  a  for a particular a.
 P  a.
Invalid conclusion
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Converse Error - Example
• If my car was Ferrari, it would be able to travel
at over a hundred miles per hour.
• I clocked my car at 101 miles per hour.
• Therefore, my car is a Ferrari.
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Inverse Error
• The conditional statement is not logically equivalent to its
inverse.
Formal Version
x, if P  x  then Q  x  .
~ P  a  for a particular a.
 ~Q  a  .
Informal Version
If x makes P( x) true, then x makes Q( x) true.
a does not make P( x) true.
 a does not make Q( x) true.
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Inverse Error - Example
• If I hit my professor with a cream pie, she will
flunk me.
• I will not hit my professor with a cream pie.
• Therefore, she will not flunk me.
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Example – pg. 143 # 15
• The argument may be valid by universal modus
ponens or universal modus tollens; or the
argument is invalid and exhibit the converse or
inverse error. State which are valid and which are
invalid. Justify your answers.
– Any sum of two rational numbers is rational.
– The sum r + s is rational.
–  The numbers r and s are both rational.
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Example – pg. 143 # 18
• The argument may be valid by universal modus
ponens or universal modus tollens; or the
argument is invalid and exhibit the converse or
inverse error. State which are valid and which are
invalid. Justify your answers.
– If an infinite series converges, then the terms go to 0.

n
– The terms of the infinite series  n  1 do not go to 0.
n 1

n
–  The infinite series  n  1 does not converge.
n 1
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