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CmSc 175 Discrete Mathematics
Lesson 07: Arguments with Quantified Statements
1. Rule of universal instantiation
If some property is true of everything in a domain, then
it is true of any particular thing in the domain.
This is the fundamental tool of deductive reasoning.
2. Universal Modus Ponens
The rule of universal instantiation can be combined with modus ponens to obtain the rule
called universal modus ponens:
 x, if P(x) then Q(x)
P(a) for a particular a
 Q(a).
(the symbol is read "therefore"
The argument consists of two premises at least one of which is universally quantified
The first and the second premises are called major and minor premises respectively.
Example:
1. If a number is even, then its square is even
2. 6 is a particular number that is even
62 is even
As a formal argument:
 x, if even(x) then even (x2)
even(6)
even(62)
3. Universal Modus Tollens
When we combine universal instantiation with modus tollens we have universal modus
tollens, the heart of proof by contradiction.
 x, if P(x) then Q(x)
 Q(a) for a particular a
  P(a).
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Example:
1. If a number is divisible by 6 then it is divisible by 2.
2. 15 is a particular number that is not divisible by 2
15 is not divisible by 6
As a formal argument:
 x, if div_by_6(x) then div_by_2 (x)
not div_by_2(15)
not div_by_6(15)
4. Converse and Inverse errors
Converse Error (Quantified Form): When we are given the premises "For all x, if P(x)
then Q(x)”, and "Q(a)" -- then P(a) is an invalid conclusion.
Inverse Error (Quantified Form): When we are given the premises "For all x, if P(x)
then Q(x)" and " ~P(a)" -- then ~Q(a) is an invalid conclusion.
Venn Diagrams: A particularly helpful technique for evaluating the validity of an
argument having quantified statements is to use Venn Diagrams. Depict each predicate
P(x) as a statement of membership of element x in some circular region P; and depict "if
P(x) then Q(x)" by sketching circle P strictly inside of circle Q; etc. If the conclusion
statement holds true for every value of x that makes the premise statements
(geometrically) true, then the argument is valid.
Examples
a. Converse error
All human beings are mortal
Felix is mortal
Felix is a human being <<<<<< Invalid conclusion
Venn diagram:
Felix
mortal
mortal
Felix
human
human
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Felix is somewhere in the circle representing the mortal beings. However Felix is
not necessarily within the circle representing humans
b. Inverse error
All human beings are mortal
Q is not human
Q is not mortal <<<<<< Invalid conclusion
Q
Q
mortal
human
mortal
human
Q is somewhere outside the circle of human beings, however Q is not necessarily outside
the circle of mortal beings
Diagrams for valid arguments
a. Universal modus ponens
All human beings are mortal
Socrates is human
Socrates is mortal
mortal
Socrates
human
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Socrates is in the circle of human beings. The circle of human beings is inside the circle
of mortal beings. Therefore Socrates is in the circle of mortal beings
b. Universal modus tollens
All human beings are mortal
Zeus is not mortal
Zeus is not human
Zeus
mortal
human
Zeus lies somewhere outside the circle of mortal beings. Since the circle of humans is
inside the circle of mortal beings, Zeus is outside the circle of human beings.
5. Exercises
Identify the type of the argument if valid, or the type of the error in case of invalid
argument
All nerds are good at math.
Buffy is good at math.
 Buffy is a nerd.
Animals at the bottom of the food chain are very nervous.
People are not at the bottom of the food chain.
 People are not nervous
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All trees have leaves.
Roses have leaves.
 Roses are trees.
Pigs can’t fly.
Wilbur is a pig.
 Wilbur can’t fly.
Pigs can’t fly.
Tweety can fly.
 Tweety isn’t a pig.
Every adult is eligible to vote.
John is eligible to vote.
 John is an adult.
Any odd integer x can be written in the form x = 2k+1 for some integer k
y is an odd integer
 y = 2k+1 for some integer k
My professors are happy when I pay close attention to their lectures.
My professors are not happy today
 I don’t pay close attention to their lectures today.
People who are good at logic make good programmers
Buffy would not be a good programmer
 Buffy is not good at logic
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