The Building Blocks of Arguments
•  Predicates
•  Implications
•  Logical Pitfalls
Predicates
•  Predicate: P(x), statement that
incorporates a variable(s) (x), such that
it whenever x is specified P(x) becomes
unambiguously T or F.
•  Domain of P(x) is the set of x’s for
which P(x) can be evaluated
Predicates: Examples
•  P(n) is “n is even”
–  Domain: integers
–  Evaluate: 1, 4, 5, 8, 15, 105, 2000
•  P(x): x>15
•  R(x): (x>5)∧(x<20)
–  Evaluate: 1, 10, 100
Predicates: Negation
•  P(x): (x<0)
–  P(x)’: (x<0)’ ≅ (x≥0)
•  P(x): (x<5)∧(x>20)
–  P(x)’: ((x<5)∧(x>20))’
(x<5)’∨(x>20)’
(x≥5)∨(x≤20)
Predicates and Domains
•  Predicates partition the domain
T
P(x)
P(x)
F
•  “Does the predicate become true for elements in
the domain?”
–  Sometimes
–  Never
–  Always
Some Important Symbols
•  “element of”
•  “for all”
–  “For students (s) in this class (C)…”
•  “there exists”
–  “For students (s) in this class (C), there
exists a number (i) which is a valid student
ID (I)” “Quantifiers”
Also: “Quantified predicate”
Examples
•  D = {-2, -1, 0, 1, 2}
•  Translate to English and decide T/F
Negating Quantified Predicates
•  Opposite of “exists” is “there does not
exist”
–  This becomes “for all” in the negated
predicate
•  Opposite of “for all” is “there exists an
exception” –  This become “exists” in negative predicate
Examples
•  D = {-2, -1, 0, 1, 2}
•  Translate to English and decide T/F
Example: Multiple Quantifiers
Set of all integers
Implications
•  Implication: statement of the form “if p
is true then q is true”
–  p, q – propositions
•  p – hypothesis
•  q - conclusion
–  Shorthand “if p then q”
–  Shorthand “p -> q”
–  The implication has lower precedence than
other logical operators
–  Sometimes called “conditionals”
Examples
•  Indentify domain and two predicates to
put into canonical form
–  “If a real number has a real square root,
then it is positive”
–  “If a real number satisfies x*x–x=6, then
x=3”
–  “If an integer is even, then 2^n-1 is a
multiple of 3”
–  “If an integer ends in 3, then it’s a
multiple of 3”
Truth Tables
•  Two cases to consider
Consider all p, q
Consider only p->q
Prove implication:
Show that this can never happen
Disprove implication:
Show that this can happen (e.g. find example)
Implications and Games
•  “A policeman walks into a bar…”
–  Law: If you are drinking you are 21
Al
Betty
Cindy
19
Coke
Beer
Dan
25
Implications and English
•  Car dealer (A): “If you did not buy your
car from us, you paid too much”
–  Identify p and q
•  Which statements are consistent or
inconsistent
–  You
–  You
–  You
–  You
bought from A and got a good deal
bought from A and paid too much
didn’t buy from A and paid too much
didn’t buy from A and got a good deal
Examples
•  T/F?
If F give counter example
Negation of Implication
•  The negation of p->q is p∧q’
–  How do we prove this?
Not an implication!
Negation of Quantified Implication
Practice - Negation
•  “If you by an extended warranty, then
nothing will go wrong with your ipod”
•  “If Chris gets a flu shot then we will
not get sick”
•  “For all triangles t, if t has 3 equal
sides then it has 3 equal angles”
•  “For all x in {1, 2, 3, 4, 5}, x^2 is
positive”
Other Implications
•  Consider the implication (I):
•  The converse of I is:
•  The inverse of I is:
•  The contrapositive is:
Important Properties
•  An implication and its contrapositive are
logically equivalent
•  The converse and the inverse of an
implication are logically equivalent
•  An implication is not logically equivalent
to its inverse.
Prove these
Car Dealer Again
•  Car dealer (A): “If you did not buy your
car from us, you paid too much”
–  Identify p and q
•  Identify the relationship
–  “If you
–  “If you
us”
–  “If you
–  “If you
us”
bought from us, you got a good deal”
paid too much, you did not buy from
got a good deal you bought from us”
got a good deal, you did not buy from
Biconditional
•  P <-> q means p->q and q->p
•  English: “p if and only if q”
•  Example: –  Suppose CS2100 is restricted majors
–  Suppose CS2100 is required for majors
–  Write the above in as quantified
implications on the domain “students”
–  Write the biconditional statement
Prove that if p<->q, then p,q are logically equivalent
Implications, Predicates & Domains
•  Implications are often used to indicate
properties of subsets of domains
–  Two options:
•  Use the restricted domain
•  Construct a logical that is conditional on membership
in the subset
•  Rewrite the following as implications
(specify the new old and new domain)
•  For all even integers m, m ends with the digits 0, 2,
4 or 8
•  For all x > 0, x^2 > x
•  For every positive odd integer n, n^3 – n is divisible
by 4
More Liars
•  A says “If B is truthful, so am I”. B
says “At least one of us is lying”. –  Who (if anyone) is truthful?
Validity of Arguments
•  P->q
–  If p is true, then q is true
–  p is true only if q is true
–  For q to be true, it is sufficient that q is
true
–  For p to be true, it is necessary that q be
true
•  “therefore”, an implication that is
reached as part of a proof or argument
Reaching Conclusions
•  “Your mother is at work and your
father is at work”
–  “therefore, your father is at work”
•  “If you eat your peas or you carrots,
then you are done”
–  “I ate my peas, therefore I am done”
Typical Fallacies
•  The converse fallacy is an invalid form
of reasoning given by:
•  The inverse fallacy is an invalid form of
reasoning given by: