1. Introduction to Predicates and Quantified Statements II
Lecture 8
Section 2.2
Mon, Jan 24, 2005

2. Negations of Universal Statements
The negation of
x  S, P(x)
is the statement
x  S, P(x).
If “x  R, x2 > 10” is false, then “x  R, x2  10” is true.

3. Negations of Existential Statements
The negation of
x  S, P(x)
is the statement
x  S, P(x).
If “x  R, x2 < 0” is false, then “x  R, x2  0” is true.

4. Example: Negation of a Universal Statement
p = “Everybody likes me.”
Express p as
x  {all people}, x likes me.
p is the statement
x  {all people}, x does not like me.
p = “Somebody does not like me.”

5. Example: Negation of an Existential Statement
p = “Somebody likes me.”
Express p as
x  {all people}, x likes me.
p is the statement
x  {all people}, x does not like me.
p = “Everyone does not like me.”
p = “Nobody likes me.”

6. Negation of a Universal Conditional Statement
The negation of x  S, P(x)  Q(x) is the statement
x  S, (P(x)  Q(x))
which is equivalent to
x  S, P(x)  Q(x).

7. Example
Find the error(s) in the following “solution.”

8. The Operators, , , and 
Let D = {x1, x2, …, xn}.
The statement x  D, P(x) is equivalent to
P(x1)  P(x2)  …  P(xn).
The statement x  D, P(x) is equivalent to
P(x1)  P(x2)  …  P(xn).

9. The Operators, , , and 
That is why x  D, P(x) is false if P(x) is false for a single x  D.
And that is why x  D, P(x) is true if P(x) is true for a single x  D.

10. Supporting Universal Statements
Consider the statement
“All crows are black.”
Let C(x) be the predicate “x is a crow.”
Let B(x) be the predicate “x is black.”
The statement can be written formally as
x, C(x)  B(x).
Question: What would constitute evidence in support of this statement?

11. Supporting Universal Statements
The statement is logically equivalent to
x, ~B(x)  ~C(x).
Question: What would constitute evidence in support of this statement?