Chapter 1
Preliminaries
1.1
Quantifiers
1
Sentential logic/propositional calculus:
A, B
are absolute statements about the state of affairs. Any expression like
A
is (globally)
true or false.
How to express more delicate ideas: relations between specific objects/individuals, etc?
Predicate logic (aka 1st order logic):
A
(
x
)
, B
(
x
) are statements about a variable
x
. It may be that
A
(
n
) is true but
A
(
m
) is
false!
So how to express when something is always true or sometimes true or never true?
Use quantifiers.
Definition 1.1.1
(Universal quantifier)
.
If
A
(
x
) is true for every possible value of
x
(under discussion), we say
∀
x, A
(
x
).
Example 1.1.1.
x
2
≥
0
,
∀
x
∈
R
. TRUE
∀
x, a < b
=
⇒
a
2
< b
2
. FALSE
∀
x
≥
0
, a < b
=
⇒
a
2
< b
2
. TRUE
∀
x
∈
(0
,
1)
,
∀
n, x
n
< x
. TRUE (assume
n
∈
N
).
1
May 2, 2007
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Math 413
Preliminaries
Definition 1.1.2
(Existential quantifier)
.
If
A
(
x
) is true for at least one allowable value
of
x
, we say
∃
x, A
(
x
), or
∃
x
such that
A
(
x
).
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 Fall '11
 Wong
 Calculus, Logic, Quantification, Model theory, Firstorder logic, Sentential logic/propositional calculus

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