Quantifiers - University of Nebraska Omaha

MATH 2230 — FALL 2013 15
that it is true. Thus, if a formula has a “free” instance of a variable,
then it does not make much sense to ask if it is true 12 .
Why do I add the adjective “free” when talking about a variable?
Note that not every occurrence of a variable is the same as any other.
Compare the following two formulas:
(a) x is a prime number bigger than 1000,
(b) there is a prime number x such that x>1000.
Again, if I ask you if the first formula is true, than you will reply
“depends on x”. But the same question on the second formula will
be clearly answered with “sure, this formula is true”! So what is the
di↵erence? Both formulas involve a variable x, but
• the first formula “speaks” about x, while
• the second formula “speaks” about 1000, the x here could be
totally eliminated. (We could have just say “there is a prime
number bigger than 1000.)
Look carefully what happens here. We have formula
x is a prime number bigger than 1000
(i.e., formula in (a)) and then, essentially, we add in front of it the
phrase “there is an x such that” getting
there is an x such that x is a prime number bigger than 1000
(and you should agree that this is the formula in (b)). In general, if we
have a formula with a free variable x, then by adding “there is an x such
that” in front of it we obtain a formula where x is not free anymore.
In situations like that we say that “the variable x got bounded by the
existential quantifier”.
In a parallel way we may consider adding “for all x we have that” in
front of a formula with a free variable x, asinthesentence
• for all real numbers x we have that x 2 0.
This operation also creates a formula where x is not free anymore
and we say then that “the variable x got bounded by the universal
quantifier”.
Thus we have two 13 quantifiers:
• the existential quantifier 9x (it is read “there is an x such that”,
“for some x we have”),
• the universal quantifier 8x (it is read “for all x we have”, “for
every x we have”).
12 unless there are some conventions around, see later
13 we may consider more quantifiers but these two are the most important