Quantification --- Forming Propositions from Predicates
A predicate with variables is not a proposition. For example,
the statement x > 1 with variable x over the universe of real numbers
is neither true nor false since we don't know what x is. It can be true
or false depending on the value of x.
For x > 1 to be a proposition either we substitute a specific number for x
or change it to something like "There is a number x for which x > 1 holds",
or "For every number x, x > 1 holds".
More generally, a predicate with variables
(called an
For example, x > 1 becomes 3 > 1 if 3 is assigned to x, and it becomes a true statement, hence a proposition.
In general, a quantification is performed on formulas of predicate logic (called wff ), such as x > 1 or P(x), by using quantifiers on variables. There are two types of quantifiers: universal quantifier and existential quantifier.
The universal quantifier turns, for example, the statement x > 1 to "for every object x in the universe, x > 1", which is expressed as "x x > 1". This new statement is true or false in the universe of discourse. Hence it is a proposition once the universe is specified.
Similarly the existential quantifier turns, for example, the statement x > 1
to "for some
object x in the universe, x > 1", which is expressed as "
x x > 1." Again, it is true or false in the universe of discourse, and hence
it is a proposition once the universe is specified.
The Universal Quantifier The expression: x P(x), denotes the universal quantification of the atomic formula P(x). Translated into the English language, the expression is understood as: "For all x, P(x) holds", "for each x, P(x) holds" or "for every x, P(x) holds". is called the universal quantifier, and x means all the objects x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for every object x in the universe. For example, "All cars have wheels" could be transformed into the propositional form, x P(x), where:
The Existential Quantifier The expression: xP(x), denotes the existential quantification of P(x). Translated into the English language, the expression could also be understood as: "There exists an x such that P(x)" or "There is at least one x such that P(x)" is called the existential quantifier, and x means at least one object x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for at least one object x of the universe. For example, "Someone loves you" could be transformed into the propositional form, x P(x), where:
Existential Quantifier and Connective OR
If all the elements in the universe of discourse can be listed, then the existential quantification
xP(x) is equivalent to the disjunction:
P(x1) P(x2)
P(x3)
... P(xn).
For example, in the above example of x P(x),
if we knew that there were only 5 living creatures in our universe of discourse
(say: me, he, she, rex and fluff), then we could also write the statement as:
P(me) P(he) P(she)
P(rex) P(fluff)
An appearance of a variable in a
For example,
in x P(x, y), the variable x
is bound while y is free.
In x [ y
P(x, y)
Q(x, y) ] , x and the y in P(x, y)
are bound, while y in Q(x, y) is free, because the scope of
y is P(x, y).
The scope of
x is [ y
P(x, y)
Q(x, y) ] .
How to read quantified formulas
When reading quantified formulas in English, read them from left to right.
x can be read as "for every object x
in the universe the following holds" and x can be read as
"there erxists an object x in the universe which satisfies the following" or "for some object
x in the universe the following holds".
Those do not necessarily give us good English expressions. But they are where we can start. Get the correct
reading first then polish your English without changing the truth values.
For example, let the universe be the set of airplanes and let F(x, y) denote
"x flies faster than y". Then
xy F(x, y)
can be translated initially as "For every airplane x the following holds: x is faster
than every (any) airplane y".
In simpler English it means "Every airplane is faster than every airplane (including itself !)".
x y F(x, y)
can be read initially as "For every airplane x the following holds: for some airplane y,
x is faster than y".
In simpler English it means "Every airplane is faster than some airplane".
xy F(x, y)
represents "There exist an airplane x which satisfies the following: (or such that) for every airplane
y, x is faster than y". In simpler English it says
"There is an airplane which is faster than every airplane" or "Some airplane is
faster than every airplane".
x y F(x, y)
reads "For some airplane x there exists an airplane y such that x is faster than
y", which means "Some airplane is faster than some airplane".
Order of Application of Quantifiers
When more than one variables are quantified in a wff such as
y x
P( x, y ), they are applied from the inside, that is, the one closest to
the atomic formula is applied first. Thus
y x
P( x, y )
reads y [ x
P( x, y ) ] ,
and we say "there exists an y such that for every x, P( x, y ) holds" or "for some y, P( x, y ) holds
for every x".
The positions of the same type of quantifiers can be switched
without affecting the truth value
as long as there are
no
quantifiers of the other type between the ones to be interchanged.
For example
x y
z P(x, y , z)
is equivalent to
y x
z P(x, y , z),
z y
x P(x, y , z),
etc.
It is the same for the universal quantifier.
However, the positions of different types of quantifiers can
not be switched.
For example
x y
P( x, y ) is not equivalent to
y x
P( x, y ). For let P( x, y ) represent x
< y for the set of numbers as the universe, for example. Then x
y
P( x, y ) reads "for every number x, there is a number
y that is greater than x", which is true, while y x
P( x, y ) reads "there is a number
that is
greater than
every (any) number",
which is not true.