L1.34
Notes: Logic (continued)
The expression x>3 is not a proposition.
It contains a free variable.
However, if we specify that
x
is 1, that is, if we
bind
x
to the value 1,
then x>3 becomes a proposition. It’s false in this case.
Expressions like x>3, containing an unbound or free variable, are called
propositional
functions
.
We often denote them like this: P(x): x>3 or
sometimes we simply say “Let P(x) denote x>3.” While P(x) is a
propositional function,
P(9) or P(1) are propositions.
P(9) is 9>3 and is
True. P(1) is 1>3 and is False. In effect, the substitution of a number for
x
in
the propositional function has the effect of binding the variable
x
to that
number.
We can do this with two (or more) variables:
Let Q(x,y) be x>y.
It is a propositional function in two variables.
Q(3,2) is a proposition.
It has a truth value.
Q(x,2) is a propositional function of one variable.
In the above discussion P and Q (by themselves) are called
predicates
.
x
and
y
are
variables
.
Predicates and propositional functions go beyond numbers.
We can make
them do anything we want.
For instance:
Let P(x): x murdered Nicole
Then P(OJ) is a proposition, namely OJ murdered Nicole.
In this case notice that P(3) wouldn’t make sense, any more than OJ>3
would make sense in the earlier cases. Thus we note that predicates and their
propositional function require a
UNIVERSE OF DISCOURSE (often
denoted as U)
. U
is the collection of entitities which the variables in a
propositional function are allowed to be bound to. When you have
predicates, propositional functions and variables, there
must
be a universe of
discourse. It may be implicit. It may consist of all possible entitities in the
universe. But whatever it is, it must be there, and it is your right to demand
to know what the universe of discourse is when shown a predicate used as a
propositional function.
Let’s move on.
Let the Universe U be numbers.
Consider the statement:
1
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“For every number x, x=3.”
Is it a proposition?
Can we determine whether
it is true or false?
The answer is yes, we can.
It is a proposition and it is
False in this case. We have succeeded in binding the variable by use of the
words “For every.”
Logically we proceed this way.
Let P(x): x>3
We write
2200
xP(x), meaning
2200
x(x>3), meaning in words, for every number x,
x>3.
More accurately,
2200
x means “For every element in the Universe.”
2200
xP(x) IS TRUE IF AND ONLY IF P(x) IS TRUE FOR EVERY
SUBSTITUTION FOR x FROM U.
Consider in the same vein the statement: “There is a number x such that
x>3.” Is that a proposition?
Of course.
It’s True. Letting P(x): x>3, we
denote this expression as
5
xP(x) or
5
x(x>3) meaning, there is an element in
the Universe such that it is larger than 3.
5
xP(x) IS TRUE IF AND ONLY IF P(x) IS TRUE FOR AT LEAST ONE
SUBSTITUTION FOR x FROM U.
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 Spring '08
 WATKINS
 Logic, Quantification, Universal quantification, Existential quantification, propositional function, propositional functions

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