In the context of discussing the subsets of a given universe,
U, a set can be thought of as being a function which accepts
as valid input any element of U and for each such input
produces a truth value True or False. Similarly,
a binary relation on U can be thought of as a function which
accepts as input any pair from U and gives a value of
either True or False. Put more symbolically,
we let 
be the set of all ordered pairs from U
and then a binary relation on U is a function 
. A similar situation
applies to relations of more than two arguments. That is to
say, for instance, that a three argument relation on U is
a function from 
to 
.
Thus, when a language is interpreted, variable free terms
just represent elements of the universe and the truth value
of an atomic sentence is just the relation named by its
leading symbol applied to the input given by its arguments.
The propositional connectives are then defined by truth
functions. That is to say by functions which use truth
values for both input and output. The function 
for
instance is the function which produces the value False
on input True and yields True on input False.
This can be summarized by saying that is defined by the
following truth table:
With this definition, the truth value of 
is just
this function applied to the truth value of A. Each of
the other propositional connectives has a similar table.
Here is the table for 
:
The first two columns of this table give all possible
combinations of True and False for the
two variables and the third column gives the corresponding
function value. We see that 
is False
just in case both arguments are False.
Here is the table for the other two connectives, 
and
:
Note in particular that False False
is defined to be True. Some people may object to this
but they should again recall Humpty's dictum. By 
, we mean that proposition which is False only in the
case when it has a True hypothesis and a False
conclusion.
The truth functional nature of the two quantifiers, 
and 
, is somewhat trickier to capture. Take for instance
. Our intention is that 
is True
if there
is some element of the universe which makes A(x) True.
So that 
is True of the integers
because 2 + 2 = 4.
Now consider the real numbers and the sentence, 
. We know from the intermediate value theorem
that this polynomial has a root between -2 and 0.
However, we have no name for this root so we can not point to
a sentence in the language of arithmetic which makes this
existential statement true. Given an interpretation, what we
can do is choose a variable not given a value by the interpretation,
something original such as c, and extend the interpretation
to include letting c stand for a real root of this polynomial. Thus
is True because there is
a value for the new variable c
which makes 
True. Similarly,
is
True because no matter what real number is used to
interpret the variable c, the sentence 
is True. Thus
the Truth values of quantified sentences depends on the
potentiality of giving truth value of sub-sentences.
