32
Hardegree, Symbolic Logic
The analogy continues. Just as we can apply functions to numbers (addition,
subtraction, exponentiation, etc.), we can apply functions to truth values. Whereas
the former are numerical functions, the latter are truth-functions.
In the case of a numerical function, like addition, the input are numbers, and
so is the output. For example, if we input the numbers 2 and 3, then the output is 5.
If we want to learn the addition function, we have to learn what the output number
is for any two input numbers. Usually we learn a tiny fragment of this in
elementary school when we learn the addition tables. The addition tables tabulate
the output of the addition function for a few select inputs, and we learn it primarily
by rote.
Truth-functions do not take numbers as input, nor do they produce numbers as
output. Rather, truth-functions take truth values as input, and they produce truth
values as output. Since there are only two truth values (compared with infinitely
many numbers), learning a truth-function is considerably simpler than learning a
numerical function.
Just as there are two ways to learn, and to remember, the addition tables, there
are two ways to learn truth-function tables. On the one hand, you can simply
memorize it (two plus two is four, two plus three is five, etc.) On the other hand,
you can master the underlying concept (what are you doing when you add two
numbers together?) The best way is probably a combination of these two tech-
niques.
We will discuss several examples of truth functions in the following sections.
For the moment, let's look at the definition of a truth-functional connective.
A statement connective is truth-functional if and only if
the truth value of any compound statement obtained by
applying that connective is a function of (is completely
determined by) the individual truth values of the con-
stituent statements that form the compound.
This definition will be easier to comprehend after a few examples have been dis-
cussed. The basic idea is this: suppose we have a statement connective, call it +,
and suppose we have any two statements, call them S
1
and S
2
. Then we can form a
compound, which is denoted S
1
+S
2
. Now, to say that the connective + is truth-
functional is to say this: if we know the truth values of S
1
and S
2
individually, then
we automatically know, or at least we can compute, the truth value of S
1
+S
2
. On
the other hand, to say that the connective + is not truth-functional is to say this:
merely knowing the truth values of S
1
and S
2
does not automatically tell us the truth
value of S
1
+S
2
. An example of a connective that is not truth-functional is discussed
later.
4. CONJUNCTION
The first truth-functional connective we discuss is conjunction, which cor-
responds to the English expression ‘and’.