Propositional Function

Now that we know something about constants and variable, let us talk about proposition a little more and then we will talk about the propositional function. Let us start with a following proposition which is, if, fact, a special theorem of arithmetic.

If 1 is a positive number and 1 < 2, then 1 is a positive number.

Obviously, this proposition is true. (Obviously- huh? since it contains constants only.)

And also, it is not difficult to see that this proposition is really boring and completely uninteresting specially from arithmetic point view. To see why this is boring and uninteresting notice the following:

You may say these are just words i.e. "you" said it is boring and to support your claim of "boringness" you threw some fancy words like truth, depends, if, then, etc. (like we haven't heard political speeches of known corrupt politicians) what? am I an idiot?

Well then, to really make sure that the above proposition is "really really" boring, let us replace components:

"1 is a positive number" and "1 < 2"

by any other propositions and what we got? The result is a series of true propositions just like the original proposition.

In order to make the above boring proposition interesting, we generalize the proposition. In other words, in order to express the fact state by our "boring proposition" in a more general form, we shall introduce the variables (say) "p" and "q", demanding that these variable symbols stands for whole proposition; variable of this kind are denoted as propositional variables (or sentential variables).

So lets do it! In the boring proposition under consideration, we replace the phrase "1 is a positive number" and formula "1<2" by "p" and "q," respectively. In this manner, we arrive at the propositonal function:

if p and q, then p

This propositional function has the property that only true propositions are obtained if arbitrary propositions are substituted in place of "p" and "q." Using this very property, we can state the above proposition is in the form of a universal proposition:

For all p and q, if p and q, then p

Note that this is an example of the law of simplification for logical multiplication. And the boring sentence considered above was merely a special instance of this universal law. Needless to say that now our "boring proposition" becomes really interesting. Since, now it asserts something about the properties of this "boring proposition."

In a similar manner, other laws of propositional logic such as law of identity, hypothetical syllogism, ... etc. can be obtained. Like our foregoing example, the laws of propositional logic asserts something about the properties of arbitrary propositions.

Russell defined propositional function (or an open proposition or condition) as follows:

In other words, it is a function whose values are proposition. Note that in a propositional function, the values must actually express propositions.

Consider the following expression:

x is human.

This expression is a propositional function since there is an undetermined variable namely, x. The expression remains propositional function so long as the variable x has no definite value or constant. Note that the above expression is neither true nor false, but when a value is assigned to x it becomes true or false proposition.

An expression of this kind, which contains variables and, on replacement of these variables by definite values or constants, becomes a proposition, is called a propositional function. Note that in mathematics, more often the word "condition" is employed in this sense; and propositional functions and propositions which are composed of mathematical symbols are usually referred as formula, such as

x + y = 5.

A propositional function becomes a proposition only after definite values or constants have been inserted in the place of the variables and the "things" denoted by those definite values or constants are said to "satisfy" the given propositional function. According to Russell, a propositional function standing all alone may be taken to be a mere condition (or schema or a mere shell) not something already significant.

For example, Socrates (say) satisfies the propositional function:

x is human.

After we assigned the value "Socrates" to the variable x, the above propositional function becomes a proposition: 

Socrates is human.

Any mathematical equation is a propositonal function as long as the variables have no definite value or constant, the equation is propositional function awaiting determination in order to become a true or false proposition.

For example, if propositional function is an "identity" it will be true when the variable is any number. (True when those numbers satisfy the identity.) For instance, the numbers 1, 2, 2.5 satisfy the propositional function:

x < 3

On the other hand, if propositional function is an equation containing one variable:

ax + c = 0

This propositional function becomes a true proposition when the variable is made equal to a "root", otherwise it becomes false.

Similarly, the equation to a curve in plane is a propositional function, true for values of the of the coordinate belonging to points on the curve, false for all other values.

Now we formalize the above discussion.

Let D be a given set. A propositional function defined on D is an expression

p(x)

which has a property that p(a) is true or false for each a ∈ D.

In other words, a propositional function, p(x), becomes a becomes a proposition (with a truth value) whenever any element a ∈ D is substituted for the variable x.

The set D is called the domain of p(x), and the set Tp of all element of the set D for which p(a) is true is called the truth set of p(x). That is,

Tp = {x : x ∈ D, p(x) is true}

or simply

Tp = {x : p(x)}

Frequently, when domain D is some set of numbers (such as R "real number", N "integers", Q "rational numbers", ...), the proposition function p(x) has the form of an equation or inequality involving the variable x.

As an example, consider the domain of x is the set of all positive integers, N. Our problem is to find the truth set, Tp, of the given propositional function p(x).

1. Given propositional function p(x) is "x + 2 > 7". The truth set of p(x) is

T_p = {x : x ∈ N, x + 2 = 7} = {6, 7, 8, ...}

Thus p(x) is true for some elements in N.

2. Given propositional function p(x) is "x + 5 < 3". The truth set of p(x) is

T_p = {x : x ∈ N, x + 5 < 3} = {} or  ∅

Thus p(x) is true no element in N.

3. Given propositional function p(x) is "x + 5 > 1". The truth set of p(x) is

T_p = {x : x ∈ N, x + 5 > 1} = N

Thus p(x) is true for all elements in N.

Let us conclude this section by saying some about propositional function in the light of traditional logic. Expressions of traditional logic such as

all A is B

are propositional functions: as we have learnt from the above discussion that A and B have to be determined as definite classes (see the theory of classes) before such expressions become true or false. The important point we would like to make is that the notion of "cases" and "instances" depends on propositional functions. To clarify this point, consider an example of generalization:

Lightning is followed by thunder.

We have number of "instances" of this statement one of which:

This is a flash of lightning and is followed by thunder.

Clearly, the above "instance" is occurence of the propositional function:

If x is a flash of lightning, x is followed by thunder.

The process of generalization consists in passing from a number of such instances to the universal truth of the proposition fucntion:

If x is a flash of lightning, x is followed by thunder.

In other words, propositional functions are always involved whenever we talk of "instances" or "cases" or "examples."

We are interested in propositional functions in two ways:

We will discuss the theory of classes and theory of relations separately. In the next section, we will talk about the former.