Conjugate Propositions

In any area of scientific inquiry, our reasoning is based explicitly or implicitly upon the laws of propositional logic. To explain the foregoing statement, Let us start with an example. Suppose we are given a proposition having the form of an implication. From this implication, we can form three further propositions namely, the converse proposition, the inverse proposition, and the contrapositive proposition. The converse, the inverse and the contrapositve propositions, together with the original proposition, are referred to as conjugate propositions.

Converse of a Conditional

The converse proposition is the result of interchanging the antecedent and the consequent  in the given (original) proposition.

The converse of a conditional proposition "if p then q" is

if q then p

Symbolically,

The converse of p → q is q → p

The fact is that a conditional proposition is not logically equivalent to its converse. That is,

p → q  ≠ q → p

Following truth table shows this fact.

As an example, consider the following proposition the implication:

if x is a positive number, then 2x is a positive number.

The converse of this proposition will be

if 2x is a positive number, then x is a positive number.

Although the above example shows that the converse proposition obtained from a true proposition is true, this is not at all true in general. Therefore, from the validity of an implication nothing definite can be inferred about the validity of the converse. This point, I think, needs a little explanation. Let us start with repeating myself that this example shows that the converse of a true position is true. But , as we can see from the above truth-table, this not a general rule. To see that this is not a general rule, it is sufficient to replace "2x" by "x²" in the above propositions (both - original and converse). The original proposition becomes:

if x is a positive number, then is a x² positive number.

and remains true. On the other hand, the converse proposition becomes:

if x² is a positive number, then x is a positive number.

and becomes false.

Now the question is how to interpret this sort of situation. Well! if it happens that the conditional and its converse propositions are both true, then the fact of their simultaneous truth can be expressed by joining the antecedent and consequent of any one of the two propositions by the words "if, and only if." That is,

x is a positive number if, and only if, 2x is a positive number.

And certainly, we can also interchange the two sides of the equivalence.

2x is a positive number if, and only if, x is a positive number.

Now we generalize the foregoing discussion by saying that other ways of saying the same thing:

1. The relation of consequence holds between these two propositions in both directions.

2. The two propositions are equivalent. [See Equivalence.]

3. Each of the two propositions represents a necessary and sufficient condition for the other. [See Necessary and Sufficient Conditions.]

Inverse of a Conditional

The inverse proposition is obtained by replacing both the antecedent and the consequent of the given proposition by their negation.

The Inverse of a conditional proposition "if p then q" is

if ~p then ~q

Symbolically,

The inverse of p → q is ~p → ~q

The fact is that a conditional proposition is not logically equivalent to its inverse. That is,

p → q  &ne ~p → ~q

Following truth table shows this fact.

As an example, consider the following proposition the implication:

if x is a positive number, then 2x is a positive number.

The inverse of this proposition will be

if x is not a positive number, then 2x is not a positive number.

Just like a converse proposition, the above example shows that the inverse proposition obtained from a true proposition is true, this is not at all true in general. Therefore, from the validity of an implication nothing definite can be inferred about the validity of the inverse proposition. To make sure that this is not a general rule, just like we handled converse proposition, it is sufficient to replace "2x" by "x²" in the above propositions (both - original and inverse). The original proposition becomes:

if x is a positive number, then is a x² positive number.

and remains true. On the other hand, the converse proposition becomes:

if x is not positive number, then x² is not a positive number.

and becomes false.

I

Contrapositive of a Conditional

The contrapositive is the result of interchanging the antecedent and the consequent  in the inverse proposition. Therefore, the contrapoitive proposition is the converse of the inverse proposition and also the inverse of the converse proposition.

The contrapositive of a conditional proposition "if p then q" is

If ~q then ~p

Symbolically,

The contrapositive of p → q is ~q → ~p

The fact is that a conditional proposition is logically equivalent to its contrapositive. That is,

p → q ≡ ~q → ~p

Following truth table shows this fact.

Since we have the same truth values in column 5 and 6 so p → q and ~q → ~p are logically equivalent.

Note that this logical equivalence is the basis for one of the most important laws of deduction, known as modus tollens.

Unlike converse and inverse propositions, the situation is quite different in the case of the contrapositive proposition; whenever an implication is true, the same applies to the corresponding contrapositive proposition. As a matter of fact, this is a general of propositional logic, namely the so-called law of transposition or lately people are saying law of contraposition.

Relationship among Conditional, Contrapositive, Converse, and Inverse

Lets construct the truth table for conditional, contrapositive, converse and inverse.

From the above truth table we conclude the following facts:

    1. A conditional and its converse are not logically equivalent.

    2. A conditional and its inverse are not logically equivalent.

    3. The converse and the inverse of a conditional are logically equivalent to each other.

Something about Inference

Now that we have covered definitions, fundamental laws of propositional logic and most importantly conjugates propositions let us talk a little bit about applications of these concepts in inference in the light of the law of transportation. We know (from the above) that every implication can be represented in schematic form:

The conjugate propositions are:

According to the law of transportation any conditional proposition implies corresponding contrapositive proposition. Therefore, proposition 1 may be formulated as:

Now, using the law of transportation, we would like to show that we can form a proposition having the form of implication. This can be easily done by simply comparing propositions 1 and 2. By comparing, we see that proposition 2 has the form of an implication in which proposition, if p, then q, being its hypothesis. Since the whole implication as well as its hypothesis have been accepted as true, the conclusion, if ~q, then ~p, must be true; But that is just the contrapositve proposition in question:

This completes the demonstration.

Now let us work on this concept with concrete example. Consider the following conditional proposition:

The three conjugate propositions are:

The law of transposition says that the conditional proposition implies contrapositive proposition. Therefore, we immediately have:

Now, using the law of transposition, our goal is to show that we can form a proposition having the form of implication. Clearly, proposition 5 is in the form of an implication in which:

Since the whole implication as well as its hypothesis have been acknowledged as true, the conclusion must be true. But the conclusion is just the contrapositive proposition in question:

This completes the demonstration.

To give you an idea of "Coming Attractions," let us talk about the mechanism of the proof by means of which the proposition 6 had been demonstrated. In the above "proof", we used two rules of proof:

1. The Rule of Substitution.

2. The Rule of Detachment (Modus Ponens.)

Substitution Rule

If a true proposition of a universal character contains propositional variables, and if these variables are replaced by other propositional variables, or by propositional functions, or by proposition - always substituting equal expressions for equal variables throughout -, then the proposition obtained in this way may also be recognized as true.

Consider the following proposition:

if (if p, then q), then (if ~q, then ~p)

Applying the law of substitution, we get:

Clearly, here:

                    p stands for "x is a positive number."

                    q stands for "2x is a positive number."

Note that the above substitution rule is not quite precise. We will comeback to this concept later after looking into the propositions of "universal character."

Detachment Rule

The rule of detachment states that if two true propositions, of which one has the form of an implication while other is the antecedent of this implication, then that proposition may also be recognized as true which forms the consequent of the implication. For example, in the above example, we have applied this law on the following two propositions:

and

and derive the following proposition: