Understanding the profound rules of contraposition

The third variety of immediate inference is contraposition. Its main objective is to form an equivalent proposition for a given proposition by transposing the subject and the predicate terms by their complements. But, unlike conversion and obversion, it is not an independent form of immediate inference.

It may be defined as a series of immediate inferences consisting of Obversion and Conversion applied in a particular order to a given premise. In other words, to form the contrastive of a given proposition, we first obvert it, then convert the result and obvert the result again.

The conclusion of contraposition is called contra positive, while its premise has no corresponding name. Hence, we shall follow the general convention and call the proposition to which we apply contraposition as the premise.

The contrapositive of a given proposition may be obtained by replacing the subject term by the contradictory of the predicate term and the predicate term by the contradictory of the subject term in the given premise.

Accordingly, the contrapositive of an A-proposition, “All members are voters” is “All non-voters are non-members”. Symbolically we may restate it as follows.

A – All S is P. Premise

A – All non-P is non-S. Contrapositive

Similarly,

A – All members are voters. Premise

A – All non-voters are non-members. Contrapositive

In the above example, the subject term ‘members’ is replaced by the contradictory of the predicate term “non-voter”. Similarly the predicate term of the premise ‘voters’ is replaced by the contradictory of the subject term “non-members”. Thus, we get. “All non- others are non-members” as the contrapositive of “All members are voters.”

Similarly the contrapesitive of “All S is P” is “All not -P is non -S”. We can follow the following procedure to obtain the contrapositive of a given proposition. First obvert the premise, then convert the result and ‘obvert it again.

Contraposition of A-Proposition:

Let us find out the contraposition of an A-proposition “All S is P”. To find the contrapositive, we first apply obversion to the premise, then apply conversion to the result and obvert it again to obtain the contrapositive of the premise.

(1) All S is P Premise.

(2) No S is non-P 1, Obversion.

(3) No non-P is S 2, Conversion.

(4) All non-P is non-S 3, Obversion.

Here our premise is an A-proposition for which we wish to obtain the contrapositive. So we apply obversion to 1 which yields 2. We write the justification for (2) on the right-hand side of step 2 as shown above. This means we obtain (2) from (1) by use of obversion.

Further, we apply conversion to (2) to get (3). Finally, we write the justification for (3) on its right-hand side as “2, conversion”. This means (3) is obtained from (2) by use of conversion. Finally the application of obversion to (3) yields (4) that is the contrapositive of (1). Thus, the contrapositive of an A- proposition is an A-proposition.

Contraposition of E – Proposition:

Here our premise is an E-proposition for which we wish to find the contrapositive. Hence, we write our premise “No S is P” as step 1 and on its right hand side we write ‘premise’ as the justification for (1). Then we obtain step 2 by applying obverse to step 1.

Similarly step 3 is obtained from step 2 by use of conversion. Finally, at step 4 we get the contrapositive “some non-P is not non-S” of “No S is P” by the application of obversion to step 3. Thus, the contrapositive of an E-proposition is an O-proposition. We rewrite above reasoning in a logically systematic manner as given below

(1) No S is P – Premise

(2) All S is non-P – 1, Obversion.

(3) Some non-P is S – 2, Conversion.

(4) Some non-P is not S 3, Obversion.

Contraposition of I – Proposition:

Let “Some S is P” be our premise for contraposition. We write it in step 1. Then, step 2 is obtained from step 1 by use of obversion. Thus, step 2 we get “Some S is not non-P”, that is an O-proposition. Since an O-proposition cannot be converted, I-proposition cannot have any contrapositive. Thus, we conclude that an I-proposition has no contrapositive.

Contraposition of O – Proposition:

(1) Some S is not P – Premise.

(2) Some Sis non-P – 1, Obversion.

(3) Some non-P is S – 2, Conversion.

(5) Some non-P is not non-S 3, Obversion 3 (Contrapositive).

Let our premise be O-proposition say ‘Some S is not P’. We apply Obversion to step 1 to obtain step 2 which is an I-proposition. Then, by the application of conversion to step 2 we obtain step 3 (Some non-P is S). Finally we apply obversion to step 3 to get our desired contrapositive, namely “Some non-P is not non-S.” Thus the contrapositive of an O-proposition is an O-proposition.

Premise	Contrapositive
A All S is P	A All non-P is non-S.
E No S is P	0 Some non-P is not non-S.
I Some S is P	I proposition has no contrapositive.
0 Some Sis not P	0 Some non-P is not non-S.