The following may be taken as typical examples of the different operations included above under the name contraposition:—
| All AB is CD or de ; | |
| therefore, | (1) Anything that is either cD or dEis a or b ; |
| (2) All A is b or CD or de ; | |
| (3) Whatever is ABD or ABE is CD. | |
| Combinations of the second and third operationsgive | |
| Anything that is Ac or Ad is b or de ; | |
| Anything that is BD or BE is a or CD ; | |
| &c. | |
In all the above cases one or more terms disappear from the subject or the predicate of the original proposition, and are replaced by their contradictories in the predicate or the subject accordingly. Only in the full contrapositive, however, is every term thus transposed.
The importance of contraposition as we are now dealing with it in connexion with complex propositions is that by its means, given a universal affirmative proposition of any complexity, we may obtain separate information with regard to any term that appears in the 493 subject, or with regard to the contradictory of any term that appears in the predicate, or with regard to any combination of such terms.
Thus, given All AB is C or De, by the process described as the generalisation of the subject we have All A is b or C or De, All B is a or C or De, Everything is a or b or C or De ; the particularisation of the subject yields All ABc is De, Whatever is ABd or ABE is C, &c.; and by the combination of these processes we have All Ac is b or De, &c.
Again, the full contrapositive of the original proposition is Whatever is cd or cE is a or b ; from which we have All c is a or b or De, Whatever is d or E is a or b or C, &c.
457. Summary of the results obtainable by Obversion, Conversion, and Contraposition.—The following is a summary of the results obtainable by the aid of the processes discussed in the three preceding sections:
(1) By obversion any proposition may be changed from the affirmative to the negative form, or vice versâ.
For example, All AB is CD or EF, therefore, No AB is ce or cf or de or df ; Some P is not QR, therefore, Some P is either q or r.
(2) By the conversion of a universal negative proposition separate information may be obtained with regard to any term that appears either in the subject or in the predicate, or with regard to any combination of these terms.
For example, from No AB is CD or EF we may infer No A is BCD or BEF, No C is ABD or ABEF, No BD is AC or AEF, etc.
Also by conversion any universal negative proposition may be reduced to the following: Nothing is either X1 or X2 … or Xn.
For example, the above proposition is equivalent to the following: Nothing is either ABCD or ABEF.
(3) By the conversion of a particular affirmative proposition separate information may be obtained with regard to any determinant of the subject or of the predicate, or with regard to any combination of such determinants.
For example, from Some AB or AC is DE or DF we may infer Some A is BDE or BDF or CDE or CDF, Some D is ABE or ABF or ACE or ACF, Some AD is BE or BF or CE or CF, etc.
Also by conversion any particular affirmative proposition may be reduced to the form Something is either X1 or X2 … or Xn.
494 For example, the above proposition is equivalent to the following: Something is either ABDE or ABDF or ACDE or ACDF.
(4) By the contraposition of a universal affirmative proposition separate information may be obtained with regard to any term that appears in the subject, or with regard to the contradictory of any term that appears in the predicate, or with regard to any combination of these terms.
For example, from All AB is CD or EF we may infer All A is b or CD or EF, All c is a or b or EF, All Be is a or CD, All ce is a or b, All Adf is b, &c.
Also by contraposition any universal affirmative proposition may be reduced to the form Everything is either X1 or X2 … or Xn.
For example, the above proposition is equivalent to the following: Everything is a or b or CD or EF.
(5) By the contraposition of a particular negative proposition separate information may be obtained with regard to any determinant of the subject or with regard to the contradictory of any alternant of the predicate or with regard to any combination of these.
For example, from Some AB or AC is not either D or EF we may infer Some A is not either bc or D or EF, Some d is not either a or bc or EF, Some Ae or Af is not either bc or D, &c.
Also by contraposition any particular negative proposition may be reduced to the form Something is not either X1 or X2 … or Xn.
For example, the above proposition is equivalent to the following: Something is not either a or bc or D or EF.
EXERCISES.
458. No citizen is at once a voter, a householder, and a lodger; nor is there any citizen who is none of the three.
Every citizen is either a voter but not a householder, or a householder and not a lodger, or a lodger without a vote.
Are these statements precisely equivalent? [V.]
In may be shewn that each of these statements is the logical obverse of the other. They are, therefore, precisely equivalent.