CHAPTER III.

IMMEDIATE INFERENCES FROM COMPLEX PROPOSITIONS.

454. The Obversion of Complex Propositions—The doctrine of obversion is immediately applicable to complex propositions; and no modification of the definition of obversion already given is necessary. From any given proposition we may infer a new one by changing its quality and taking as a new predicate the contradictory of the original predicate. The proposition thus obtained is called the obverse of the original proposition.

The only difficulty connected with the obversion of complex propositions consists in finding the contradictory of a complex term; but a simple rule for performing this process has been given in section [426]:—Replace all the simple terms invoked by their contradictories, and throughout substitute alternative combination for conjunctive and vice versâ.

Applying this rule to AB or ab, we have (a or b) and (A or B), that is, Aa or Ab or aB or Bb ; but since the alternants Aa and Bb involve self-contradiction, they may by rule (5) of section [433] be omitted. The obverse, therefore, of All X is AB or ab is No X is Ab or aB.

As additional examples we may find the obverse of the following propositions: (1) All A is BC or DE ; (2) No A is BcE or BCF ; (3) Some A is not either B or bcDEf or bcdEF.

(1) All A is BC or DE yields No A is (b or c) and at the same time (d or e), or, by the reduction of the predicate to a series of alternants, No A is bd or be or cd or ce.

(2) No A is BcE or BCF. Here the contradictory of the 489 predicate is (b or C or e) and (b or c or f), which yields b or Cc or Cf or ce or ef. Cc may be omitted by rule (5) of section [433]; also ef by rule (7), since ef is either Cef or cef. Hence the required obverse is All A is b or Cf or ce.

(3) Some A is not either B or bcDEf or bcdEF. The obverse is Some A is b and (B or C or d or e or F) and (B or C or D or e or f); and by the application of the rules summarised in section [433] this will be found to be equivalent to Some A is bC or bDF or bdf or be.

455. The Conversion of Complex Propositions.—Generalising, we may say that we have a process of conversion whenever from a given proposition we infer a new one in which any term that appeared in the predicate of the original proposition now appears in the subject, or vice versâ.

Thus the inference from No A is BC to No B is AC is of the nature of conversion. The process may be simply analysed as follows:—

No A is both B and C,
therefore, Nothing is at the same time A, B, and C,
therefore, No B is both A and C.

The reasoning may also be resolved into a series of ordinary conversions:—

No A is BC,
therefore (by conversion), No BC is A,
that is, within the sphere of C, no B is A,
therefore (by conversion), within the sphere of C, no A is B,
that is, No AC is B,
therefore (by conversion), No B is AC.

Or, it may be treated thus,

No A is BC,
therefore, by section [445], rule (1), No AC is BC,
therefore, also by section [445], rule (1), No AC is B,
therefore (by conversion), No B is AC.

Similarly it may be shewn that from Some A is BC we may infer Some B is AC.

Hence we obtain the following rule: In a universal negative or a particular affirmative proposition any determinant of the subject may be transferred to the predicate or vice versâ without affecting the force of the assertion.

We have just shewn how from

No A is BC,

we may obtain by conversion

No B is AC.

490 Similarly, we may infer

No C is AB,
No AB is C,
No AC is B,
No BC is A.

The proposition may also be written in the form

There is no ABC,
or, Nothing is at the same time A, B, and C.

The last of these is a specially useful form to which to bring universal negatives for the purpose of logical manipulation.

In the same way from Some A is BC or BD we may infer

Some AB is C or D,
Some AC or AD is B,
Some B is AC or AD,
Some C or D is AB,
Some BC or BD is A,
Something is ABC or ABD.

There is no inference by conversion from a universal affirmative or from a particular negative.

456. The Contraposition of Complex Propositions.—According to our original definition of contraposition, we contraposit a proposition when we infer from it a new proposition having the contradictory of the old predicate for its subject. Adopting this definition, the contrapositive of All A is B or C is All bc is a.

The process can be applied to universal affirmatives and to particular negatives. By obversion, conversion, and then again obversion, it is clear that in each of these cases we may obtain a legitimate contrapositive by taking as a new subject the contradictory of the old predicate, and as a new predicate the contradictory of the old subject, the proposition retaining its original quality. For example: All A is BC, therefore, Whatever is b or c is a ; Some A is not either B or C, therefore, Some bc is not a.

The above may be called the full contrapositive of a complex proposition. It should be observed that any proposition and its full contrapositive are equivalent to each other; we can pass back from the full contrapositive to the original proposition.

In dealing with complex propositions, however, it is convenient to give to the term contraposition an extended meaning. We may say that we have a process of contraposition when from a given proposition we infer a new one in which the contradictory of any term that appeared in the predicate of the original proposition now appears 491 in the subject, or the contradictory of any term that appeared in the subject of the original proposition now appears in the predicate.

Three operations may be distinguished all of which are included under the above definition, and all of which leave us with a full equivalent of the original proposition, so that there is no loss of logical power.

(1) The operation of obtaining the full contrapositive of a given proposition, as above described and defined.[496]

[496] In some cases we may desire to drop part of the information given by the complete contrapositive. Thus, from All A is BC or E may infer Whatever is be or ce is a ; but in a given application it may be sufficient for us to know that All be is a.

(2) An operation which may be described as the generalisation of the subject of a proposition by the addition of one or more alternants in the predicate. Thus, from All AB is C we may infer All A is b or C ; from Some AB is not either C or D we may infer Some A is not either b or C or D.

For inferences of this type the following general rule may be given: Any determinant may be dropped from the subject of a universal affirmative or a particular negative proposition, if its contradictory is at the same time added as an alternant in the predicate.

This rule may be established as follows: Given All AB is C (or Some AB is not C)—and these may be taken, so far as the rule in question is concerned, as types of universal affirmatives and particular negatives respectively—we have by obversion No AB is c (or Some AB is c), and thence, by the rule for conversion given in section [455], No A is Bc (or Some A is Bc); then again obverting we have All A is either b or C (or Some A is not either b or C), the required result.

It will be observed that, as stated at the outset, these operations leave us with a proposition that is equivalent to our original proposition. There is, therefore, no loss of logical power.

By the application of the above rule with regard to all the explicit determinants of the subject any universal affirmative proposition may be brought to the form Everything is X1 or X2 … or Xn ; and it will be found that by means of this transformation, complex inferences are in many cases materially simplified.

(3) An operation which may be described as the particularisation of the subject of a proposition by the omission of one or more alternants in the predicate. Thus, from All A is B or C we may infer All Ab is C ; from Some A is not either B or C we may infer Some Ab is not C.

492 For inferences of this type the following general rule may be given: Any alternant may be dropped from the predicate of a universal affirmative or a particular negative proposition, if its contradictory is at the same time introduced as a determinant of the subject.[497]

[497] The application of this rule again leaves us with a proposition equivalent to our original proposition. The following rule, which may be regarded as a corollary from the above rule, or which may be arrived at independently, does not necessarily leave us with an equivalent: If a new determinant is introduced into the subject of a universal affirmative proposition (see section [449]) every alternant in the predicate which contains the contradictory of the determinant may be omitted. Thus, from Whatever is A or B is C or DX or Ex, we may infer Whatever is AX or BX is C or D.

The application of this rule may sometimes result in the disappearance of all the alternants from the predicate; and the meaning of such a result is that we now have a non-existent subject.

Thus, given All P is ABCD or Abcd or aBCd, if we particularise the subject by making it PbC, we find that all the alternants in the predicate disappear. The interpretation is that the class PbC is non-existent, that is, No P is bC ; a conclusion which might of course have been obtained directly from the given proposition.

This rule is the converse of that given under the preceding head; and it follows from the fact that the application of that rule leaves us with an equivalent proposition.

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.

LetV = voter,  v = not voter;
H = householder,  h = not householder;
L = lodger,l = not lodger.

495 The first of the given statements is No citizen is VHL or vhl ; therefore (by obversion), Every citizen is either v or h or l and is also either V or H or L ; therefore (combining these possibilities), Every citizen is either Hv or Lv or Vh or Lh or Vl or Hl.
But (by the law of excluded middle), Hv is either HLv or Hlv ; therefore, Hv is Lv or Hl. Similarly, Lh is Vh or Lv ; and Vl is Hl or Vh.
Therefore, Every citizen is Vh or Hl or Lv, which is the second of the given statements.
Again, starting from this second statement, it follows (by obversion) that No citizen is at the same time v or H, h or L, l or V ; therefore, No citizen is vh or vL or HL, and at the same time l or V ; therefore, No citizen is vhl or VHL, which brings us back to the first of the given statements.

459. Given “All D that is either B or C is A,” shew that “Everything that is not-A is either not-B and not-C or else it is not-D.” [De Morgan.]

This example and those given in section [466] are adapted from De Morgan, Syllabus, p. 42. They are also given by Jevons, Studies, p. 241, in connexion with his Equational Logic. They are all simple exercises in contraposition.
We have What is either BD or CD is A ; therefore, All a is (b or d) and (c or d); therefore, All a is bc or d.

460. Infer all that you possibly can by way of contraposition or otherwise, from the assertion, All A that is neither B nor C is X. [R.]

The given proposition may be thrown into the form

Everything is either a or B or C or X ;

and it is seen to be symmetrical with regard to the terms a, B, C, X, and therefore with regard to the terms A, b, c, x. We are sure then that anything that is true of A is true mutatis mutandis of b, c, and x, that anything that is true of Ab is true mutatis mutandis of any pair of the terms, and similarly for combinations three and three together.
We have at once the four symmetrical propositions:

All A is B or C or X ;  (1)
All b is a or C or X ;  (2)
All c is a or B or X ;  (3)
All X is a or B or C.  (4)

496 Then from (1) by particularisation of the subject:

All Ab is C or X ;  (i)

with the five corresponding propositions;

All Ac is B or X ;  (ii)
All Ax is B or C ;  (iii)
All bc is a or X ;  (iv)
All bx is a or C ;  (v)
All cx is a or B.  (vi)

By a repetition of the same process, we have

All Abc is X (which is the original proposition over again);(α)
and corresponding to this:All Abx is C ;(β)
All Acx is B ;(γ)
All bcx is a.(δ)

It will be observed that the following are pairs of full contrapositives;—(1) (δ), (2) (γ), (3) (β), (4) (α), (i) (vi), (ii) (v), (iii) (iv).

A further series of propositions may be obtained by obverting all the above; and as there has been no loss of logical power in any of the processes employed we have in all thirty propositions that are equivalent to one another.

461. If AB is either Cd or cDe, and also either eF or H, and if the same is true of BH, what do we know of that which is E? [K.]

Whatever is AB or BH is (Cd or cDe) and (eF or H);
therefore, Whatever is AB or BH is CdeF or cDeF or CdH or cDeH ;
therefore, Whatever is ABE or BHE is CdH ;
therefore, All E is ah or b or CdH.

462. Given A is BC or BDE or BDF, infer descriptions of the terms Ace, Acf, ABcD. [Jevons, Studies, pp. 237, 238.]

In accordance with rules already laid down, we have immediately—

Ace is BDF ;
Acf is BDE ;
ABcD is E or F.

463. Find the obverse of each of the following propositions:—
(1) Nothing is A, B, or C ;
(2) All A is Bc or bD ;
(3) No Ab is CDEf or Cd or cDf or cdE ;
(4) No A is BCD or Bcd ;
(5) Some A is not either bcd or Cd or cD. [K.]

497 464. Shew that the two following propositions are equivalent to each other:—No A is B or CD or CE or EF ; All A is bCde or bcEf or bce. [K.]

465. Contraposit the proposition, All A that is neither B nor C is both X and Y. [L.]

466. Find the full contrapositive of each of the following propositions:
(1) Whatever is B or CD or CE is A ;
(2) Whatever is either B or C and at the same time either D or E is A ;
(3) Whatever is A or BC and at the same time either D or EF is X ;
(4) All A is either BC or BD. [De Morgan.]

467. Find the full contrapositive of each of the following propositions:—
All A is BCDe or bcDe ;
Some AB is not either CD or cDE or de ;
Whatever is AB or bC is aCd or Acd ;
Where A is present along with either B or C, D is present and C absent or D and E are both absent ;
Some ABC or abc is not either DEF or def. [K.]

468. What information can you obtain about Af, Be, c, D, from the proposition All AB is CD or EF? [M.]

469. Establish the following: Where B is absent, either A and C are both present or A and D are both absent; therefore, where C is absent, either B is present or D is absent. [K.]

470. Establish the following: Where A is present, either B and C are both present or C is present D being absent or C is present F being absent or H is present; therefore, where C is absent, A cannot be present H being absent. [K.]

471. Given that Whatever is PQ or AP is bCD or abdE or aBCdE or Abcd, shew that (1) All abP is CD or dE or q ; (2) All DP is bC or aq ; (3) Whatever is B or Cd or cD is a or p ; (4) All B is C or p or aq ; (5) All AB is p ; (6) If ae is c or d it is p or q ; (7) If BP is c or D or e it is aq. [K.]

472. Bring the following propositions to the form Everything is either X1 or X2 … or Xn:—
Whatever is Ac or ab or aC is bdf or deF ;
Nothing that is A and at the same time either B or C is D or dE. [K.]

473. Shew that the results in section [447] follow from those in section [446] by the rules of contradiction and contraposition. [K.]