486 The above may seem to imply that an alternative synthesis may be expressed in a greater number of ways than a conjunctive synthesis. This, however, is not the case. It has been shewn that an alternative synthesis may be expressed by a hypothetical or by the denial of a conjunctive (that is, by a true disjunctive). But corresponding to this, a conjunctive synthesis may be expressed by the denial of a hypothetical or by the denial of an alternative. Thus, representing the denial of a proposition by a bar drawn across it, we have
xy = x̅ or y̅ = If x, y̅ ;
xy = x or y = If x, y.
Mr Johnson shews that any ordinary proposition with a general term as subject may be regarded as a compound proposition resulting from the conjunctive or alternative combination of singular (molecular) propositions, with a common predication, but different subjects. Let S1, S2, … S∞ represent a number of different individual subjects; and let S represent the aggregate collection of individuals S1, S2, … S∞. Then
S1 and S2, and S3 … and S∞ = Every S ;
S1 or S2, or S3 …… or S∞ = Some S.
480 “Thus we arrive at the common logical forms, Every S is P, Some S is P. The former is an abbreviation for a determinative, the latter for an alternative, synthesis of molecular propositions.”487
487 Mind, 1892, p. 25. Mr Johnson of course recognises that a quantified subject-term (all S) is not usually a mere enumeration of individuals first apprehended and named. But he points out that “however the aggregate of things, to which the universal name applies, is mentally reached, the propositional force for purposes of inference or synthesis in general is the same” (p. 28).
In other words,
Every S is P = S1 is P and
S2 is P and S3 is P … and
S∞ is P ;
Some S is P = S1 is P or
S2 is P or S3 is P … or
S∞ is P.
443. The Opposition of Compound propositions.—The rule for obtaining the contradictory of a complex term given in section 426 may be applied also to compound propositions. Thus, the contradictory of a compound proposition is obtained by replacing the constituent propositions by their contradictories and everywhere changing the manner of their combination, that is to say, substituting conjunctive combination for alternative and vice versâ.488 The following are examples: All A is B and some P is Q has for its contradictory Either some A is not B or no P is Q ; Either some A is both B and C, or all B is either C or both D and E has for its contradictory No A is both B and C, and some B is not either C or both D and E.
488 It has been shewn in the preceding section that the words all and some are abbreviations of conjunctive and alternative synthesis respectively. Hence the rule that, in the ordinarily recognised propositional forms, contradictories differ in quantity as well as in quality is itself only a particular application of the general law here laid down.
It follows, as in section 427, that there is a duality of formal equivalences in the case of compound propositions, each equivalence yielding a reciprocal equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versâ.
444. Formal Equivalences of Compound Propositions.—The laws relating to the conjunctive or alternative synthesis of propositions are practically identical with those relating to the conjunctive or alternative combination of terms; and we have accordingly the following propositional equivalences corresponding to the equivalences of terms given in section 433. The symbols here stand for propositions, not terms; and negation is represented by a bar over the proposition denied. 481
| (1) | x (y or z) = xy or xz, | ⎱ | Laws of Distribution ; |
| (2) | x or yz = (x or y) (x or z), | ⎰ | |
| (3) | xx = x, | ⎱ | Laws of Tautology (Law of Simplicity and Law of Unity) ; |
| (4) | x or x = x, | ⎰ | |
| (5) | x = x or yy = (x or y) (x or y), | ⎱ | Laws of Development and Reduction ; |
| (6) | x = x (y or y) = xy or xy, | ⎰ | |
| (7) | x or xy = x, | ⎱ | Laws of Absorption ; |
| (8) | x (x or y) = x | ⎰ | |
| (9) | x or y = x or xy, | ⎱ | Law of Exclusion and Law of Inclusion. |
| (10) | xy = x (x or y),489 | ⎰ |
489 It is not maintained that
all the above laws are ultimate or even independent of one another.
The synthesis of propositions is admirably worked out by Mr Johnson in
his articles on the Logical Calculus (Mind, 1892). He
gives five independent laws which are necessary and sufficient
for propositional synthesis. These laws are briefly enumerated below;
for a more complete exposition the reader must be referred to Mr
Johnson’s own treatment of them.
(i) The Commutative Law: The order of pure synthesis is
indifferent (xy = yx).
(ii) The Associative Law: The mode of grouping in pure
synthesis is indifferent (xy . z = x . yz).
(iii) The Law of Tautology: The mere repetition of a
proposition does not in any way add to or alter its force (xx = x).
(iv) The Law of Reciprocity: The denial of the denial of a
proposition is equivalent to its affirmation (x̅ =
x). “In this principle are included the so-called Laws of
Contradiction and Excluded Middle, viz., ‘If x,
then not not-x’, and ‘If not not-x, then
x’.”
(v) The Law of Dichotomy: The denial of any proposition is
equivalent to the denial of its conjunction with any other proposition
together with the denial of its conjunction with the contradictory of
that other proposition (x = xy xy̅).
“This is a further extension of the Law of Excluded Middle, when
applied to the combination of propositions with one another. The
denial that x is conjoined with y combined with the
denial that x is conjoined with not-y is equivalent to
the denial of x absolutely. For, if x were true, it must
be conjoined either with y or with not-y. This law,
which (it must be admitted) looks at first a little complicated, is
the special instrument of the logical calculus. By its means we may
always resolve a proposition into two determinants, or conversely we
may compound certain pairs of determinants into a single
proposition.”
445. The Simplification
of Complex Propositions.—The terms of a complex proposition
may often be simplified by means of the rules given in the preceding
chapter, and the force of the assertion will remain unaffected. For
the further simplification of complex propositions the following rules
may be added:
(1) In a universal negative or a particular affirmative
proposition any determinant of the subject may be indifferently
introduced or omitted as a determinant of the predicate and vice
versâ.
482 To say that No AB is AC is the same as to say that No AB is C, or that No B is AC. For to say that No AB is AC is the same thing as to deny that anything is ABAC ; but, as shewn in section 429, the repetition of the determinant A is superfluous, and the statement may therefore be reduced to the denial that anything is ABC. And this may equally well be expressed by saying No AB is C, or No B is AC.490
Again, Some AB is AC may be shewn to be equivalent to Some AB is C, or to Some B is AC ; for it simply affirms that something is ABAC, and the proof follows as above.
(2) In a universal affirmative or a particular negative proposition any determinant of the subject may be indifferently introduced or omitted as a determinant of any alternant of the predicate.
All A is AB may obviously be resolved into the two propositions All A is A, All A is B.491 But the former of these is a merely identical proposition and gives no information. All A is AB is, therefore, equivalent to the simple proposition All A is B. Similarly, All AB is AC or DE is equivalent to All AB is C or DE.
491 The resolution of complex propositions into a combination of relatively simple ones will be considered further in the following section.
Again, Some A is not AB affirms that Some A is a or b ;492 but by the law of contradiction No A is a ; therefore, Some A is not B, and obviously we can also pass back from this proposition to the one from which we started. Similarly, Some AB is not either AC or DE is equivalent to Some AB is not either C or DE.
(3) In a universal affirmative or a particular negative proposition any alternant of the predicate may be indifferently introduced or omitted as an alternant of the subject.
If All A is B or C, then by the law of identity it follows that Whatever is A or B is B or C ; it is also obvious that we can pass back from this to the original proposition.
Again, if Some A or B is not either B or C, then since by the law of identity All B is B it follows that Some A is not either B or C ; and it is also obvious that we can pass back from this to the original proposition.
(4) In a universal affirmative or a particular negative proposition the contradictory of any determinant of the subject may be indifferently introduced or omitted as an alternant of the predicate, and vice versâ.
483 By this rule the three following propositions are affirmed to be equivalent to one another: All AB is a or C ; All B is a or C ; All AB is C ; and also the three following: Some AB is not either a or C ; Some B is not either a or C ; Some AB is not C.
The rule follows directly from rule (1) by aid of the process of obversion (see chapter 3).
(5) In a universal negative or a particular affirmative proposition the contradictory of any determinant of the subject may be indifferently introduced or omitted as an alternant of the predicate.
By this rule the two following propositions are affirmed to be equivalent to one another: No AB is a or C ; No AB is C ; and also the two following: Some AB is a or C ; Some AB is C.
The rule follows directly from rule (2) by obversion.
(6) In a universal negative or a particular affirmative proposition the contradictory of any determinant of the predicate may be indifferently introduced or omitted as an alternant of the subject.
This rule follows from rule (3) by obversion.
446. The Resolution of Universal Complex Propositions into Equivalent Compound Propositions.—We may enquire how far complex propositions are immediately resolvable into a conjunctive or alternative combination of relatively simple propositions. Universal propositions will be considered in this section, and particulars in the next.
Universal Affirmatives. Universal affirmative complex
propositions may be immediately resolved into a conjunction of
relatively simple ones, so far as there is alternative combination in
the subject or conjunctive combination in the predicate. Thus,
(1) Whatever is P or Q is R = All P is R and all Q is
R ;
(2) All P is QR = All P is Q and all P is R.
Universal Negatives. Universal negative complex propositions
may be immediately resolved into a conjunction of relatively simple
ones, so far as there is alternative combination either in the subject
or in the predicate. Thus,
(3) Nothing that is P or Q is R = No P is R and no Q is
R ;
(4) No P is either Q or R = No P is Q and no P is
R.
So far as there is conjunctive combination in the subject or
alternative combination in the predicate of universal affirmative
propositions, or conjunctive combination either in the subject or in
the predicate of universal negative propositions, they cannot be 484 immediately493
resolved into either a conjunctive or an alternative combination of
simpler propositions. It may, however, be added that propositions
falling into this latter category are immediately implied by certain
compound alternatives. Thus,
(i) All PQ is R is implied by All P is R or all Q is
R ;
(ii) All P is Q or R is implied by All P is Q or all
P is R ;
(iii) No PQ is R is implied by No P is R or no Q
is R ;
(iv) No P is QR is implied by No P is Q or no P is
R.
493 It will be shewn subsequently that even in these cases universal complex propositions may be resolved into a conjunction of relatively simpler ones by the aid of certain immediate inferences.
447. The Resolution of
Particular Complex Propositions into Equivalent Compound
Propositions.—Particular complex propositions cannot be
resolved into compound conjunctives, but they may under certain
conditions be immediately resolved into equivalent compound
alternative propositions in which the alternants are relatively
simple. This is the case so far as there is alternative combination in
the subject or conjunctive combination in the predicate of a
particular negative, or alternative combination either in the subject
or in the predicate of a particular affirmative. Thus,
(1) Some P or Q is not R = Some P is not R or some Q is
not R ;
(2) Some P is not QR = Some P is not Q or some P
is not R ;
(3) Some P or Q is R = Some P is R or some Q
is R ;
(4) Some P is Q or R = Some P is Q or some P is
R.
Particular complex propositions cannot be immediately resolved into
compound propositions (either conjunctive or alternative) so far as
there is conjunctive combination in the subject or alternative
combination in the predicate if the proposition is negative, or so far
as there is conjunctive combination either in the subject or in the
predicate if the proposition is affirmative. In these cases, however,
the complex proposition implies a compound conjunctive proposition,
though we cannot pass back from the latter to the former. Thus,
(i) Some PQ is not R implies Some P is not R and Some Q
is not R ;
(ii) Some P is not either Q or R implies Some
P is not Q and some P is not R ;
(iii) Some PQ is R implies
Some P is R and some Q is R ;
(iv) Some P is QR implies
Some P is Q and some P is R.
It must be particularly noticed that, although in these cases the 485 compound proposition can be inferred from the complex proposition, still the two are not equivalent. For example, from Some P is Q and some P is R it does not follow that Some P is QR, for we cannot be sure that the same P’s are referred to in the two cases.
All the results of this section follow from those of the preceding section by the application of the rule of contradiction to the propositions themselves and the rule of contraposition to the relations of implication between them.
448. The Omission of
Terms from a Complex Proposition.—From the two preceding
sections we may obtain immediately the following rules for inferring
from a given proposition another proposition in which certain terms
contained in the original proposition are omitted:
(1) Any determinant may be omitted from an undistributed
term ;494
(2) Any alternant may be omitted from a distributed term.495
494 The subject of a particular or the predicate of an affirmative proposition.
495 The subject of a universal or the predicate of a negative proposition.
For example,—
Whatever is A or B is CD, therefore, All A is C ;
Some AB is CD, therefore, Some A is C ;
Nothing that is A or B is C or D, therefore, No A is C ;
Some AB is not either C or D, therefore, Some A is not C.
The above rules may also be justified independently, as will be shewn in the following section. The results which they yield must be distinguished from those obtained in section 445. In the cases discussed in that section, the terms omitted were superfluous in the sense that their omission left us with propositions equivalent to our original propositions; but in the above inferences we cannot pass back from conclusion to premiss. From Some A is C, for example, we cannot infer that Some AB is C.
449. The Introduction of
Terms into a Complex Proposition.—Corresponding to the
rules laid down in the preceding section we have also the following:
(1) Any determinant may be introduced into a distributed
term ;
(2) Any alternant may be introduced into an undistributed
term.
These rules, and also the rules given in the preceding section, may be established by the aid of the following axioms: What is true of all (distributively) is true of every part ; What is true of part of a part is true of a part of the larger whole.
486 When we add a determinant to a term, or remove an alternant, we
usually diminish, and at any rate do not increase, the extension of
the term; when, on the other hand, we add an alternant, or remove a
determinant, we usually increase, and at any rate do not diminish, its
extension. Hence it follows that if a term is distributed we may add a
determinant or remove an alternant, whilst if a term is undistributed
we may add an alternant or remove a determinant. Thus,
All A is CD, therefore, All AB is C ;
No A is C, therefore, No AB is CD ;
Some AB is C, therefore, Some A is C or D ;
Some AB is not either C or D, therefore, Some A is not C.
From the above rules taken in connexion with the rules given in
section 445 we may obtain the following corollaries:
(3) In universal affirmatives, any determinant may be introduced
into the predicate, if it is also introduced into the subject; and any
alternant may be introduced into the subject if it is also introduced
into the predicate.
Given All A is C, then All AB is C by rule (1) above;
and from this we obtain All AB is BC by rule (2) of section
445.
Again, given All A is C, then All A is B or C ; and
therefore, by rule (3) of section 445, Whatever is A or B is B or C.
(4) In universal negatives any alternant may be introduced into
subject or predicate, if its contradictory is introduced into the
other term as a determinant.
Given No A is C, then No AB is C ; and, therefore, by
rule (5) of section 445, No AB is b or C.
Again, given No A is C, then No A is BC ; and, therefore, by rule
(6) of section 445, No A or b is BC.
In none of the inferences considered in this section is it possible to pass back from the conclusion to the original proposition.
450. Interpretation of Anomalous Forms.—It will be found that propositions which apparently involve a contradiction in terms and are thus in direct contravention of the fundamental laws of thought—for example, No AB is B, All Ab is B—sometimes result from the manipulation of complex propositions. In interpreting such propositions as these, a distinction must be drawn between universals and particulars, at any rate if particulars are interpreted as implying, while universals are not interpreted as implying, the existence of their subjects.
487 It can be shewn that a universal proposition of the form No AB is B or All Ab is B must be interpreted as implying the non-existence in the universe of discourse of the subject of the proposition. For a universal negative denies the existence of anything that comes under both its subject and its predicate; thus, No AB is B denies the existence of ABB, that is, it denies the existence of AB. Again, a universal affirmative denies the existence of anything that comes under its subject without also coming under its predicate; thus, All Ab is B denies the existence of anything that is Ab and at the same time not-B, that is, b ; but Ab is Ab and also b, and hence the existence of Ab is denied.
Since the existence of its subject is held to be part of the implication of a particular proposition, the above interpretation is obviously inapplicable in the case of particulars. Hence if a proposition of the form Some Ab is B is obtained, we are thrown back on the alternative that there is some inconsistency in the premisses; either some one individual premiss is self-contradictory, or the premisses are inconsistent with one another.
EXERCISES.
451. Shew that if No A is bc or Cd, then No A is bd. [K.]
452. Give the contradictory of each of the following propositions:—(1) Flowering plants are either endogens or exogens, but not both; (2) Flowering plants are vascular, and either endogens or exogens, but not both. [M.]
453. Simplify the following
propositions:—
(1) All AB is BC or be or CD or cE or DE ;
(2) Nothing
that is either PQ or PR is Pqr or pQs or pq or prs or qrs or pS or
qR. [K.]
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.
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 dE is 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 operations give | |
| 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.]