Part 3 — CHAPTER II.
Reading edition
COMPLEX PROPOSITIONS AND COMPOUND PROPOSITIONS.
440. Complex Propositions.—A complex proposition may be defined as a proposition which has a complex term either for its subject or its predicate. The ordinary distinctions of quantity and quality may be applied to complex propositions; thus All AB is C or D is a universal affirmative complex proposition. Some AB is not EF is a particular negative complex proposition. In the following pages propositions written in the indefinite form will be interpreted as universal, so that AB is CD will be understood to mean that all AB is CD. It is to be added that in dealing with complex propositions we interpret particulars as implying, but universals as not implying, the existence of their subjects in the universe of discourse.
441. The Opposition of Complex Propositions.—The opposition of complex terms has been already dealt with, and the opposition of complex propositions in itself presents no special difficulty. It must, however, be borne in mind that as we interpret particulars as implying the existence of their subjects, but universals as not doing so, we have the following divergences from the ordinary doctrine of opposition: (1) we cannot infer I from A, or O from E; (2) A and E are not necessarily inconsistent with each other; (3) I and O may both be false at the same time. The ordinary doctrine of contradictory opposition remains unaffected. The following are examples of contradictory propositions: All X is both A and B, Some X is not both A and B ; Some X is Y and at the same time either P or Q or R, No X is Y and at the same time either P or Q or R.
442. Compound Propositions.485—A compound proposition may be defined as a proposition which consists in a combination of other propositions. The combination may be either conjunctive (i.e., when two or more propositions are affirmed to be true together) or alternative (i.e., when an alternative is given between two or more propositions); for example, All AB is C and some P is not either Q or R is a compound conjunctive proposition; Either all AB is C or some P is not either Q or R is a compound alternative proposition. Propositions conjunctively combined may be spoken of as determinants of the resulting compound proposition; and propositions alternatively combined may be spoken of as alternants of the resulting compound proposition. In what follows, both conjunctive and alternative propositions are interpreted as being assertoric.
Only two types of compound propositions are here recognised, the conjunctive and the alternative. Pure hypothetical propositions are compound, but (except in so far as we interpret hypotheticals and alternatives differently in respect of modality) they are equivalent to alternative propositions, and may be regarded as constituting one mode of expressing an alternative synthesis. Thus (taking x and y as symbols representing propositions, and x and y as their contradictories) the hypothetical proposition If x then y expresses an alternative between x and y and is, therefore, equivalent to the alternative proposition x or y. Combinations of the true disjunctive type (for example, not both x and y) may also be regarded as a mode of expressing an alternative synthesis; thus, the true disjunctive proposition just given is equivalent to the alternative proposition x or y.486
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.
“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
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.
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.
(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⎰
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.
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.
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.
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 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.
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 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
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.
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.
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.]