Studies and Exercises in Formal Logic

No doubt if immediate inferences are no more than verbal transformations, then they can all be based on the principle of identity as interpreted by Mill, namely, on the principle that whatever is true in one form of words is true in any other form of words having the same meaning. But if conversion (or any other form of immediate inference) is more than mere verbal transformation, the equivalence of the convertend and the converse is just what we have to shew; they are not merely two different forms of words having the same meaning.

423. The Laws of Thought and Mediate Inferences.—Mansel expresses the view that syllogistic reasoning—and indeed all formal reasoning whatsoever—can be based exclusively on the laws of 467 identity, contradiction, and excluded middle. The principle of identity is, he says, immediately applicable to affirmative moods in any figure, and the principle of contradiction to negatives.464 His proof of this position consists in quantifying the predicates of the propositions constituting the syllogism, and then making use—for affirmatives—of the axiom that “what is given as identical with the whole or a part of any concept, must be identical with the whole or a part of that which is identical with the same concept,” and—for negatives—of the axiom that “some or all S, being given as identical with all or some M, is distinct from every part of that which is distinct from all M.”

These formulae, however, go distinctly beyond the laws of identity and contradiction as ordinarily stated. They may indeed be regarded as equivalent to the dictum de omni et nullo, adapted so as to be applicable to syllogisms made up of propositions with quantified predicates; and if it is assumed that the dictum is only another form of stating the laws of identity and contradiction then the question needs no further discussion. Only in this case we must no longer express the law of identity either in the form “What is true is true,” or in the form “A is A”; nor the law of contradiction either in the form “If a judgment is true, its contradictory is not true,” or in the form “A is not not-A.” The laws as thus formulated cannot be regarded as adequate expressions of the axiom upon which syllogistic reasoning proceeds. They do not bring out the function of the middle term which is the characteristic feature of the syllogism, nor could the rules of the syllogism be deduced from them.

Of course syllogistic reasoning, like all other reasoning, presupposes the laws of thought, and in the process of indirect reduction, which occupies a not unimportant place in the doctrine of the syllogism, these laws come in explicitly.

It is not necessary to consider in detail formal inferences belonging to the logic of relatives, e.g., B is greater than C, A is greater than B, therefore, A is greater than C. Here we require the principle that whatever is greater than anything that is greater than a third thing is itself greater than the third thing; and it would be still more difficult than in the case of the dictum de omni et nullo to evolve this principle immediately out of the three laws of thought.

A GENERALIZATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX PROPOSITIONS.465

CHAPTER I.

THE COMBINATION OF TERMS.

424. Complex Terms.—A simple term may be defined as a term which does not consist of a combination of other terms. We denote a simple term by a single letter; for example, A, P, X. The combination of simple terms yields a complex term; and the combination may be either conjunctive or alternative.

A complex term resulting from the conjunctive combination of other terms may be called a conjunctive term, and it will be found convenient to denote such a term by the simple juxtaposition of the other terms involved.466 This kind of combination is sometimes called determination; and we may speak of the elements combined in a conjunctive term as the determinants of that term. Thus, A and B are the determinants of the conjunctive term AB.

A complex term resulting from the alternative combination of other terms may be called an alternative term; and we may speak of the elements combined in such a term as the alternants of that term. Thus, A and B are the alternants of the alternative term A or B.467

469 In the following pages, in accordance with the view indicated in section 191, the alternants in an alternative term are not regarded as necessarily exclusive of one another (except of course where they are formal contradictories). Thus, if we speak of anything as being A or B we do not intend to exclude the possibility of its being both A and B. In other words, A or B does not exclude AB.

It is necessary at this point to consider briefly the logical signification of the words and, or. In the predicate of a proposition their signification is clear; they indicate conjunctive and alternative combination respectively; for example, P is Q and R, P is Q or R. But when they occur in the subject of a proposition there is in each case an ambiguity to which attention must be called.

Thus, there would be a gain in brevity if we could write a proposition with an alternative term as subject in the form P or Q is R. This last expression would, however, more naturally be interpreted to mean P is R or Q is R, the force of the or being understood, not as yielding a single categorical proposition with an alternative subject-term, but as a brief mode of connecting alternatively two propositions with a common predicate. Hence, when we intend the former, the more definite mode of statement, Whatever is either P or Q is R, or Anything that is either P or Q is R, should be adopted.

There is also ambiguity in the form P and Q is R. This would naturally be interpreted, not as a single categorical proposition with a conjunctive subject-term (PQ is R), but as a brief mode of connecting conjunctively two propositions with a common predicate, namely, P is R and Q is R. In order, therefore, to express unambiguously a proposition with a conjunctive subject-term, it will be well either to adopt the method of simple juxtaposition without any connecting word as, for example, PQ is R, or else to employ one of the more cumbrous forms, Whatever is both P and Q is R, or Anything that is both P and Q is R.468

425. Order of Combination in Complex Terms.—The order of 470 combination in a complex term is indifferent whether the combination be conjunctive or alternative.469

Thus, AB and BA have the same signification. It comes to the same thing whether out of the class A we select the B’s or out of the class B we select the A’s.

Again, A or B and B or A have the same signification. It is a matter of indifference whether we form a class by adding the B’s to the A’s or by adding the A’s to the B’s.

426. The Opposition of Complex Terms.—However complex a term may be, the criterion of contradictory opposition given in section 40 must still apply: “A pair of contradictory terms are so related that between them they exhaust the entire universe to which reference is made, whilst in that universe there is no individual of which both can be affirmed at the same time.” In what follows it will be found convenient to denote the contradictory of any simple term by the corresponding small letter. Thus for not-A we may write a, and for not-B we may write b.

Now whatever is not AB must be either a or b, whilst nothing that is AB can be either a or b. Hence

⎰ AB,  
⎱ a or b,

constitute a pair of contradictories. Similarly,

⎰ A or B,
⎱ ab,  

are a pair of contradictories. And the same will hold good if A and B stand for terms which are already themselves complex (although relatively simple as compared with AB or A or B).

If, then, two terms are conjunctively combined into a complex term (of which they will constitute the determinants), the contradictory of this complex term is found by alternatively combining the contradictories of the two determinants. And, conversely, if two terms are alternatively combined into a complex term (of which they will constitute the alternants), the contradictory of this complex term is found conjunctively combining the contradictories of the two alternants.

In each case, we substitute for the relatively simple terms involved their contradictories, and (as the case may be) change 471 conjunctive combination into alternative combination, or alternative combination into conjunctive combination.

Hence the successive application of the above rule, for finding the contradictory of a complex term where we are dealing with a single pair of determinants or alternants, will result in our ultimately substituting for each simple term involved its contradictory, and reversing the nature of their combination throughout.470 We may, therefore, lay down the following rule for obtaining the contradictory of any complex term: Replace each constituent simple term by its contradictory and throughout substitute conjunctive combination for alternative combination and vice versâ.471 This rule is of simple application, and it is of fundamental importance in the treatment of complex propositions adopted in the following pages.

Thus, the contradictory of A or BC

is a and (b or c),
i.e., ab or ac ;

and the contradictory of ABC or ABD

is (a or b or c) and (a or b or d),

which, by the aid of rules presently to be given, is reducible to the form

a or b or cd.

It is possible for two complex terms to be formally inconsistent or repugnant without being true contradictories. This will be the case if they contain contradictory determinants without between them exhausting the universe of discourse. The terms AB and bC afford an example: nothing can be both AB and bC (for, if this 472 were so, something would be both B and not-B), but we cannot say à priori that everything is either AB or bC (since something may be Abc, which is neither AB nor bC).

427. Duality of Formal Equivalences in the case of Complex Terms.—It will be shewn in the following sections that certain complex terms are formally equivalent to other complex terms or to simple terms (for example, A or aB = A or B, A or AB = A); and it is important to notice at the outset that such formal equivalences always go in pairs. For if two terms are equivalent, their contradictories must also be equivalent; and hence, applying the rule for obtaining contradictories given in the preceding section, we are enabled to formulate the simple law that to every formal equivalence there corresponds another formal equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versâ.472 This law may be more precisely established as follows:—A formal equivalence that holds good for any given set of terms must equally hold good for any other set of terms; and, therefore, whatever holds good for the terms A, B, &c. must hold good for their contradictories a, b, &c. Hence, given any equivalence, we may first replace each simple term by its contradictory, and then take the contradictory of each side of the equivalence. The result of this double transformation will be that we shall obtain another equivalence in which every conjunctive combination has been replaced by an alternative combination, and conversely, while the term-symbols involved have remained unchanged. This proves what was required.

The application of the above law will be fully illustrated in the sections that immediately follow.

428. Laws of Distribution.—In order to combine a simple term conjunctively with an alternative term, we must conjunctively combine it with every alternant of the alternative.473 A and (B or C)474 denotes whatever is A and at the same time either B or C, and hence is equivalent to AB or AC. It follows that in order to combine two alternative terms conjunctively, we must conjunctively combine every alternant of the one with every alternant of the other. Thus, 473 (A or B)(C or D) denotes whatever is either A or B and at the same time either C or D, and is equivalent to AC or AD or BC or BD.475

We have then

A(B or C) = AB or AC,

and applying the law of duality of formal equivalences given in the preceding section, we have at once another equivalence, namely,

A or BC = (A or B)(A or C).476

These two equivalences are called by Schröder the Laws of Distribution.477 They are of the greatest importance in the manipulation and simplification of complex terms.

429. Laws of Tautology.—The following rules may be laid down for the omission of superfluous terms from a complex term:
 (a) The repetition of any given determinant is superfluous.
 Out of the class A to select the A’s is a process that leaves us just where we began. In other words, what is both A and A is identical with what is A. Thus, such terms as AA, ABB, are tautologous; the former merely denotes the class A, and the latter the class AB. Hence the above rule, which is called by Jevons the Law of Simplicity.478
 (b) The repetition of any given alternant is superfluous.
 To say that anything is A or A is equivalent to saying simply that it is A. Hence such terms as A or A, A or BC or BC, are tautologous; and we have the above rule, which is called by Jevons the Law of Unity.479

It will be seen by reference to the rule given in section 427 that the Law of Simplicity (AA = A) and the Law of Unity (A or A = A) are reciprocal; that is, the former is deducible from the latter and vice versâ. For the only difference between them is that conjunctive combination in the one is replaced by alternative combination in the other.480

474 430. Laws of Development and Reduction.—Important formal equivalences are yielded by the laws of contradiction and excluded middle.

Applying further the Laws of Distribution given in section 428 we have the following:

A = A or Bb = (A or B) (A or b),
A = A (B or b) = AB or Ab.

These may be taken as formulae for the development and the reduction of terms. Thus, the substitution of (A or B) (A or b) for A may be called the development of a term by means of the law of contradiction; and the substitution of AB or Ab for A the development of a term by means of the law of excluded middle. In both the above cases the term A is developed with reference to the term B. Similarly by developing A with reference to B and C, we should have (A or B or C) (A or B or c) (A or b or C) (A or B or c) if we make use of the law of contradiction, or ABC or ABc or AbC or Abc if we make use of the law of excluded middle. Development by means of the law of excluded middle is the more useful of the two processes in the manipulation of complex terms, and it may be understood that this is meant when the development of a term is spoken of without further qualification.

Conversely, the process of passing from (A or B) (A or b) to A, or from AB or Ab to A, may be called the reduction of a term by means 475 of the law of contradiction or the law of excluded middle, as the case may be.

Following Jevons, we may speak of an alternative term of the type AB or Ab as a dual term, and of the substitution of A for AB or Ab as the reduction of a dual term.481

431. Laws of Absorption.—It may be shewn that any alternant which is merely a subdivision of another alternant may be indifferently introduced or omitted from a complex term. Thus, AB being a subdivision of A, the terms A or AB and A are equivalent. This rule (which is called by Schröder the Law of Absorption482) may be established as follows: By the development of A with reference to B, A or AB becomes AB or Ab or AB ; but, by the law of unity, this is equivalent to AB or Ab ; and by reduction this is equivalent to A.

Applying the rule given in section 427 we obtain a second law of absorption, namely, A (A or B) = A, which is the reciprocal of the first law of absorption, A or AB = A.

432. Laws of Exclusion and Inclusion.—The contradictory of any alternant in a complex term may be indifferently introduced or omitted as a determinant of any other alternant; that is to say, the terms A or aB and A or B are equivalent. This may be established as follows: By the law of absorption A or aB is equivalent to A or AB or aB, and by reduction this yields A or B. The above equivalence may be called the Law of Exclusion on the ground that by passing from A or B to A or aB we make the alternants mutually exclusive.

The reciprocal equivalence A (a or B) = AB may be expressed as follows: The contradictory of any determinant in a complex term may be indifferently introduced or omitted as an alternant of any other determinant. This equivalence may be called the Law of Inclusion on the ground that by passing from AB to A (a or B) we make the determinants collectively inclusive of the entire universe of discourse.

433. Summary of Formal Equivalences of Complex Terms.—The following is a summary of the formal equivalences contained in the five preceding sections (those that are bracketed together being 476 in each case related to one another reciprocally in the manner indicated in section 427):—

434. The Conjunctive Combination of Alternative Terms.—The first law of distribution gives the general rule for the conjunctive combination of alternatives. But with a view to such combination special attention may be called (i) to the second law of distribution, namely, (A or B) (A or C) = A or BC ; and (ii) to the equivalence (A or B) (AC or D) = AC or AD or BD, which may be established as follows: By the first law of distribution (A or B) (AC or D) is equivalent to AAC or ABC or AD or BD ; but by the law of simplicity AAC = AC, and by the law of absorption AC or ABC = AC ; hence our original term is equivalent to AC or AD or BD, which was to be proved.

From the above equivalences we obtain the two following practical rules which are of great assistance in simplifying the process of conjunctively combining alternatives:
 (1) If two alternatives which are to be conjunctively combined have an alternant in common, this alternant may be at once written down as one alternant of the result, and we need not go through the form of combining it with any of the remaining alternants of either alternative;
 (2) If two alternatives are to be conjunctively combined and an alternant of one is a subdivision of an alternant of the other, then the former alternant may be at once written down as one alternant of the result, and we need not go through the form of combining it with the remaining alternants of the other alternative.483

 477

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 479 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