W. E. Johnson, Logic: Part I (1921)
§ I. It has been customary to apply the phrase Laws of Thought to three specific formulae; but the application of the phrase should be extended to cover all first principles of Logic. By first principles we mean certain propositions whose truth is guaranteed by pure reason. It is often too hastily said that logic as such is not concerned with truth but only with consistency; as if a conclusion were guaranteed by formal logic merely because it is consistent with any arbitrarily assumed premisses. But this entirely misrepresents the function of formal logic, which is not permissive, but rather prohibitive. It guarantees the truth -- not of any proposition that is consistent with the premisses -- but only of the proposition whose contradictory is inconsistent with the premisses. And even this statement goes too far; for logic does not allow any arbitrarily chosen premisses to be taken as true; and thus the only conclusions that it can be said in any sense to guarantee are those which have been correctly inferred from premisses that are themselves true. When consistency is placed in a kind of antithesis to truth, it seems often to be assumed that logic is indifferent to truth. That the reverse is the case is shown by the consideration that to say that a conclusion is validly drawn from given premisses is tantamount to asserting the truth of a certain composite proposition, viz. that the premisses imply the conclusion.
CHAPTER XIV
LAWS OF THOUGHTIn enunciating and formulating the fundamental principles of Logic, we shall not enter into the question whether they are all independent of one another, nor into the problem as to how a selection containing the smallest possible number could be made amongst them from which the remainder could be formally derived. This problem is perhaps of purely technical interest, and the attempt at its solution presents a fundamental, if not insuperable, difficulty: namely, that the procedure of deriving new formulae from those which have been put forward as to be accepted without demonstration, is governed implicitly by just those fundamental logical principles which it is our aim to formulate explicitly. We can, therefore, have no assurance that, in explicitly deriving formulae from an enumerated set of first principles, we are not surreptitiously using the very same formulae that we profess to derive. If this objection cannot be removed, then the supposition that the whole logical system is based on a few enumerable first principles falls to the ground.
§ 2. The charge has been brought against all the fundamental principles of Formal Logic that they are trivial; or otherwise that they are nothing but truisms. Now a truism may be defined as a proposition which is (1) true, and (2) accepted by everybody on mere inspection as true; and these are just the characteristics required of a fundamental principle of logic. Hence to charge the fundamental formulae with being mere truisms is not to condemn them, but to admit that they are fitted to fulfil the function for which they are intended. This function is to enable us to demonstrate further formulae, some of which, though true, are not accepted by everybody on mere inspection as true. It is an actual fact that by means of truisms and truisms alone we can demonstrate truths which are not truisms. The above and similar criticisms directed against the fundamental formulae of Logic will be best met by directly examining this or that formula so as to bring out its precise significance in view of the different points of view from which it has been criticised; and we shall adopt this plan as occasion offers.
§ 3. Before enunciating the fundamental principles in detail, we will enquire into what is implied in speaking of them as 'Laws.' The word law is closely connected with the notion of an imperative; and many logicians of the present day hold that the so-called laws of thought are no more imperatives than are the axioms of arithmetic or geometry. With this view I agree, inasmuch as the axioms of mathematics can themselves be regarded as having an imperative aspect; but this is because all truth may be so regarded. The idea of truth and falsity, in my view, carries with it the notion of an imperative, namely of acceptance and rejection -- a corollary from the theory which insists on the reference of judgment and assertion to the thinker. For it is only so far as assertion is recognised to be a mental act, that the notion of an imperative becomes relevant. An imperative of reason implies a restraint upon the voluntary act of assertion -- a restraint which does not, however, infringe the freedom that characterises every volition, since the obligation to think in accordance with truth is self-imposed. Any study of which imperatives constitute the subject-matter has been called a normative science, and normative sciences have been contrasted with positive sciences. But from a certain point of view every science may be said to exercise an imperative function in so far as any mistake or confusion in the judgments of the ordinary man is corrected or criticised by the scientist as such. Every science therefore can without any confusion of thought be regarded as normative; which is only another way of saying that the notions of truth and falsity as predicable of propositions carry with them the notions 'to be accepted' or 'to be rejected' understood as imperatives. But an explanation can be given for the restricted use of the term normative to logic, aesthetics and ethics: viz., that, while each deals with a certain kind of mental fact, it does not deal with it merely as fact. Every science which deals with man, either in his individual or social capacity, takes as its topic the description of mental facts -- including an analysis of how men think, feel and act; but such a descriptive study of our thoughts, feelings and actions (including their causal relations) treated generally, historically or speculatively, is to be distinguished from the study of precisely the same facts in relation to certain norms or standards, and from the critical examination of these norms or standards themselves. The division of sciences in general into normative and positive is, therefore, unsound, inasmuch as all sciences may be regarded as normative in the sense that they are potentially corrective of mistaken, false or obscure views. This division (into normative and positive) is therefore properly restricted to sciences dealing with psychological material; thus the positive or descriptive treatment of mind -- in its thinking, feeling or acting aspect -- is (like all sciences) normative in the sense of being potentially corrective of false judgments on the topics directly dealt with; while the treatment in Logic, Aesthetics and Ethics of these same processes is normative in the more special sense that these sciences examine and criticise the norms of thought, feeling or action themselves. Within the range for which the antithesis between normative and positive holds, the distinction between a descriptive or causal account of psychological or sociological matters, and an examination of standards or norms, is now-a-days of the first importance, inasmuch as the substitution of causal description in the place of evaluation of standard has been woefully common in works which profess to found Ethics upon psychology or sociology.
§ 4. To return to the consideration of the principles which exercise an imperative function. The fundamental formulae for conjunctive and composite propositions have been given in the chapter on compound propositions; these must be included in the general consideration of the Laws of Thought. Certain of these laws, and in particular the Reiterative, Commutative and Associative laws of Conjunction are -- not only the materials which explicitly compose the logical system -- but are also implicitly used in the process of building up the system. Thus, for example, we not only explicitly formulate the Reiterative Law, but in making repeated use of this or of any other law, we are implicitly using the Reiterative Principle itself. This will be seen to hold in the same way of the Commutative and Associative principles of Conjunction. Finally, inasmuch as the system is developed by means of inference, the essential principles of implication are not only explicitly formulated in the formulae for composite propositions, but also implicitly used in constructing the logical system itself.
§ 5. The next set of laws to be considered will be those which express the nature of Identity, since this is a formal conception which applies with absolute universality to all possible objects of thought whatever the category to which they may belong. Identity is a relation, and as such has certain properties which are exhibited in what we shall call the Laws of Identity. Relations in general may be classified according to the formal properties they possess, irrespectively of the terms related. It will be necessary here to introduce and define three of these properties, viz., transitiveness, symmetry and reflexiveness. Using the symbols x, y, z, to stand for the terms of any relation, and the symbols r^ and rv for any relation and its converse, then,
(1) the relation r^ is called transitive : when 'x is r^ to y' and 'y is r^ to z' together implies 'x is r^ to z' for all cases of x, y, z; for example, ancestor, greater than, causing, implying;
(2) the relation r^ is called symmetrical: when 'x is r^ to y' implies 'y is r^ to x'; in other words, when 'x is r^ to y' implies 'x is rv to y,' for all cases of x and y; for example, cousin, incompatible with, other than;
(3) the relation r^ is called reflexive: when 'x is r^ to x' for all cases of x; for example, compatriot of, simultaneous with, homogeneous with.
Now the three Laws of Identity are most simply expressible by the statement that identity is (1) transitive, (2) symmetrical, (3) reflexive; or otherwise, for every object of thought (represented by the symbols x, y, z).
It will be observed that there are a host of other relations which have these same three properties; e.g. contemporaneous, homogeneous, compatriot, numerically equal, equal in magnitude, etc., but analysis of every such relation shows it to contain a reference to some identical element, upon which these formal properties depend.
- Transitive Law: If x is identical with y, and y is identical with z; then x is identical with z.
- Symmetrical Law: If x is identical with y, then y is identical with x.
- Reflexive Law: x is identical with x.
§ 6. The phrase 'Law of Identity' has been traditionally used for one of the three fundamental logical principles, known as the Laws of Identity, of Non-Contradiction, and of Excluded Middle, to which the term 'Laws of Thought' has been usually restricted; but, since these three laws relate exclusively to propositions, whereas the conception of identity applies to all objects of thought, I propose to substitute for the traditional terminology, the Principles 'of Implication,' 'of Disjunction' and 'of Alternation' respectively; and to insert a fourth, to be called the 'Principle of Counter-implication.' The four together will be entitled 'the Principles of Propositional Determination.' The four laws are thus brought into line with the four forms of composite proposition discussed in a preceding chapter. The composite propositions expressed in their general form, i.e. in terms of two independent components p, q, are of course not guaranteed as true by pure logic; in other words, they require material or experiential certification as opposed to merely formal or rational certification. The principles, on the other hand, are those cases of the composite propositions, expressed in their quite general form, the truth of which is guaranteed by pure logic. For the purposes of formulating the principles on the lines of the four composite functions, we may slightly modify the expression of these latter as follows:
(1) Implicative Function: If P is true, then Q is true. (2) Counterimplicative Function: If P is false, then Q is false. (3) Disjunctive Function: Not both P true and Q true. (4) Alternative Function: Either P true or Q true. The principles are obtained by substituting P for Q in the implicative and counterimplicative functions, and P-false for Q-true in the disjunctive and alternative functions. Thus:
Principles of Propositional Determination
(P being any proposition)where the words 'must be' and 'cannot be' serve to indicate that the principles are formally or rationally certified.
(1) Implicative: It must be that if P is true, then P is true, (2) Counterimplicative: It must be that if P is false, then P is false, (3) Disjunctive: P cannot be both true and false, (4) Alternative: P must be either true or false, This formulation uses a single proposition P together with the two adjectives true and false, in preference to the more usual mode of expression which employs two propositions, P and not-P, and a single adjective 'true'; as in the following:
- If P is true, then P is true.
- If not-P is true, then not-P is true.
- P and not-P cannot both be true.
- Either P or not-P must be true.
There are several reasons for adopting the former of these two modes of formulation in preference to the latter. In the first place it uses the comparatively simple notion of P being false instead of the rather awkward notion of not-P being true. Secondly it enables us to define 'contradiction' by means of the principles, which would be impossible without a circle if we introduced the contradictories P and not-P into the formulation. In the third place, the introduction of the phrases P-true and P-false is in accordance with the fact that the adjectives true and false are the first characteristics by which the nature of the proposition as such is to be understood. A closer analysis of this formulation of the alternative and disjunctive principles will throw further light on the nature of the antithesis between the adjectives true and false. We have emphasised the point that these adjectives are predicable only of propositions; in other words 'anything that is true or false is a proposition'; the principle of alternation adds to this statement its complementary, viz., 'anything that is a proposition is true or false.' It is clear, of course, that these two statements are not synonymous. Again, the principle of disjunction states that the adjectives true and false are incompatible; and this again goes beyond what is explicitly involved in the statement that they are predicable exclusively of propositions.
The most obvious immediate application of these principles is obtained by taking P to stand for a definite singular proposition: 's is p,' where 's' stands for a uniquely determined or singular subject, and 'p' for any adjective. Then P-false becomes ' s is not-p.' In this application, the four principles may be called the Principles of Adjectival Determination, and assume the following form:
Principle of Implication: If s is p, then s is p. Principle of Counterimplication: If s is not-p, then s is not-p. Principle of Disjunction: s cannot be both p and not-p. Principle of Alternation: s must be either p or not-p.
In this application, the principles are expressed in terms of any adjectives p and not-p predicated of any subject; instead of being expressed in terms of the adjectives true and false predicated of any proposition P. In ordinary logical text-books the 'Laws of Thought' are almost always expressed in this specialised form; but, by this mode of enunciation, the generality which characterises the formulation in terms of propositions is lost; for when 'adjectives predicated of any subject' is substituted for 'propositions,' we have only a special case from which the general could not have been derived. It is convenient for many purposes to use the term 'predication' to stand for 'adjective' or 'proposition'; thus we may include both the general and the special formulae of determination in the abbreviated forms 'If p then p'; 'If not-p then not-p'; ' Not-both p and not-p'; 'Either p or not-p'; where p stands for any predication. The two sets of formulae might again be expressed -- without any modification of meaning -- in the form of universals, since P stands for any proposition, and s for any subject; thus:
Generalised Form for Propositional Determination Generalised Form for Adjectival Determination If any proposition is true, it is true
If any proposition is false, it is false
No proposition can be both true and false
Any proposition must be either true or falseIf anything is p, it is p
If anything is not-p, it is not-p
Nothing can be both p and not-p
Anything must be either p or not-pA comparison between these two generalised formulations of the principles will bring out the important distinction between false and not-true and again between true and not-false. According to the principles of adjectival determination 'Anything must be either true or not-true'; whereas of propositions we can say 'Any proposition must be either true or false.' Now, since it is only propositions of which truth is properly predicable, therefore of anything that is not a proposition the adjective true must be denied; thus we must say 'The table is not true' on the elementary ground that 'the table' is not a proposition; but we cannot say that 'The table is false,' because it is only propositions which can be said to be false. Thus the principles of propositional determination force upon us the notable consideration that the word false does not really mean the same as not-true. To have expressed the principle of alternation in the form 'Anything must be either true or false' without the necessary restriction to a proposition would have been actually wrong. On the other hand, the form 'Any proposition must be either true or not-true' is not sufficiently determinate, for this alternative would hold of any subject whatever and fails to express the alternative peculiar to the proposition itself. On the same ground, the disjunctive principle is not properly expressed in the form 'No proposition can be both true and not-true.' This affords another, and, in my view, the most important justification for formulating the principles in terms of the adjectives true and false instead of in terms of the propositions P and not-P. In illustration of the above discussion we may point to the analogy between the four adjectives true, false, not-true, not-false and the four adjectives male, female, not-male, not-female. The antithesis between male and not-male or again between female and not-female is applicable to any subject whatever, but that between male and female is applicable exclusively to organisms. Analogously the antithesis between true and not-true or again between false and not-false is applicable to any subject-term whatever, but that between true and false is applicable exclusively to propositions.
§ 7. Now the so-called Law of Identity -- which I have expressed in the form 'If P is true then P is true' where P stands illustratively for any proposition -- has been the favourite object of attack by critics of the principles of formal logic, on the score of its insignificance or triviality. The reason why the formula appears to have little or no significance is that its implicans is literally identical with its implicate. It will be found, however, that the necessary condition for the explicit use of the relation of identity is that the identified element should have entered into different contexts. It must be noted that this necessary reference to difference of context does not render the relation of identity other than absolute, i.e. it in no way implies that in its two occurrences the identified element is partly identical and partly different. The application of this genera condition to the Principle of Implication requires us to contemplate the proposition 'P is true' as one that may have been asserted in different connections or on different occasions or by different persons. Then, since the formula 'If P is true then P is true' is to be understood as logically general, its full import can be expressed in the form: 'If the asserting of P in any one context is true, then the asserting of P in any context whatever is true.' If this analysis be accepted, it will be found that the principle could not have been enunciated except for the possibility of identifying an assertum or proposition as distinct from the various attitudes (belief, interrogation, doubt, denial) which might have been adopted towards it on different occasions by the same or different persons. One important element of meaning, therefore, implicit in the formula is that it tacitly implies the identifiability of a proposition as such.
Turning now to the adjective 'true' as it occurs in our analysis of the formula, let us contrast it with certain adjectives that are predicable of things in general. The principle -- that what can be asserted in one context as true must be asserted in any other context as true -- is more familiarly particularised in the form 'any proposition that is once true is always true'; that is to say that 'true' as predicable of any proposition is unalterable; whereas there are certain adjectives and relations predicable of things in general which may characterise them only temporarily. Contrasting, for instance, the Principle of Propositional Determination 'If any proposition is true it is true' with the Principle of Adjectival Determination 'If anything is p it is p,' we find that in the former the copula 'is' is to be interpreted without reference to the present or any other assigned time; whereas in the latter the adjective/ may be alterable, so that the copula 'is' must here be understood as referring to definitely assigned time. In the case of anything that is at an assigned moment of time p, the principles of logic do not entitle us to assert that it will be or has always been p. Taking as examples 'The water has a temperature of 30° C or 'Mr B. is at home,' we must say on the one hand that if these propositions are true at any time, they are true at all times. But we must not say that if the predicate 'having a temperature of 30° C or the relation 'being at home' is true of a given subject at one time, it will be true at all times. This obvious comment would not have been required if language had distinguished in the mode of the verb 'to be' between a timeless predication and a tense (present, past or future). Certain logicians have, however, deliberately denied the dictum that what is once true is always true, and their denial appears to be due to a confusion between the time at which an assertion is made, and the time to which an assertion refers; or as Mr Bosanquet has neatly put it -- 'between the time of predication and the time in predication.' Others, i.e. the Pragmatists, have made the denial of this dictum a fundamental factor in their philosophy, inasmuch as they have taken the term 'true' to be virtually equivalent to 'accepted,' whereas everybody else would agree that the term is equivalent rather to the phrase 'to be accepted.' Again, the dictum would not have been confidently admitted in the days before the principles of Logic had been formulated by Aristotle, when the antithesis between the immutability of truth and the mutability of things appears to have presented an insurmountable problem. Since, then, it has been disputed from three different points of view that the truth or falsity of a proposition is independent of the time of assertion, the first Law of Thought -- my interpretation of which brings out clearly this quality of truth -- is effectively freed from the charge of triviality.
But not only must we interpret the principles as implying the unalterability of truth, for further, according to the principle of disjunction, a proposition cannot be both true and false; and this is to be interpreted to imply that, if a proposition is true in any one sense, there can be no sense of the word true in which it could be false, or other than true. On this interpretation the principle would be opposed by those philosophers who employ the words relative and absolute, or similar terms, to distinguish two kinds of truth. In consistency with this philosophical position, the term 'true' must be said to have two meanings, so that one and the same proposition might be true in one meaning of the term 'true' and false in another. It would seem that it is only on this theory that philosophers could maintain that a certain proposition such as 'Matter exists' is true in or for science, and at the same time false from the point of view of philosophy. According to the view, however, of those who maintain rigidly the validity of the Principles of Determination, it cannot be said that the same proposition is true in one sense and false in another sense, although it may be said, of course, that one sense given to a certain collocation of words would yield a true proposition, while another sense given to the same collocation of words would yield a false proposition. We must deny for instance that 'Matter exists' can be true in one sense and false in another sense, though we do not for a moment dispute that 'Matter exists' in one sense may be true while 'Matter exists' in another sense may be false. It is noteworthy that the confusion here is exactly parallel to that between the time of predication and the time in predication. Thus the assertion 'Mr Brown is at home' cannot be true at one time and false at another; though that 'Mr Brown is at home at one time' may be true and that 'Mr Brown is at home at some other time' may be false.
§ 8. We propose now to consider the Principles of Adjectival Determination with a view to giving added significance to the predicational factor by bringing out the relation of an adjective to its determinable. For this purpose the principles will be reformulated as follows:
(1) Principle of Implication: If s is p, where p is a comparatively determinate adjective, then there must some determinable, say P, to which p belongs, such that s is P.
(2) Principle of Counterimplication: If s is P, where P is a determinable, then s must be p, where p is an absolute determinate under P.
(3) Principle of Disjunction: s cannot be both p and p', where p and p' are any two different absolute determinates under P.
(4) Principle of Alternation: s must be either not-P; or p or p' or p'' . . . continuing the alternants throughout the whole range of variation of which P is susceptible -- p, p', p'' . . . being comparatively determinate adjectives under P.
For convenience of reference, these formulae may be elliptically restated as follows:
- If s is p, then s is P.
- If s is P, then s is p.
- s cannot be both p and p', nor p' and p'', nor p and p'' . . . .
- s must be either not-P or p or p' or p'' or p''' . . . .
Contrasting this reformulation with the original formulation of the principles of adjectival determination, it will be observed that, while the predications of s are more precise, they are not so palpably obvious. The force of the first principle is that, if a subject is of such a kind that a certain determinate adjective can be predicated of it, then this presupposes that the subject belongs to a certain category such that it may be compared in character with other subjects belonging to the same category, the ground of comparison being equivalent to the determinable. The second principle states that any subject whose character is so far known that a certain determinable adjective can be predicated of it, must in fact be characterised by some absolutely determinate value of that determinable, and that, although, in many cases, such a precise determination of character is impossible, yet the postulate that in fact the subject has some determinate character is one that reason seems to demand. The third principle as reformulated gains in significance, as compared with the mere disjunction of p with the indeterminate not-p, since now it precludes the possibility of conjoining an indefinite number of pairs of predicates, which are here exhibited as determinate and positive. In fact, in the principle of disjunction in its original form (according to which p cannot be joined with not-p) not-p should signify -- not merely some or any adjective other than p -- but some adjective that is necessarily incompatible with p, and the only such adjectives are those other than p which belong to the same determinable.
The significance of the Principle of Alternation in its new form requires special discussion. It is developed from the dichotomy 'Any subject must be either not-P or P' where P stands for any determinable. This again assumes that 'Some subjects are not P,' i.e. that there are subjects belonging to such a category that the determinable adjective P is not predicable of them. The negative here must be termed a pure negative, in the sense that not-P cannot be resolved into an alternation of positive adjectives. For example, in the statement: 'Material bodies are not conscious' the negative term 'not conscious' does not stand for any single positive determinable which would generate a series of positive determinates. We ought in fact to maintain that 'not-conscious' is not properly speaking an adjective at all; for in accordance with the reformulated Principle of Adjectival Implication, every adjective that can be predicated of a subject must be a more or less determinate value of some determinable1. Eliminating, then, the negative not-P from the predicate, the reformulated Principle of Adjectival alternation may now be expressed in the form: 'Any subject that is P must be either p or p' or p'' or . . .' where the alternative predication p or p' or p'' or . . . is restricted to subjects of which the determinable P is predicable. If we compare this form of the Principle of Alternation with the Principle of Counterimplication, viz., If s is characterised by P, it must be characterised by one or other determinate value of P, there would appear to be no obvious difference between them. The Principle of Alternation, however, supplements that of Counterimplication by implicitly postulating that the range of possible variation of the determinable can be apprehended in its completeness. The question whether we can in this way apprehend the complete range of possible variation of any determinable must be examined in detail. Consider the determinable 'integral number' which is always predicable of a collection or aggregate as such. Of a 'collection' we can in the first place assert universally that 'it is either zero or greater than zero,' and this it is to be observed, goes beyond the mere assertion that it is 'either zero or not zero.' Again we may assert that any collection is 'either zero or one or more than one,' where the alternant 'more than one' is not merely negative, but positive -- though comparatively indeterminate. Proceeding in this way, we may resolve exhaustively the range of possible variations of number by an enumerated and finite series of positively indicated alternants: 'zero or one or two or three or . . . or n or greater than n.' What is here said of integral number holds of quantity in general, and may also be applied to any determinable (continuous or discrete) whose determinates have an order of betweenness and can therefore be serially arranged. For example, the range of hue can be exhaustively resolved into the nine alternants 'red or between red and yellow, or yellow, or between yellow and green, or green, or between green and blue, or blue, or between blue and violet, or violet.'
We may summarise (with some additional comments) what has been said with respect to the Principles of Adjectival Determination, formulated with reference to the determinable. (1) If s is p, then s is P. This postulates that whenever a comparatively determinate predication is asserted, then a determinable to which the determinate belongs can always be found; but it must be pointed out that language does not always supply us with a name for the determinable. (2) If s is P, then s is p. This postulates that in actual fact every adjective is manifested as an absolute determinate; it is to be supplemented, however, by the recognition that for a continuously variable determinable it is impossible actually to characterise a given subject by a precisely determinate adjective. (3) s cannot be both p and p'. This asserts that any two different determinates are incompatible; but, inasmuch as we are unable practically to characterise an object determinately (in the case of a continuously variable determinable), we must apply the formula to the case where p and p' (though only comparatively determinate) are figuratively speaking 'outside one another.' To represent this figurative analogy, suppose a point (a, b, c or d) to represent an absolute determinate, and the segment of a line (p or p') to represent a comparative determinate:
then, if b is between a and c, and only then, can we assert that the (comparative) determinates p and p' are codisjunct or incompatible. (4) Any s that is P must be either p or p' or . . . . Here the predications p, p', p'', etc. need not be absolute determinates, but to render the principle practically significant it is necessary that we should be able to compass in thought the entire stretch or range of variation of which P is susceptible.
p p' _________________________ a b c d § 9. We will now pass to the principles according to which the manifested value of any one variable is determined by its connection with the manifested values of other variables. These principles may be expressed in forms analogous to those of adjectival determination and will be entitled the Principles of Connectional Determination. They embody the purely logical properties of the causal relation; but the notion of cause and effect -- being properly restricted to phenomena temporally alterable -- will be replaced by the wider notion of determining and determined. The characters whic may be said jointly to determine some other characte correspond to what is commonly called the cause, while any character which is thereby determined corresponds to an effect. Analysis of the general conception of causal connection reveals two complementary aspects which may be thus expressed: (a) wherever, in two instances, there is complete agreement as regards the cause-factors, there will be agreement as regards any effect-factor; and (b) wherever, in two instances, there is any (partial) difference as regards the cause-factors, there will be some difference in one or other of the effect-factors. In formulating the Principles of Connectional Determination, such symbols as P, Q, R, T, will be introduced to represent the characters that are connectionally determined, along with A, B, C, D, to represent those which connectionally determine the former. Thus the conjunction abcd would correspond to a cause-complex, and pqrt to an effect-complex.
Principles of Connectional Determination
(1) Principle of Implication. Taking any determinable P, the determinate value which it assumes in any manifestation is determined by the conjunction of a finite number of determinables A, B, C, D (say), such that any manifestation that has the determinate character abed (say) will have the determinate character p (say).
(2) Principle of Counterimplication. Taking any determinable A, the determinate value which it assumes in any manifestation determines (in conjunction with other factors) a conjunction of a finite number of determinables P, Q, R, T (say), such that if, for instance, some manifestation having the determining character a has the determined character pqrt, then any manifestation that has the (different) character a' will have one or other of the different characters p' or q' or r' or t' (say).
(3) Principle of Disjunction. P being one of the characters determined by the conjunction of the determining characters A, B, C, D, there can be no three instances characterised respectively by
abcd~p, a'bcd~p', a''bed~p. (4) Principle of Alternation. On the same hypothesis, it must be that either 'every abed is p' or 'every abed is p' ' or 'every abed is p'' ' or, etc., where the range of alternation covers all possible determinate values of P.
The Principle of Implication postulates that the determinate value assumed by any variable is dependent in any instance not upon an indefinite number of conditions which might in some sense be exhaustive of the whole state of the universe, but upon a set of conditions that are capable of enumeration. The theoretical and practical possibility of enumerating the factors which together constitute the determining complex, enables us to express the nature of reality in universal propositions of the form 'every abed is p.' If the character p could be predicated universally only of a class determined by an infinite number of conjoined characters, reality could not be described by means of universal propositions; or in other words, nature would not present uniformities which could be comprehended by thought; in short there would be nothing that could be called Laws of Nature. Hence the significance of our first principle is that reality presents uniformities that can be comprehended in thought, and that, whatever variable aspect of the universe we may be concerned with, a uniformity or law could be found such that from it the value of the variable in any manifestation could be inferred from knowledge (at least theoretically possible) of the values assumed by other variables. The Principle of Implication represents that more familiar aspect of the so-called Law of Causation expressed in terms of agreement: that in any two instances where there is complete agreement as regards the cause complex, there will be agreement as regards the effect; or, still more colloquially, the same cause entails the same effect.
Turning now to the Principle of Counterimplication, this represents the other and complementary aspect of causation; namely that of difference. It postulates that we can by enumeration exhaust the characters that are determined in their variation by any cause complex; just as we assumed that the cause complex in the previous principle could be exhaustively described. In other words the effect, determined by any variation in the causal or determining complex, does not permeate the whole universe, but is restricted to some assignable sphere. This important postulate being presumed, the principle proceeds to state that, if any variable presents a different value in two instances, indications of this difference will be shown in one or other of the variables that are affected or determined by the given variable. This principle is therefore complementary to the preceding one; whereas the Principle of Implication asserts that where there is agreement in the cause there will be agreement in the effect, the Principle of Counterimplication asserts that where there is difference in the cause there will be a difference in the effect. It may perhaps even be said that in the popular conception of cause this latter aspect -- viz. of difference -- is more prominent than the former, viz. agreement. Here we must point out that the principles are not parallel, inasmuch as complete agreement in the cause is required to ensure agreement in the effect, whereas any partial difference in the cause will entail some difference in the effect.
A word must be said about the strictly formal relations between these Principles of Implication and Counterimplication. By what is familiarly known as inference by contraposition, the proposition 'Every abed is p' is equivalent to the proposition 'Every p' is either a' or b' or c' or d'.' Similarly the proposition 'Every a' is p' or q' or r' or t' ' is equivalent to 'Every pqrt is a.' Applying this formal contraposition to the formulae for cause and effect, we see that the proposition that 'The same cause always entails the same effect' is logically equivalent to 'Any difference in the effect would entail some difference in the cause'; and again the proposition that 'Any difference in the cause will entail some difference in the effect' is logically equivalent to 'The same effect always entails the same cause.' It will be thus seen that the implicative and counterimplicative principles are not obtainable one from the other as equivalents by contraposition, but are complementary to one another, so that taken together they represent the relation between cause and effect as reciprocal. Take the one aspect of this relation; then plurality of cause holds in the sense that the effect may be partially the same in two instances where the cause is different; and plurality of effects holds in the same sense, namely, that the cause may be partially the same in two instances where the effect is different. Take the other aspect of the relation: thus, when the effect is completely and determinately characterised the character of the cause is thereby uniquely determined, just as when the cause is completely and determinately characterised the character of the effect is thereby uniquely determined. Thus, whether we are considering the relation of cause to effect or of effect to cause, the principles postulated will be in terms of complete agreement or of partial difference.
We pass now to the Disjunctive Principle. In order to expound this we must consider three instances of ABCD which agree as regards the determinate values of all but one, viz. A, of these determinables. Then, taking into consideration the Counterimplicative Principle, a difference as regards A in two instances would entail some difference in one or other of the characters that are determined by the complex ABCD. The principle then states that, if, in some pair of instances, a variation in the determining factor A entails a variation in the selected character P, then any further variation in A would entail a further variation in this same character P; whereas if, in two instances, a variation in A entails no variation in P, then any further variation in A would entail no further variation in the same character P. It is essential to note that the Disjunctive Principle could not have been formulated as a disjunction of two types of instance, such as abcd~p and a''bcd~p. This disjunction would be equivalent to asserting that a variation in any determining factor such as A would entail a variation in any or every determined factor such as P; whereas the Counterimplicative Principle has laid down only that a variation in A would entail a variation in one or other of the determined characters and not necessarily in every one of them. The Principle of Disjunction then supplements that of Counterimplication by maintaining that if some one variation in A entails a variation in the selected character P, then any variation in A would entail a variation in the same character P. It might be supposed, in the case where a variation of A entails no variation in P, that P is not causally connected with A, and that therefore A could be eliminated. But the mere elimination of A is not in general permissible, since the character P in some one of its determinate values requires that A should be manifested in some or other of its determinate values; though, as regards the determinate value of P, it may be a matter of indifference what specific value A has. Since in this case A cannot be eliminated, it would be symbolically requisite to express the relation of determination for the case under consideration -- not in the form 'bcd determines p' -- but in the form 'Abcd determines p,' where the significance of the symbol A is that any determinate value may from instance to instance be manifested without affecting the determinate value p.
Fourthly, the Alternative Principle of Connectional Determination, asserts an alternation of universal propositions, and of course goes beyond any statement that could be derived from the Principle of Adjectival Alternation, in which the alternative is in the predicate. Thus the latter states the universal proposition that 'Every abed is p or p' or p'' or . . .' whereas the principle under present consideration states an alternation between the universal propositions 'Every abcd is p' or 'Every abcd is p' ' or . . . .
These principles will be very much more fully discussed when we deal with the topic of formal or demonstrative induction; they have been introduced at this early stage of our logical exposition in order to indicate the nature of the transition from the Principles of Propositional Determination which are purely axiomatic, to those of Adjectival Determination under a determinable, which have the character partly of axioms and partly of postulates, and from these again to the Principles of Connectional Determination which may be taken as pure postulates.
§ 10. The formulation of the principles of connectional determination has an important bearing upon the problem of internal and external relations. In controversies on this topic it appears to be agreed that the division of relations into internal and external is both exclusive and exhaustive; and yet there seems to be no agreement as to what precisely the distinction is. One school holds that all relations are internal; the other that all are external. But on the face of it it would appear that some must be internal, others external; for otherwise it would seem impossible to give meaning to the distinction. It will be found, however, that those who deny external relations doubt, for instance, not whether spatial and temporal relations are properly to be called external, but rather whether space and time are themselves real in the sense that the real can be truly characterised by spatial and temporal relations; those on the other side who deny internal relations apparently hold that the independent otherness of the terms of the relation renders the relation external, inasmuch as the specific and variable relation of one term to another is not that which determines or is determined by the mere existence of the one or of the other term. The adherents then of the exclusively internal view of relations hold that the relation and its terms are mutually determinative, and the adherents of the exclusively external view, that the relation and its terms are mutually non-determinative or independent. Now it appears to me that the root misunderstanding amongst the two schools of philosophy on this point is, not as to what is meant by an internal as contrasted with an external relation, but rather what is the nature of the terms between which the relation is supposed to subsist. The one school maintains that the relation subsists between the characters of the two related terms; the other that it subsists between the terms themselves. According to the former contention, relations are internal in the sense that they depend wholly upon the character of the terms related; according to the latter, they are external in the sense that they do not depend at all upon the mere existence of the terms qua existents. In this connection there is a further source of confusion, namely as to whether in the character of a term are to be included such relations as those of space and time, these being admittedly external, in contrast to qualities proper which are admittedly internal.
At this point I will state my solution of the problem, which will appear so simple that it would seem difficult to account for the origin of the controversy. I hold, then, that relations between adjectives as such are internal; and those between existents as such are external. In this account, adjectives are to include so-called external relations, even the characterising relation itself, as well as every other relation. The otherness which distinguishes the 'this' from the 'that' is the primary and literally the sole external relation, being itself direct and underived. And this relation is involved in every external relation. In fact, qua existent, the 'this' and the 'that' have no specific relation. The specific external relations that hold of one to another existent are derivative from their characters, in the wider sense of character. Thus, the relation of the 'this' to the 'that' obtained from the fact that 'this is blue and that is green,' is derived from the nature of the qualities blue and green. Again the relation of proximity or remoteness obtained from the fact that 'this is here and that is there,' is derived from the positions of the 'this' and the 'that' by which their specific spatiality is characterised. The most important application of the distinction is to causal and other forms of connectional determination. Here the primary relation called cause is that between the character, dating, and locating, of two occurrences, from which the relation between the occurrences themselves is derived, the former being internal and the latter external. If there were no such internal causal relation, nothing could be stated as to the relation of event to event, except that the one is invariably accompanied by the other in a certain assignable spatial relation of space and time; and even this external relation is derived from the internal relation subsisting between the temporal and spatial positions occupied by the two events. If, however, all spatial properties were relative, as is maintained by Einstein and his followers, there would be no spatial relations other than internal, in fact nothing to distinguish a space from that which occupies it. The principles of connectional determination have therefore been expressed directly in terms of the characters by which the manifestations of reality may be described, from which must be derived the external relations between such manifestations themselves. It will have been observed that the correlative notions of determination and dependence enter into the formulation of the principles as directly applicable to the characters of manifestations and therefore only derivatively to the manifestations themselves. Hence the potential range for which these principles hold extends beyond the actually existent into the domain of the possibly existent. In this way the universality of law is wider than that of fact. While the universals of fact are implied by the universals of law, the statement of the latter has intrinsic significance not involved in that of the former.
Notes 1 A negative predication of this type has sometimes been called Privative; but unfortunately the term privative has also been used in an opposite sense, namely, for a predication applied to a subject belonging to a category for which the positive adjective is normally applicable; as when we predicate of a person that he is blind or that he is (temporarily) unconscious.