THE THREE FUNDAMENTAL LAWS OF THOUGHT.
414. The three Laws of Thought.—The so-called fundamental laws of thought (the law of identity, the law of contradiction, and the law of excluded middle) are to be regarded as the foundation of all reasoning in the sense that consecutive thought and coherent argument are impossible unless they are taken for granted. The function which they thus perform is, however, negative rather than positive. Whilst constituting necessary postulates, apart from which our thought would become chaotic, they do not by themselves advance us on our way. On the one hand, we cannot without their support proceed a step in reasoning; on the other hand, if we were to rely on their aid alone, thought would immediately come to a standstill.
This is at any rate the view taken of the three laws in the exposition that follows. It is true that many logicians have ascribed to them functions of a more positive character, and—starting from the position that they are the fundamental assumptions of logic—have gone on to regard them as the basis upon which alone all logical doctrine, at any rate in its more formal aspect, can be established. The attempt to justify this view has necessitated reading into the laws much more meaning than they can properly be made to contain, and their interpretation has in consequence become highly complex and even confused.
At the outset the question arises whether the laws are to be regarded as referring to terms (or concepts) or to judgments. My own view is that, in all three cases, the latter reference is the more fundamental, but that a reference of the former kind is involved secondarily. This I shall endeavour to bring out in dealing with 451 the laws individually. The distinction is one to which considerable importance is rightly attached by Sigwart.
The question of the mutual relations between the three laws may be briefly touched upon before we proceed to consider the laws separately and in detail; it is not, however, a question that can be disposed of until a later stage. The main point to which attention may conveniently be called at once is that it is only in relation to the other laws that the full force of each of them can be brought out. The laws of identity and contradiction may be regarded as positive and negative statements of the same principle, namely, the unambiguity of the act of judgment; and the laws of contradiction and excluded middle are supplementary to one another in so far as between them they express the nature of negation. At the same time, an endeavour will be made to establish the independence of the laws in the sense that they cannot be deduced one from another.
415. The Law of Identity.—Following Sigwart, I think it most convenient to interpret this law as expressing the unambiguity of the act of judgment. Truth is something fixed and invariable. In the words of Mr Bradley, “Once true always true, once false always false. Truth is not only independent of me, but it does not depend upon change and chance. No alteration in space or time, no possible difference of any event or context, can make truth falsehood. If that which I say is really true, then it stands for ever” (Logic, p. 133).[454] Hence, since a judgment is the expression of truth, the content of a judgment is fixed and invariable; and only when our judgments are so regarded can our thoughts and reasonings be valid. It is in this sense that the law of identity is a fundamental principle of logic (which is the science of valid thought and reasoning); for it is clear that if for a given judgment we were allowed—when it suited us—to substitute another, or if the content of a given judgment could be regarded as now this and now something else, all thought would become chaotic and reasoning would be a sham. Of the validity of no single step of reasoning 452 could we be sure, since as we took the step the content of the original judgment might change, and on this ground it would be open to anyone to admit the original judgment and at the same time deny the inference attempted to be drawn from it.
[454] Compare what has been already said in section [50] about the universality of judgments. In particular, the bearing of Mr Bosanquet’s distinction between the time of predication and the time in predication must be borne in mind. When we say that the truth affirmed in any judgment is independent of time, we mean the time of predication, and we assume that the judgment is fully expressed: in order that it may be fully expressed, the time in predication, if any, must be made explicit.
It may be said that, as thus interpreted, the law of identity merely states that we cannot both affirm a judgment and deny it, and that this is what is expressed in the law of contradiction. There is force in this, to the extent that the laws of identity and contradiction may be regarded as expressing the positive and negative aspects of the same principle. It is, as Sigwart has said, only through the rejection of simultaneous affirmation and negation that we become conscious of the unambiguity of the act of judgment. At the same time, the positive formulation of the principle in the form of the law of identity—apart from its negative formulation in the form of the law of contradiction—is justifiable and helpful.
The unambiguity of the act of judgment may be expressed somewhat differently (and its positive aspect, as distinct from what is expressed by the law of contradiction, may thereby be made more clear) by saying that the repetition of a judgment neither adds to nor alters its force. On this basis we may perhaps justify the passage of thought which consists in the repetition, not of a complete judgment, but of part of its content only. In other words, we may thus justify formal reasoning, so far as it involves mere elimination; and in the majority of formal reasonings elimination is involved, though it may be questioned whether mere elimination from a single proposition (as in passing from All S is MP to All S is P) is by itself entitled to the name of reasoning at all.
Mill gives an enunciation of the law of identity which must be distinguished from the above: “Whatever is true in one form of words is true in every other form of words which conveys the same meaning” (Examination of Sir William Hamilton’s Philosophy, p. 466). This is a postulate which it is necessary to make in connexion with the use of language as an instrument of thought. So long as the judgment expressed is the same, the form of expression which we give to it is immaterial; and, since in logical doctrine we cannot explicitly recognise more than a limited number of distinct propositional forms, we have to claim to be allowed to substitute for any non-recognised form that recognised form which 453 expresses the same judgment. Mill’s postulate, however, goes beyond the law of identity regarded as expressing the unambiguity of the act of judgment, and it cannot be regarded as equally fundamental. It is sometimes given as the justification of immediate inferences: to this point we shall return.
We may now turn to the law of identity in the form in which it is more ordinarily stated, namely, A is A, Everything is what it is. This form is open to criticism if regarded as professing to give information with regard to objects. In another sense, however, it may be taken to express an unambiguity of terms or concepts which is involved in the unambiguity of the act of judgment. For it is clear that unless in any given process of thought or reasoning our terms or concepts have a fixed signification and reference, the unambiguity of the act of judgment cannot be realised. We have here the secondary reference to terms or concepts which is contained in all the laws of thought in addition to their primary reference to judgments.
As the repetition of a judgment neither adds to nor alters its force, so we may say the same of terms (or concepts), meaning thereby that to refer to anything as both A and A is the same thing as to refer to it simply as A. This yields Boole’s fundamental equation x2 = x (which itself admits of a twofold interpretation according as x stands for a term or a proposition).
The reasons why we should not interpret the formula A is A as expressing a judgment respecting the object A have to be considered. The fundamental difficulty is that this so-called judgment is, if interpreted literally, not thinkable at all. For all actual thought implies difference of some kind. Whenever we think of anything, it is as distinguished from something else, or as having properties in common with other things, or at any rate as itself existing at different times. Hence in no case can we think pure identity.
There are two ways of avoiding this difficulty.
(a) We may say that what is intended by identity is not pure identity, but exact likeness in some assigned respect or respects, the likeness sometimes amounting, so far at any rate as our apprehension is concerned, to indistinguishableness except in the property of occupying different portions of space (as, for example, in the case of a number of pins or bullets of the same make and size). On this interpretation, the law of identity may be regarded 454 as equivalent to Jevons’s principle of the Substitution of Similars—“Whatever is true of a thing is true of its like”—or to the axiom that “Things that are equal to the same thing are equal to one another.” Mansel indeed explicitly gives this axiom as equivalent to the law of identity.
It seems clear, however, on reflection that it is a misnomer to speak of these principles as laws of identity, and that at any rate they cannot be adequately expressed by the bare formula A is A. Nor can any analogous interpretation be given to the laws of contradiction and excluded middle. We must, therefore, reject this interpretation of the law of identity regarded as one of the three traditional laws of thought.
(b) We may attempt to evade the difficulty by explaining that by identity we mean continuous identity, as when I say “This pen is the same as the one with which I was writing yesterday.” Here there is no longer pure identity, since there is a difference of time.
If, adopting this interpretation, we mean by the law of identity that what is true of anything at a given time is true of it at other times also, we have no self-evident law, but a fallacy. For the properties of objects are not constant. In other words, the possession by an object of any given property is not, like the truth of a judgment (fully expressed), independent of time.
We must then by the law of identity, as thus interpreted, mean to assert not any identity of properties, but the identity of the subject of properties amidst all the changes that may take place in the properties themselves. This may be regarded as a theory as to the nature of individuality and continuous identity in the midst of change, and is of great importance in its proper place. But it cannot properly stand as one of the traditional laws of thought which constitute the foundation of logical doctrine.
416. The Law of Contradiction.—The principle of contradiction is best regarded as expressing one aspect of the relation between contradictory judgments, namely, that they cannot both be true. The essential characteristic of a judgment is that it claims to be true. But we cannot declare anything to be true without implicitly declaring something else to be false. All affirmation implies denial; and we cannot clearly grasp the import of any given judgment unless we understand precisely what it denies.
The relation between a judgment and its denial is made explicit by the law of contradiction and the law of excluded middle, the 455 first of which declares that two contradictory judgments cannot both be true, and the second that they cannot both be false.
It is clear that the law of contradiction, as thus interpreted, does not carry us very far, and that it cannot fulfil the function, which Hamilton assigned to it, of serving as the principle of all logical negation. It serves, however, to express the significance of negation, and at the same time to set forth (from a different point of view from that taken by the law of identity) a fundamental postulate which must be granted if our processes of thought and reasoning are to be valid. For validity of thought and reasoning demand that false judgments shall be refuted; and only by the help of the law of contradiction is any such refutation possible. The refutation requires that another judgment contradictory of the first shall be established; but this would go for nothing, if two contradictories could be true together.
The law of contradiction thus takes its place by the side of the law of identity as a first principle of dialectic and reasoning: not indeed advancing us on our way, but serving as a postulate, without which it would not even be possible for us to make a start.
We may pass to a consideration of the formula A is not not-A, by which the law of contradiction is more usually expressed. Here, as Sigwart points out, we have no longer an expression of a relation between two judgments, but an affirmation that in a given judgment the predicate must not contradict the subject; and inasmuch as denial and contradiction have primarily no meaning except in relation to judgments, this interpretation of the principle of contradiction can at any rate not be regarded as equally fundamental with that which we have previously given. At the same time, it is clear that if any A were not-A, then, understanding not-A to denote whatever does not belong to the class A, we should have two contradictory judgments, for we should be able to assert of something both that it belonged to the class A and that it did not belong to the class A.
The formula A is not not-A need not, therefore, be rejected, if its secondary character is recognised.
Mill’s attitude towards the law of contradiction involves an apparent inconsistency. He begins by regarding it as a mode of defining negation.[455] It is, he says, a mere identical proposition that if the negative be true, the affirmative must be false; for the 456 negative asserts nothing but the falsity of the affirmative, and has no other sense or meaning whatever. He goes on, however, both in the Logic and in the Examination of Sir William Hamilton’s Philosophy, to speak of the law as a generalisation from experience. He finds its original foundation in the fact that belief and disbelief are two different mental states, excluding one another, this being a fact which we obtain by the simplest observation of our own minds. We observe, moreover, that light and darkness, sound and silence, equality and inequality, in short any positive phenomenon whatever and its negative, are distinct phenomena, pointedly contrasted, and the one always absent when the other is present. From all these facts the law of contradiction is, in Mill’s opinion, a generalisation.
[455] Logic, ii. 7 § 5.
Two distinct points appear to be involved in this argument. As regards the reference to belief and disbelief, we must agree that the foundation of the law of contradiction is to be found in the nature of judgment. The essential characteristic of a judgment is that it claims to be true, and the affirmation of a truth implies by its very nature a denial. It is, however, difficult to see where any generalisation comes in here.
The other point that Mill raises, namely, the fact that all our knowledge is of contrasts is a generalisation which is ordinarily known as the psychological law of relativity. The fact, however, that we cannot apprehend light except as distinguished from darkness, sound except as distinguished from silence, etc., cannot be regarded as equivalent to the law of contradiction. What that law asserts is, as Mill himself puts it, that “the same proposition cannot be both false and true.”
Boole maintains that “the axiom of metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for anything to possess a quality and at the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x2 = x.” The law of contradiction is expressed in Boole’s system in the form x(1 − x) = 0, where x may stand either for the truth of a judgment or for a term; and it is of course clear that x(1 − x) = 0 follows from x2 = x. It will, however, be observed that the converse also holds good, so that the question as to which of the two laws is really the more fundamental remains open to discussion. Apart from this, any attempt to deduce the law of contradiction from any other principle whatsoever is open to the fundamental objection that unless the law of contradiction is 457 accepted as a postulate no single step in reasoning is possible: for as soon as it is open to us to affirm a judgment and at the same time to deny it, it is à fortiori open to us to affirm a judgment and to deny any inference that may be drawn from it. To the question of the interdependence of the laws of thought we shall return.
It has been denied that the law of contradiction is a necessary law of thought, on the ground that not only do we often meet with self-contradiction, but that sometimes people have even boasted of holding contradictory opinions.[456] If, however, the law of contradiction is to be rejected, it must be shewn not merely that we sometimes contradict ourselves, but that we do so with perfect clearness of thought, and that we do not thereby stultify ourselves.
[456] Compare Bain, Logic, Deduction, p. 223.
The mere fact of our holding contradictory opinions goes for nothing so long as the self-contradiction is not realised by us. In such cases it may be assumed that one or other of the contradictory doctrines will be given up as soon as the contradiction between them is made manifest. If the truth of both is still maintained, it will probably be found that there is some reservation—as, for example, by means of a distinction between different kinds of truth, one doctrine being held to be true literally and the other in some poetical or allegorical sense—whereby consistency is restored at the expense of ambiguity and want of clearness. Apart from some explanation of this kind, the problem of accounting for the way in which some of us appear to hold inconsistent beliefs is one for the psychologist rather than the logician. The ultimate explanation must be sought in confusion of thought, or lack of intellectual sincerity, or in these two causes combined. From a logical point of view to rest in an unresolved contradiction is to stultify ourselves and to confess failure.
417. The Sophism of “The Liar.”—The sophism known as Ψευδόμενος or The Liar has been thought by some writers to present an exception to the universal applicability of the law of contradiction.[457]
[457] Compare Ueberweg, Logic, p. 245.
“Epimenides, the Cretan, says that all Cretans are liars. He is, therefore, himself a liar. Hence what he says is not true, and the Cretans are not liars. But if so, his statement may be accepted, and they are liars. And so on, ad infinitum.”
The solution is simple if we interpret the statement of Epimenides 458 to mean merely that Cretans usually speak falsehood. Let his assertion then be understood in a stricter sense than this, and as meaning that Cretans are always in all things liars, that no assertion made by a Cretan is ever by any chance true.
Again the solution is simple if we merely suppose the assertion false. Epimenides here speaks falsely, but Cretans frequently or sometimes speak the truth. We are obviously confusing the contradictory with the contrary if we pass from the position that it is not true that Cretans are always in all things liars to the position that what a Cretan says must therefore be true.
The sophism is a little more puzzling if we begin by assuming it to be true that Cretans never speak the truth. Such an assumption contains no self-contradiction, and there is therefore nothing to prevent our taking it as our starting-point. This being so, let Epimenides make his assertion. Because it is true, here is a Cretan who has spoken the truth, and therefore it is false. Its own truth proves its own falsity. But, again, because it is true, Epimenides cannot be speaking the truth, and therefore it is false. Once more its own truth proves its own falsity.
The argument may also be put as follows. Assume it to be true that Cretans are always in all things liars, and then let Epimenides, the Cretan, make this assertion. Either he speaks truly or he speaks falsely. But if he speaks truly, it thereby follows that he speaks falsely; whilst, on the other hand, if he speaks falsely, he merely affords additional evidence of the truth of what he says.
The problem offering itself for solution is how an apparently valid argument can thus yield as its result nothing but a bare contradiction. The explanation is that we have commenced with premisses that are implicitly contradictory, and that our subsequent reasoning has fulfilled its proper function in making the contradiction explicit. There is nothing self-contradictory in assuming that Cretans never speak the truth; but having commenced with this assumption, we cannot without implicit contradiction suppose a Cretan to make the assertion. In other words, the two premisses—Cretans are always in all things liars; and Epimenides, the Cretan, said so—cannot be true together.
418. The Law of Excluded Middle.—The law of excluded middle supplements the law of contradiction in explaining the nature of the relation between two contradictory judgments. The law of contradiction tells us that, of two contradictory judgments one or other 459 must be false, the truth of either implying the falsity of the other; the law of excluded middle tells us that of two contradictory judgments one or other must be true, the falsity of either implying the truth of the other. It is only by the aid of the two laws combined that the meaning of negation can be fully expressed.
Sigwart regards the law of excluded middle as a derivative principle dependent upon the principle of contradiction and another principle which he designates the principle of twofold (or double) negation. He observes that to interpret the nature of negation completely we must add to the principle of contradiction the further principle that the negation of the negation is affirmative, that to deny a negation is equivalent to affirming the same predicate of the same subject. To this further principle he gives the name of double negation; and it is, he says, only because the denial of the negation is the affirmation itself that there is no medium between affirmation and negation.
The deduction is as follows. Let X = A is B, and X = A is not B. The principle of contradiction tells us that of the two judgments X and X, one is necessarily false. It follows that one is necessarily true. For if I deny X then by so doing I maintain X, while if I deny X then (by the principle of double negation) I maintain X. Therefore, the denial of both is equivalent to the affirmation of both, that is, it involves a contradiction. Hence there is no middle statement between affirmation and negation.
In criticism of the above it may be questioned whether the bare law of contradiction justifies us in passing explicitly from the denial of X to the affirmation of X. Sigwart’s own statement of the principle of contradiction is that X and X cannot be true together. This enables us to pass from the affirmation of X to the denial of X, or from the affirmation of X to the denial of X ; but nothing more. There appears, moreover, to be a want of symmetry in Sigwart’s treatment of the matter. He makes the law of contradiction yield (1) affirmation of X is denial of X, (2) affirmation of X is denial of X, (3) denial of X is affirmation of X ; while the principle of double negation yields only (4) denial of X is affirmation of X.
All four of these relations are required in order that the nature of contradiction may be fully expressed; but unless we sum up all four in a single statement, it seems better to express (1) and (2) by means of the principle of contradiction, and (3) and (4) by means of a second principle, whether we call the latter by the name of the 460 principle of excluded middle or by any other name. It will be observed that we can express (1) and (2) together in the form Not both X and X, and (3) and (4) together in the form Either X or X.
Sigwart’s principle of double negation thus appears to express one-half of what is ordinarily expressed by means of the law of excluded middle; and its separate recognition may be regarded as unnecessary. I agree with Sigwart, however, in holding that the law of excluded middle does no more than help to unfold the meaning of negation.
It is not necessary to occupy space in discussing the relation of the formula Every A is B or not-B to the principle of excluded middle as above described. This formula expresses a secondary relation between so-called contradictory terms which follows from the corresponding, but more fundamental, relation between contradictory judgments.
For what is ordinarily known as the law of excluded middle, Jevons proposes the name law of duality.[458] This he does on the ground that the law in question asserts that at every step there are two possible alternatives, and hence gives to all the formulae of reasoning a dual character. The law of duality occupies an important position in Jevons’s system of formal logic, which is based on the repeated application of the principle of dichotomal division. It may, however, be questioned whether, as thus employed, the law of duality ought not to include the law of contradiction as well as the law of excluded middle. It is as important at each stage that the alternatives are exclusive as that they are exhaustive.
[458] Principles of Science, 1, § 3.
419. Grounds on which the absolute universality and necessity of the law of excluded middle have been denied.—The universal applicability of the law of excluded middle has been more frequently denied than that of either of the two laws previously discussed. The denial usually depends upon a confusion between contradictory opposition and contrary opposition. It is said, for example, that there is a mean between greater and less. This is true; but the law of excluded middle does not exclude the possibility of such a mean. That law does not tell us that a given quantity must be either greater or less than another given quantity; it only tells us that it must be either greater or not greater.
Closely connected with this is the case where our inability 461 (through lack of the requisite knowledge or power of discernment) to decide in favour of either of two contradictory alternatives is supposed to yield a third alternative; as, for example, where to the two alternatives “guilty” and “not guilty” is added the third alternative “not proven.” “Guilty” and “not guilty,” considered purely in relation to the supposed culprit, are true contradictories, and they admit of no mean. But “proved to be guilty” and “proved to be not guilty” are contraries, not contradictories; and it is here that the third alternative “not proven” comes in.
Some difficulty may also arise from ambiguity or uncertainty in the use of language. Thus it may perhaps be said that a prisoner may be neither “guilty” nor “not guilty,” but “partially guilty.” By “guilty,” however, we must understand either “entirely guilty” or “guilty in any degree”; and whichever of these meanings we adopt the difficulty is resolved.
We may deal similarly with the question whether an action occupying a finite interval of time for its completion has or has not taken place when it is actually proceeding; for example, whether a battle has or has not been fought when it is half through, or whether the sun has or has not risen when half its circumference is above the horizon.
The difficulties which arise in such cases as these are really verbal difficulties.
Other difficulties arising from uncertainty as to the precise range of application of terms are partly verbal and partly dependent upon our imperfect powers of discrimination. We may perhaps hesitate to say of a given colour whether it is “blue” or “green,” and therefore whether it is “blue” or “not blue.” If, however, by means of the spectrum or otherwise we are able to determine quite precisely what we mean by “blue,” the difficulty is obviated.
Mill remarks, on a different ground from any of the above, that the principle of excluded middle is not true unless with a large qualification. “A proposition must be either true or false, provided that the predicate be one which can in any intelligible sense be attributed to the subject. ‘Abracadabra is a second intention’ is neither true nor false. Between the true and the false there is a third possibility, the unmeaning” (Logic, ii. 7, § 5).
The reply to this is that the law of excluded middle applies only to propositions properly so-called, that is, to propositions regarded as the verbal expressions of judgments, a condition which clearly is 462 not satisfied by a sentence (falsely called a proposition) which is unmeaning. If we define a proposition as the verbal expression of a judgment, then an “unmeaning proposition”—a mere fortuitous jumble of words that conveys nothing to the mind—is in reality a contradiction in terms.
By an “unmeaning proposition” in the above argument we have understood a so-called proposition which has no meaning for the person who utters it or for anyone else. To a given individual a statement made by someone else may be unmeaning because he does not understand the force of the terms employed; but this in no way affects the principle that the statement will as a matter of fact be either true or false.
Whilst, however, every judgment must be either true or false, it is quite possible that unsuitable questions may be put, the correct answers to which will be negative, but will be felt to be barren and insignificant because anyone who understands the meaning of the terms employed will recognise at once that the predicate cannot in any intelligible sense be attributed to the subject.[459]
Is virtue circular? This question is felt to be absurd; but it is not unmeaning. By saying that anything is circular we mean that it has some figure and that its figure is circular. If, therefore, the question of circularity is raised in connexion with something that is immaterial, and therefore has no figure at all, the answer must be in the negative.[460]
[460] Compare Bradley, Principles of Logic, p. 145. Mr Bradley puts the question, “When a predicate is really known not to be ‘one which can in any intelligible sense be attributed to the subject,’ is not that itself ground enough for denial?”
This point may perhaps hardly seem worth raising. It helps, however, to explain how Mill is led to his denial of the universal applicability of the law of excluded middle. In his criticism of Hamilton’s doctrine of noumena the question is raised whether matter in itself has a minimum of divisibility or is infinitely divisible. Mill’s answer is that although we appear here to have contradictory alternatives, both may have to be rejected, since divisibility may not be predicable at all of matter in itself. In other words, the proposition that matter in itself has a minimum of divisibility is neither true nor false, but unmeaning.
It is to be observed, however, that “having a minimum of 463 divisibility” and “being infinitely divisible” are not contradictories except within the sphere of the divisible. If a wider point of view be taken, the contradictory of “having a minimum of divisibility” must be expressed simply in the form “not having a minimum of divisibility,” the latter including the case of “infinite divisibility,” and also that of “the absolute inapplicability of the attribute of divisibility.”
420. Are the Laws of Thought also Laws of Things?—On the view taken of the laws of thought in the preceding pages, the question whether these laws are also laws of things must be regarded as somewhat misleading. We have described the laws as postulates which are fundamental in all valid thought and reasoning, and we have regarded them as concerned essentially with judgments. Our results may be very briefly summarised as follows.
The truth affirmed in any judgment, when fully expressed, is independent of time and context. It is accordingly not open to us to accept a judgment at one stage of an argument or course of reasoning and reject it at another. This unambiguity of the fact of judgment is declared by the law of identity, and again by the law of contradiction, the one looking at the question from the positive, and, the other from the negative, point of view. Again, all judgment involves both affirmation and denial; and the force of any judgment is not fully grasped by us until we realise clearly what it denies as well as what it affirms. The law of contradiction, in conjunction with the law of excluded middle, has the function of making explicit what we mean by denial. The three laws may be expressed by these formulae: I affirm what I affirm, and deny what I deny ; If I make any affirmation, I thereby deny its contradictory ; If I make any denial, I thereby affirm its contradictory.
It follows that we cannot make any progress in material knowledge except in subordination to these laws. But at the same time they do not directly advance our knowledge of things. They are distinctly laws relating to judgments, and not directly to the things about which we judge.
No doubt when it is said that the laws of thought are also laws of things, the laws are contemplated in what we have regarded as their secondary forms: A is A ; A is not not-A ; Everything is A or not-A. But even so it is difficult to give them any meaning regarded as real propositions. By “A” we mean “A” neither more nor less; and by “not-A” we mean “that which is not A but includes everything else.” The laws do not profess to give any 464 material knowledge, and their validity is in no way dependent upon material conditions.
The question raised in this section has in substance been already dealt with in rather more detail in special connexion with the law of identity.
421. Mutual Relations of the three Laws of Thought.—If the validity of the ordinary processes of immediate inference is granted, it can be shewn that the three laws of thought mutually involve one another.
Starting from the hypothetical proposition,
| If A is true then C is true | (i), |
we obtain as its (true) disjunctive equivalent,
| It cannot be that A is true and C is not true | (ii), |
and as its alternative equivalent,
| Either C is true or A is not true | (iii). |
If now for C, we write A we have the following set of equivalent propositions:
If A is true, it is true ;
It cannot be that A is both true and not true ;
A is either true or not true ;
and these are expressions of the law of identity, the law of contradiction, and the law of excluded middle respectively.
It has been already shewn in section [108] that a similar result is obtainable if we write S for P in the following trio of equivalent propositions:
Every S is P ;
Nothing is both S and not P ;
Everything is P or not S.
These results indicate the close relations that exist between the three laws. But it is a mistake to suppose that we can regard one only of them as fundamental and the two others as deducible from this one. For the laws of thought stand at the foundation of all proof, and they must be postulated in order that the equivalences above assumed may themselves be shewn to be valid.
422. The Laws of Thought in relation to Immediate Inferences.—Granting that the laws of thought stand at the foundation of all proof, it is a further question what inferences, if any, can be shewn to be valid by their aid alone.
Hamilton claims that the law of identity is the principle of all logical affirmation, the law of contradiction of all logical negation, 465 and the law of excluded middle of all logical disjunction. By logical affirmation we may here understand affirmation which can be based on purely formal considerations without reference to the matter of thought, and we may interpret logical negation and logical disjunction similarly. The three laws of thought are accordingly held by Hamilton to justify what we have elsewhere called formal propositions, according as they are affirmative, negative, or disjunctive respectively. The division into affirmative, negative, and disjunctive is, however, of the nature of a cross division; and the question arises where we are to place formal hypotheticals such as the following:—If it is true that whatever is S is P, then it is true that whatever is not P is not S ; If it is true that all S is M and that all M is P, then it is true that all S is P. Apparently, since they are affirmative, they are to be brought under the law of identity; and inasmuch as the principle of any formal inference whatsoever may be expressed in a formal proposition similar in character to the above propositions, we find that Hamilton practically lays down the doctrine that in the three laws of thought (if not in the law of identity alone) we have a sufficient foundation upon which to base all logical inference.
This doctrine may, in the first place, be briefly considered with special reference to immediate inferences.
It may be granted that the process of obversion can be based exclusively on the laws of contradiction and excluded middle. From All S is P we pass to No S is not-P by the law of contradiction; and from No S is P we pass to All S is not-P by the law of excluded middle.
But it is a different matter when we pass to the consideration of the processes of conversion and contraposition; and it will be found that attempts to base these processes exclusively on the three laws of thought usually resolve themselves either into bare assertions or else into practical denials that conversion and contraposition are processes of inference at all.
De Morgan observes, “When any writer attempts to shew how the perception of convertibility ‘A is B gives B is A’ follows from the principles of identity, difference, and excluded middle, I shall be able to judge of the process; as it is, I find that others do not go beyond the simple assertion, and that I myself can detect the petitio principii in every one of my own attempts” (Syllabus of Logic, p. 47).
466 The test that I should be disposed to apply to any attempted proof of the validity of the process of conversion is to ask wherein the principle involved in the proof makes manifest the inconvertibility of an O proposition, and the illegitimacy of the simple conversion of A. It is clear that we have no right to assume that any self-evident principles that we may call to our aid[461] are equivalent to the law of identity.
[461] For example,—If one class is wholly or partially contained in a second, then the second is at least partially contained in the first; If one class is wholly excluded from a second, then the second is wholly excluded from the first.
The following attempt to establish the conversion of A and of I by means of the law of identity may be taken as an example: “Every affirmative proposition may be considered as asserting that there are certain things which possess the attributes connoted both by the subject and the predicate—the class SP. Hence the principle of identity justifies the conversion of an affirmative proposition. For if there are S’s which possess the attribute P, the principle of identity necessitates that some of the objects which possess that attribute are S’s.” The law of identity is referred to here, but we may fairly ask in what form that law really comes in. Does the argument amount to more than that as thus analysed the validity of the conversion in question is self-evident?[462] Might we not for the words “the principle of identity necessitates” substitute the words “it is self-evident”?[463]
[462] In so far as the argument is intended to amount to more than this, it contains a petitio principii.
[463] Compare, further, the discussion of the legitimacy of conversion in section [99].
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.”
[464] Prolegomena Logica, p. 222.
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.