CHAPTER IX.
THE CHARACTERISTICS OF INFERENCE.
376. The Nature of Logical Inference.—The question as to the nature and characteristics of inference, so far as its solution depends on the more or less arbitrary meaning that we choose to attach to the term “inference,” is a merely verbal question. The controversies to which the question has given rise do not, however, depend mainly on verbal considerations; and the fact that they partly do so has increased rather than diminished the difficulties with which the problem is beset.
It will be generally agreed that inference involves a passage of thought from a given judgment or combination of judgments to some new judgment. This alone, however, is not sufficient to constitute inference in the logical sense. The formation of new judgments by the unconscious association of ideas is a psychological process which might be brought under the above description; but it is not what we mean by logical inference.
(1) It is, in the first place, an essential characteristic of logical inference that the passage of thought should be realised as such. The connexion between the judgment or judgments from which we set out and the new judgment at which we arrive must be one of which we are, at any rate on reflection, explicitly conscious.
(2) But this again is not in itself sufficient. There must further be an apprehension that the passage of thought is one that is valid ; there must, in other words, be a recognition that the acceptance of the judgment or judgments originally given 414 constitutes a sufficient ground or reason for accepting the new judgment.
In logical inference, then, I do not merely pass from P to Q ; I realise that I am doing so. And I apprehend further that the truth of P being granted, the truth of Q necessarily follows. For logical inference, in short, it is required that there should be a logical relation between a premiss or premisses and a conclusion, not merely a psychological relation between antecedents and consequents in a train of thought.
This distinction between the logical and the psychological may be briefly illustrated by reference to what are known as acquired perceptions. Psychologists are, for example, agreed that our perception of distance through the sense of sight or the sense of sound is not immediate, but acquired in the course of experience. Here then we have a case in which one perception generates another; but there is no conscious passing from premisses to a conclusion, and nothing that can properly be called inference. Hence we must reject Mill’s dictum that “a great part of what seems observation is really inference” (Logic, iv. 1, § 2), so far as the dictum is based—as to a large extent it is—on the position that a great part of our perceptions are acquired, not immediate. Here, as well as in connexion with some of his other and more important logical doctrines, Mill is open to the charge of failing to distinguish between the cause of a belief and its ground or reason.
377. The Paradox of Inference.—The description of logical inference given in the preceding section leads up immediately to the fundamental difficulty which any discussion of the subject must inevitably bring to the forefront. We are in fact face to face with what has aptly been designated the “paradox of inference.” On the one hand, we are to advance to something new; the conclusion of an inference must be different from the premisses, and hence must go beyond the premisses. On the other hand, the truth of the conclusion necessarily follows from the truth of the premisses, and the conclusion must therefore in some sense be contained in the premisses.
There may appear to be a contradiction here; and this view 415 tends to be confirmed when it is found that the two characteristics of inference referred to are by different schools of logicians used in such a way as between them to deprive the category of inference of any content whatsoever.
On the one hand, by laying stress on the characteristic of novelty, we may be led to doubt whether formal inference of any description can properly be so called. For in all such inference the conclusion is implicitly contained in the premisses, and in uttering the premisses we have virtually committed ourselves to the conclusion. How then can we be said to make any advance to what is really new?
On the other hand, by laying stress on the characteristic of necessity, we may be led to doubt whether any inductive inference can properly be so called. For in such inference the falsity of the conclusion is not demonstrably inconsistent with the truth of the premisses. We may hold that if the premisses are true the conclusion will be true. But can we hold that it must be true, unless we also hold that in affirming the premisses we have virtually affirmed the conclusion too? And then we are back on the other horn of the dilemma.
This is not the place at which to discuss the difficulty from the point of view of inductive logic. We must, however, attempt a solution from the point of view of formal logic.
378. The nature of the difference that there must be between premisses and conclusion in an inference.—In order to find a solution of the difficulty, so far as formal inference is concerned, we must pursue our analysis further. We have said that the conclusion must be different from the premiss or premisses. But we have not yet asked what must be the nature of the difference or wherein it must consist; and it is on the answer to this question that everything turns.
If we consider two sentences we shall find that they may differ from one another from three distinct standpoints, representing three degrees of difference.
(1) In the first place, two sentences may differ from one another from the verbal standpoint only; that is to say, though different in the words of which they are made up, they may have the same meaning, and what the one is intended to convey 416 to the mind may be precisely what the other is intended to convey. In this case, regarded as propositions and not as mere sentences, they cannot be said to be really different at all; for they do not represent different judgments.
This (to take an example from Jevons) applies to two such sentences as Victoria is the Queen of England, Victoria is England’s Queen. It applies also to a statement expressed in a given language and the same statement translated into a second language, assuming that an absolutely literal translation is possible.
It has indeed been maintained by some writers that a difference of expression necessarily involves some difference of thought. But this at any rate appears not to be the case where one single word is substituted for another completely coincident with it both in denotation and in connotation (as thought by the speaker). Where one complex term is substituted for another (for example, England’s Queen for Queen of England) there may no doubt be involved some change in the order of thought; but this does not necessitate any change of meaning in the thought considered as a whole. Again we ought perhaps not to say that the same proposition expressed in two different languages has absolutely the same mental equivalent, since a consciousness of the actual words of which a proposition consists may constitute part of its mental equivalent. But, as before, this makes no difference in the meaning that the proposition is intended to convey.
It should be added that when we have a judgment expressed in two different languages or in two different forms in the same language, there is (or may be) involved the further judgment that the two modes of expression are equivalent. A distinct issue is, however, here raised.[440]
[440] This issue is, I think, involved in an argument used by Miss Jones (in an article in Mind, April, 1898) in support of the doctrine that we have inference whenever we pass from a given proposition to another that is verbally different from it; for example, from Victoria is Queen of England to Victoria is England’s Queen. The passage from one of these propositions to the other is, in Miss Jones’s view, not indeed a formal immediate inference, but a syllogism in which an understood premiss has to be supplied: thus, Victoria is Queen of England, The Queen of England is England’s Queen, therefore, Victoria is England’s Queen. It may, Miss Jones adds, seem futile or even puerile to set out at length what everybody or nearly everybody knows without telling; but there may be cases (e.g., the case of a child or of a foreigner learning the English language) in which a reasoning of this kind has to be gone through.
It appears to me that there is here a failure to distinguish between two different points of view. We may no doubt draw an inference as to the equivalence of meaning of two terms or two expressions, where the whole argument is concerned with the meaning of terms or the force of expressions. Thus, to take (or, rather, adapt) another of Miss Jones’s examples, we may readily admit that there is inference if a German argues that because the word Valour is equivalent in meaning to the word Tapferkeit, and the word Bravery is also equivalent in meaning to the word Tapferkeit, therefore, the words Valour and Bravery are equivalent in meaning. Again, a child or a foreigner may arrive by a process of inference at the equivalence of such forms as Queen of England and England’s Queen. But in the syllogism given above the first premiss and the conclusion are statements of fact, while the second premiss is a statement as to modes of expression, its import being “The expression Queen of England is equivalent to the expression England’s Queen.” Hence there are more than three terms and we have not properly any syllogism at all. So far as there is inference in the case supposed, it will be something like the following,—“The form of words Queen of England is equivalent in meaning to the form of words England’s Queen,” therefore, “The judgment which is expressed in the form Victoria is Queen of England may also be expressed in the form Victoria is England’s Queen.” This is the inference, if any, that a foreigner studying the language would make; and it is very different from professing to pass from the judgment Victoria is Queen of England to the judgment Victoria is England’s Queen.
417 (2) In the second place, we may have a difference which goes beyond mere difference of expression, and constitutes a difference in subjective meaning, though there may still be no difference from the objective standpoint. In this case we have two distinct propositions, not merely two different sentences, and these propositions are the expressions of two different judgments.
This relation holds in my view between a proposition and its contrapositive; for example, between Euclid’s twelfth axiom, “If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles that are less than two right angles,” and the second part of the twenty-ninth proposition of his first book, “If a straight line fall on two parallel straight lines, it shall make the two 418 interior angles on the same side together equal to two right angles.” It cannot be said that in such a case as this we have any objective difference, any difference in the matter of fact asserted; but at the same time we hold that the two judgments to which expression is given are not to be regarded as identical quâ judgments.
To this distinction we shall return shortly from a more controversial point of view.
(3) In the third place, our sentences may differ not merely from the verbal and subjective standpoints, but also from the objective standpoint; they may affirm distinct matters of fact. As, for example, if one of them states that all potassium with which we have experimented takes fire when thrown on water, and the other that a piece of potassium with which we have not yet experimented will do the same.
Now in all three of these cases we have novelty, and the question to be decided is which of the three kinds of novelty is requisite in order that we may have inference. I hold the right answer to be that, for inference, subjective novelty is necessary and sufficient.
There is practically universal agreement that something more than mere difference of verbal expression is requisite for inference.[441]
[441] Miss Jones holds that verbal difference suffices; but this is only because she also holds, as we have seen, that we cannot have mere verbal difference, that is, difference of expression without difference of thought.
Objective novelty is certainly sufficient, but is it requisite? It is affirmed to be so by writers of the school of Mill. This may of course be a mere question of definition; that is to say, inference may be defined ab initio in such a way as to require that the conclusion reached shall express some objective fact not contained in the data on which it is based. The matter being thus decided by definition, it follows without controversy that contraposition, syllogism, and other formal inferences (so called) are not properly to be spoken of as inferences at all. But there a good deal more than a mere question of definition involved. Those who demand objective novelty appear to hold that without it we cannot have more than mere 419 verbal novelty. They overlook, or at any rate practically deny, the possibility of taking an intermediate course whereby we may have something more than verbal novelty, but something less than objective novelty.
Here then we have one form in which the point mainly at issue in regard to the nature of inference presents itself. Is it possible for two judgments to be different quâ judgments, although from the objective standpoint one of them states nothing that is not also stated by the other? Or, to put the question differently, can two judgments (or sets of judgments) be distinct as judgments although they are not logically independent, that is, although self-evident relations exist between them such that the truth of one of them involves the truth of the other?
I am ready to admit that it is no easy matter to draw a hard and fast line determining where mere verbal novelty ends and subjective novelty begins. Before attempting to deal with this difficulty, however, I will endeavour to shew that there undoubtedly are cases in which we have progress in thought without reaching anything that is objectively new.
Mill, after giving examples of so-called immediate inferences, says, “In all these cases there is not really any inference; there is in the conclusion no new truth, nothing but what was already asserted in the premisses, and obvious to whoever apprehends them” (Logic, ii. 1, § 2). Now it is certainly the case that in any formal inference the conclusion is implicitly contained in the premisses, and affirms no absolutely new fact. But it is one thing to say that a conclusion is virtually contained in certain premisses, and quite another to say that it is obvious to whoever apprehends the premisses. The identification of these two positions is one of the unfortunate consequences of taking simple conversion as the type of all immediate inference, and a single syllogism in Barbara as the type of all mediate formal inference. It may be difficult for anyone to apprehend that no S is P without at once apprehending that no P is S, or to apprehend the premisses of a syllogism in Barbara without at once apprehending the conclusion also. These cases will need discussion; but just now we are more concerned to point out 420 that there are other formal inferences against which any similar charge of obviousness cannot be brought.
All the theorems of geometry are virtually contained in certain axioms and postulates, and if we can exhaustively enumerate the axioms there is in a sense no new geometrical fact left for us to assert. Yet no one would say that the whole of geometry is at once obvious to anyone who has clearly apprehended the axioms. We shall, however, deal with syllogistic inference more in detail in a later [section]. For the present we will in the main confine ourselves to immediate inferences.
In order to shew that the conclusion of an immediate inference is not always immediately obvious to anyone who clearly apprehends the given premiss, it may be pointed out that it is Euclid’s practice to give independent proofs of contrapositives.[442] For example, the second part of Euclid I. 29 is the contrapositive of axiom 12. But it is impossible to suppose that if Euclid had regarded I. 29 as not really distinct from axiom 12, but merely as a repetition of that axiom in other words, he would have given an elaborate proof of it. The following are two other fairly simple examples of immediate inferences: Where B is absent, either A and C are both present or A and D are both absent, therefore, Where C is absent, either B is present or D is absent ; Where A is present, either B and C are both present, or C is present without D, or C is present without F, or H is present, therefore, Where C is absent, we never find H absent, A being present.
[442] See [note 4] on page 136.
In such cases as these, and they are comparatively simple ones of their kind, it cannot be maintained that the conclusion is at once obvious when the premiss is given. As a matter of fact, mistakes are not unfrequently made in immediate inferences of a still simpler and more elementary character.
379. The Direct Import and the Implications of a Proposition.—At this point a question may fairly be raised as to how we determine what is the explicit force of a given proposition, assuming the proposition to be clearly understood and fully grasped by the mind. This question is by no means easy 421 to answer, and the difficulty which it presents is the source of the doubt which sometimes arises when we attempt to draw the line between immediate inferences and mere verbal transformations.
If immediate inferences are possible, we must be able to discriminate between the direct logical import (or meaning) of a proposition and its logical implications; and it must be possible to grasp fully the meaning without at the same time necessarily realising all the implications.[443] We may begin by distinguishing between (1) the content of the judgment actually present to our mind when we utter or accept a proposition in ordinary discourse or in ordinary reading; (2) the content of the judgment which on reflection we are able to regard as constituting the full logical meaning of the proposition; (3) the content of this judgment together with the content of other judgments which it logically implies.
(1) is a psychological product which may be, and usually is, logically imperfect; that is to say, it needs to be amplified if we are fully to realise the meaning of the proposition. Such amplification cannot be regarded as constituting inference. For, in making any inference, our starting point must be the proposition considered in its logical character. The inference comes in when we pass from (2) to (3). The question, however, arises as to how far the amplification is to extend if our object is to stop short at (2). In other words, where does meaning end and implication begin?
It has been pointed out at an earlier stage that in assigning to given combinations of words their logical import there is a certain element of arbitrariness. There is often a similar element of arbitrariness in formulating the fundamental axioms of a science, as well as in framing definitions. Thus, in geometry we cannot do without some special axiom relating to parallel straight lines, but we have some choice as to what the axiom shall be. Hence what is an axiom in one system may be a theorem in another, and vice versâ. Similarly, whether Q is to be regarded as part of the meaning of P, or as an inference from P, may be relative to the interpretation 422 adopted of the schedule of propositions to which P belongs. Some illustrations of this point will be given shortly.
We have cited cases in which it appears clear that we have inference and not mere verbal transformation. But in most of these cases intermediate steps may be inserted; and if this is done to the fullest possible extent, the progress at each step may be so slight that it may not be at all easy to say wherein precisely the inference is to be found.
We must then proceed to consider the limiting cases in which there may be legitimate doubt as to whether we have inference or not. One of these cases is that of conversion. The question whether there is inference in conversion may be in itself, as Mr Bosanquet puts it, “a point of little interest” (Essentials of Logic, p. 141). Nevertheless, as a limiting case, it is not lightly to be put on one side when we are attempting to decide what fundamentally constitutes inference.
It appears to me that conversion is a process of inference if we are dealing with a schedule of propositions in which the predicative reading is adopted. In such a schedule the primary import of the various propositions involves a differentiation between subject and predicate, and to predicate P of S or to deny that P can be predicated of S is a different thing from predicating S of P or denying that S can be predicated of P. Moreover we may grasp the one relation without necessarily realising whether it does or does not involve the other. But in an equational system it is different. If two classes are affirmed to be identical it is merely a verbal question which is mentioned first, and we cannot consider that we have made any progress in thought when we merely alter the order in which they are named. It follows that we must consider that we have inference when we reduce a proposition expressed predicatively to the equational form.
In either schedule, contraposition (or a process analogous to contraposition) presents itself as an inference. In the one case, we have All S is P, therefore, Anything that is not P is not S ; in the other case, S = SP, therefore, Pʹ = PʹSʹ.
Suppose again that we have an existential schedule, and that we start from the proposition SPʹ = 0 [There is nothing that is 423 S and at the same time not P]. Here what corresponds to conversion is the passage to Either PS > 0 or S = 0 [There is something that is both P and S or else S is non-existent]; and, what corresponds to contraposition is the passage to PʹS = 0 [There is nothing that is not P and at the same time S]. Conversion, but not contraposition, now appears as a process of inference. It follows that there is inference when we pass to this schedule from either of the others, or vice versâ.
A further consequence to be drawn from the above considerations is that if propositions are given at random, inference may at the outset be required in order to adapt them to a given logical schedule, though as a rule this will not be necessary. This point has already been touched upon in section [48].
380. Syllogisms and Immediate Inferences.—In the above argument we have confined ourselves mainly to the consideration of immediate inferences. The same question in relation to the syllogism usually presents itself in a slightly different form, namely, whether every, syllogism involves a petitio principii ; and we shall discuss it in this form in the following [section]. In the meantime, we may observe that if there is no such thing as immediate inference properly so called, then the claims of the syllogism to contain inference become very hard to maintain. For by the aid of immediate inferences the premisses of a syllogism can be combined into a single proposition, and the conclusion can then be obtained as an immediate inference from the combination.[444]
As an example, we may take a syllogism in Barbara:[445]
| All M is P, | (1) | |
| All S is M, | (2) | |
| therefore, | All S is P. |
From (1),
| Everything is m or P, | |
| therefore, | Every S is m or P. |
Combining this with (2) we have
Every S is M, and also m or P ; (3)
therefore, Every S is MP (since nothing can be Mm);
therefore, Every S is P.
[445] In the argument that follows m = not-M, s = not-S.
424 All the above steps are immediate inferences, except the combination which yields (3). Hence, if we hold that syllogism is inference while so-called immediate inference is not, we must regard the whole of the inference as concentrated in the mere combination of two propositions into a single proposition; and this is hardly a position that can be accepted.
The given syllogism might also be reduced as follows:
From (1) it follows that Everything is m or P ; (4)
and from (2) we get Everything is s or M. (5)
Combining (4) and (5), Everything is (s or M) and (m or P);
therefore, Everything is sm or sP or MP ;
therefore, Every S is P.
We may note in passing that if elimination is regarded as constituting the essence of inference, then in each of the above resolutions of the syllogism all the inference is concentrated in the last step, and this again seems paradoxical.
381. The charge of Petitio Principii brought against Syllogistic Reasoning.[446]—The objection to syllogistic reasoning that it necessarily involves petitio principii is of considerable antiquity. Thus Sextus Empiricus (circa 200 A.D.), one of the Later Skeptics, seeking to disprove the possibility of demonstration, urged, as one of his arguments, that every syllogism moves in a circle, since the major premiss, upon which the proof of the conclusion depends, requires in order that it may be itself established a complete enumeration of instances, amongst which the conclusion must itself be included.[447] The same objection to the syllogism is raised by many recent logicians, including Mill and his followers. “It must,” says Mill, “be granted that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii” (Logic, ii. 3, § 2).
[446] There is a very good discussion of this question in Venn’s Empirical Logic, chapter 15. The reader may also be referred to Mansel’s edition of Aldrich, note E, and to Lotze’s Logic, §§ 98–100.
[447] Compare Ueberweg, History of Philosophy (English translation, i. p. 216).
It may be said at the outset that the plausibility of the argument by which Mill seeks to justify this position depends a 425 good deal upon a certain ambiguity that attaches to the phrase petitio principii. When the charge of petitio principii is brought against a reasoning, is it merely meant (1) that the premisses would not be true unless the conclusion also were true, or is it meant (2) that the conclusion is necessary for the proof of one of the premisses? It is clearly one thing to say that the premisses of a certain reasoning cannot be true unless the conclusion is true, and quite another to say that we cannot know the premisses to be true unless we previously know the conclusion to be so, or to say that the proof of the premisses necessitates that the conclusion shall have been already established. Only in the second of the above senses can petitio principii be regarded as a fallacy ; and any one who, seeking to prove that every syllogism is guilty of the fallacy of petitio principii merely shews that syllogistic reasoning involves petitio principii in the other sense, himself commits the fallacy of ignoratio elenchi.
In his systematic treatment of fallacies, Mill classifies petitio principii amongst fallacies of confusion, and quotes with approval Whately’s definition: it is the fallacy “in which one of the premisses either is manifestly the same in sense with the conclusion, or is actually proved from it, or is such as the persons you are addressing are not likely to know, or to admit, except as an inference from the conclusion” (Logic, v. 7, § 2 n.). This fallacy has been described as being a fallacy of proof rather than a fallacy of inference ; that is to say, it arises when we ask how a given thesis is to be established, rather than when we ask what follows from a given hypothesis. We have to enquire whether every syllogism is open to the charge of petitio principii in this sense.
It is obvious that the answer to the question in the case of any particular syllogism depends upon the grounds on which the premisses are themselves affirmed; and we may begin by calling attention to certain cases in which the justice of the charge must be admitted, the conclusion of the syllogism being regarded as a thesis to be proved.
One case is when the major premiss is an analytic proposition.[448] For if M by definition includes P amongst its 426 properties, I am not justified in saying of S that it is M unless I have already satisfied myself that it is P. The following is an example: All triangles have three sides; the figure ABC is a triangle; therefore, it has three sides.
[448] This case is noticed by Lotze, Logic, § 99.
A second obvious case of circulus in probando is where we seek to establish one of the premisses of a syllogism by means of another syllogism in which the ultimate conclusion itself appears as a premiss. For example,—All M is P (for all S is P, and all M is S); and all S is M ; therefore, all S is P.
A third case, which for our immediate purpose is more important than either of the above, is where the major premiss is an enumerative universal, summing up a number of individual instances each one of which has been separately considered. For example, All the apostles were Jews; Peter was an apostle; therefore, Peter was a Jew. A universal proposition relating to a limited class, such as the apostles, is usually established by considering the members individually; and if the truth of a universal proposition could be established in this manner only, then the charge that syllogistic reasoning necessarily involves petitio principii would not admit of refutation. This appears to be assumed in the argument of Sextus Empiricus quoted above. It is also assumed in the following dilemma, which has been given as summing up Mill’s doctrine: “If all the facts of the major premiss of any syllogism have been examined, the syllogism is needless; and if some of them have not been examined, it is a petitio principii. But either all have been examined or some have not. Therefore, the syllogism is either useless or fallacious,” Mill’s own argument may also be quoted: “We cannot be assured of the mortality of all men, unless we are already certain of the mortality of every individual man” (Logic, ii. 3, § 2).[449]
[449] Bain (Logic, Deduction, p. 208) taking as an example the syllogism, “All men are mortal, All kings are men, therefore, All kings are mortal,” asks “Supposing there were any doubt as to the conclusion that kings are mortal, by what right do we proclaim, in the major, that all men are mortal, kings included?” He then continues, “In order to say, ‘All men are mortal,’ we must have found in some other way that all kings and all people are mortal. So that the conclusion first contributes its quota to the major premiss, and then takes it back again.” The reply to Bain’s challenge is that if we are in doubt as to whether kings are mortal, we may resolve our doubt by shewing that kings belong to a class the mortality of which is admitted. The question then resolves itself into whether it is possible to establish the mortality of mankind in general without any explicit consideration of the particular case of kings.
427 It cannot, however, for a moment be allowed that universal propositions admit of proof only by enumeration. Propositions that do admit of such proof are indeed generally speaking of little importance. The syllogism is chiefly of value inferentially where the major premiss is universal in the fullest and most unlimited sense, that is, unconditionally universal, expressing a general law dependent on qualitative relations. The true character and value of such a premiss, though ordinarily written in the form All S is P, would be better brought out by the use of one of the forms Any S is P, Whatever is S is also P, It is the nature of S to be P, If anything is S it is P.[450]
[450] Sigwart holds that, in order properly to understand the value of the syllogism, we should take as our type the conditional (or, as he expresses it, the hypothetical), rather than the categorical, syllogism. We need, he says, but glance at any mathematical or physical text-book to assure ourselves that by far the greater number of propositions which are used as major premisses are hypothetical in nature, if not in expression. “Propositions such as ‘two circles which intersect have no common centre’ are hypothetical in nature; the proposition states the condition upon which the predicate is denied.… It is the same with the formulae of analytical mechanics; these and others of the same description are hypothetical judgments, and inferences are made in accordance with them by substituting definite values for the general symbols” (Logic, § 55). Sigwart perhaps attaches undue importance to the mere question of form. If our major premiss is unconditionally universal, and is understood to be so, it does not affect the character of the reasoning whether we adopt the categorical mode of expression or the conditional. Sigwart’s reason for dwelling on the hypothetical force of the major premiss is to be found largely in the trivial nature of the examples that it has been customary to give of the purely categorical syllogism.
The following may be noted as typical cases in which the grounds for accepting the truth of the premisses of a syllogism are quite independent of any explicit knowledge of the truth of the conclusion.
(1) The major premiss may itself be accepted as axiomatic, or it may be deducible (without the assistance of the conclusion) from more ultimate principles that are accepted as axiomatic. It has indeed been argued that a self-evident maxim cannot be used, or is at any rate superfluous, as a proof, because any conclusion that it might be employed to establish would be itself equally self-evident.[451] A consideration of ordinary 428 geometrical proofs will, however, at once shew that this is not necessarily the case, and that by the aid of self-evident premisses conclusions may be reached that are certainly not themselves self-evident.
[451] Compare Bailey, Theory of Reasoning, p. 74.
(2) The major premiss may be based on authority, or may be accepted on testimony; or it may be the expression of a civil law, or of a command, or of a rule of conduct;[452] and in none of these cases can it be in any degree grounded upon the conclusion.
[452] “We find,” says Sigwart, “a wide field for our inferences in the application of general laws which have their origin in our will and are meant to regulate that will. In laying down a general rule of conduct, our will determines that there shall be a universally valid connexion between certain conditions and certain modes of action. If we will the general law, it is logically necessary that we should will the particular actions prescribed by the law, if our will is to be constant and consistent, and valid for everyone who agrees in willing the general law. All penal codes in imposing a penalty of imprisonment for theft, of capital punishment for murder, lay down a series of hypothetical judgments which establish a universal connexion between committing the crime and incurring the penalty. These judgments, moreover, may also be regarded as theoretical propositions in so far as they express the general obligation of the judge to give sentence in accordance with the law” (Logic, i. p. 337).
(3) The major premiss may be an imperfect induction, based on evidence that does not include the conclusion. As an example, we may take the reasoning involved in testing the nature of a given substance in practical chemistry. In a reasoning of this kind our immediate starting point is general knowledge of the properties of chemical substances. This knowledge has been inductively obtained, but it is impossible that it should in the slightest degree depend on any antecedent acquaintance with the properties of the particular substance which is now to be investigated for the first time. Or, again, we may take astronomical inferences based on the law of universal gravitation. That law is an induction based on particular observations, but it implies an infinite number of facts that form no part of the evidence on which it is accepted as true; and many of these facts are in the first instance brought to our notice as inferences from the law, not as data leading up to it. If it is affirmed that, in cases such as these, the major premisses cannot legitimately be established independently of the conclusions syllogistically derived from them, then 429 the validity of imperfect induction as a process of arriving at knowledge must be denied.
If asked to meet the argument contained in the preceding paragraph, Mill would doubtless refer to his doctrine of the function of the major premiss in a syllogism. The real proof of the conclusion of a syllogism, he would say, is to be found, not in the major premiss itself, but in the evidence on which the major premiss is based: the major premiss is nothing more than a memorandum of evidence from which the conclusion might be directly obtained: the intervention of the major premiss is often convenient, but it is not an essential link in the passage from the ultimate data to the conclusion. In reply, it may be said that there is at any rate a shifting of the ground here, and that Mill’s doctrine, even if accepted, fails to justify the charge that every syllogism involves petitio principii ; for it is admitted that the conclusion does not itself constitute any part of the data from which the major premiss is obtained. We must, however, go further and reject the doctrine on the ground that there are at any rate some cases in which the general law expressed by the major premiss is an absolutely necessary link in the argument. Thus, to take but one illustration, there are many consequences of the law of universal gravitation which it would be quite impossible to infer directly from the evidence lying behind that law without the intervention of the law itself.
Having regard then to instances such as those adduced above, we must reject the view that syllogistic reasoning essentially involves petitio principii, in the sense of circulus in probando. Any plausibility that the opposed view may possess depends upon some confusion between the statement that every syllogism is guilty of petitio principii in the above sense and the statement that in every syllogism the premisses presuppose the conclusion in the sense that they could not be true unless the conclusion were true.
The latter statement is applicable not only to syllogistic, but to all demonstrative, inference. The question may indeed be raised whether it is not applicable to all valid inference whatsoever. It is in fact one horn of the dilemma referred to in section [377].
430 At any rate it is a misuse of language to speak of a reasoning as involving petitio principii on this ground. By petitio principii is always understood a certain form of fallacy. But in making explicit what to begin with is merely implicit there is nothing that can by any stretch of language be termed fallacious. To say that all deductive science is nothing but a huge petitio principii is clearly an absurdity. The most that can be said is that in all demonstrative reasoning (so-called) there is really no inference from premisses but only the interpretation of premisses. So far as this is a mere question of language, it may suffice to note the paradoxical conclusions to which it leads; for example, that in the whole of Euclid there is no such thing as inference or proof. So far as it is not a mere question of language, it turns on points that we have already discussed, for instance, the possibility of there being an advance in knowledge subjectively considered although from the objective standpoint the conclusions reached contain nothing new. It is unnecessary to repeat the discussion with special reference to the syllogism.