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CHAPTER VII.: REASONING IN GENERAL. - Herbert Spencer, The Principles of Psychology 
The Principles of Psychology (London: Longman, Brown, Green and Longmans, 1855).
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REASONING IN GENERAL.
§ 38. Before summing up the evidence, and presenting under its most general form the doctrine which the foregoing chapters develop in detail, it will be well briefly to glance at the current theory of reasoning, with the view of showing its insufficiency.
That so many logicians should have contended that the syllogism exhibits the process of thought by which we habitually reason, would be unaccountable, were it not for the immense influence of authority on men's opinions. Passing over the general objection, that it involves a petitio principii, and cannot therefore represent the mode by which we find our way to new truths, a cursory examination even, will suffice to show that the syllogism is a psychological impossibility. Take a case. When I say,—
and when it is asserted that this describes the mental process by which I reached the conclusion; there arises the very obvious question—What induced me to think of “All crystals”? Did the concept “All crystals,” come into my mind by a happy accident, the moment before I was about to draw an inference respecting a particular crystal? No one will assert such an absurdity. It must have been, then, that a consciousness of the particular crystal identified by me as such, was antecedent to my conception of “All crystals.” This, however, it will be said, is merely a formal objection; which may be met by putting the minor premiss first. True: but this objection is introductory to a fatal one. For the mind being, as we see, necessarily occupied about the individual crystal, before it is occupied about the class; there result the two inquiries—(1), Why, having been conscious of the individual crystal, should I, in this particular case, go on to think of the class crystals; instead of thinking of some other thing? and (2), Why, when I think of the class crystals, should I think of them as having planes of cleavage; instead of thinking of them as angular, or polished, or brittle, or having axes, or in connection with any other attribute? Is it again by a happy accident that, after the individual, the class occurs to my mind? and further, is it by a happy accident that the class is remembered as having the particular attribute I am about to predicate? No one will have the folly to say—yes. How happens it, then, that after the thought—“This is a crystal,” there arises the thought—“All crystals have planes of cleavage;” instead of some other of the thousand thoughts which mental suggestion might next produce? There is one answer, and only one. Before consciously asserting that all crystals have planes of cleavage, it has already occurred to me that this crystal has a plane of cleavage. Doubtless it is the registered experience I have had respecting the cleavage of crystals, which determines me to think of this crystal as having a plane of cleavage; but that registered experience is not present to my mind before the special predication is made; though I may become conscious of it subsequently. The process of thought which the syllogism seeks to describe, is not that by which the inference is reached, but that by which it is justified; and in its totality is not gone through at all, unless the need for justification is suggested. Each may at once convince himself of this by watching how any of his most familiar inferences originate. It is stated that Mr. So-and-so, who is ninety years old, is about to build a new mansion; and you directly say, how absurd it is that a man so near death should make such preparation for life. But how came you to think of Mr. So-and-so as dying? Did you first repeat to yourself the proposition—“All men must die?” Nothing of the kind. Certain antecedents led you to think of death as one of his attributes, without previously thinking of it as an attribute of mankind at large. To any one who considered Mr. So-and-so's folly not demonstrated, you would probably reply,—“He must die, and that very shortly:” not even then appealing to the general fact. Only on being asked why he must die, would you, either in thought or word, resort to the argument—“All men die: therefore Mr. So-and-so must die.” Obviously then, the syllogism in no way represents the ordinary inferential act; which is a single and almost unconscious intuition; but only approximately represents the process by which our inferences are, if need be, consciously verified.
As will of course be perceived, many of the formulas given in preceding chapters, are to be taken with a parallel explanation. They represent, not the primary and direct reasoning, but the secondary, and what we may call, reflex reasoning. To express any deduction by saying of the compared relations that,
is to raise the insuperable difficulty above suggested—that the class, with its appropriate predicate, cannot in order of thought precede the individual and that which we predicate of it; or, in other words—that we do not think of the class of before known relations as like the single present relation; but we think of the single present relation as like the class. Just as, before writing down the proportion I must have already recognized the unknown relation sought, as equal to the known relation premised: otherwise the writing down the premised relation would be unaccountable. Hence it is manifest, that to symbolize the deductive process in a complete manner, the inferred relation must be placed before, as well as after, the class of relations to which it is assimilated; thus—
The first of these three represents that act of thought in which, on the presentation of some object (a) there is suggested to the mind some unseen attribute (b), as possessed by it. This act is simple and spontaneous; resulting, not from a remembrance of the foreknown like relations (A : B); but merely from the influence which, as past experiences, they exercise over the association of ideas. Commonly, the inference thus determined suffices us; and we pass to some other thought: but if a doubt is internally or externally suggested, then the acts of thought represented by the rest of the symbol are gone through; and we have a process of conscious reasoning.
And here, respecting this series of mental acts, there occurs a consideration of some interest and importance. It is universally admitted that in the evolution of reasoning, induction must precede deduction—that we cannot descend from the general to the particular, until we have first ascended from the particular to the general. The fact now to be remarked is, that this is true not only of reasoning considered in its ensemble, but also, in a qualified sense, of each particular ratiocination. It was pointed out a few pages back, that as, in the development alike of the general mind and the individual mind, qualitative reasoning precedes quantitative reasoning; so, each particular act of quantitative reasoning grows out of a preceding act of qualitative reasoning: and in the present case there seems to hold the analogous law, that as, in mental progress, both general and particular, induction precedes deduction; so, every particular act of deduction properly so called, presupposes a preparatory act of induction. For may we not with propriety say, that the mental transition from the spontaneously inferred relation with which every deductive process must commence, to the class of relations it belongs to; parallels the act by which the mind originally passed from particular relations to the general relation? It is true that the particular relation is in this case not an observed one; and in so far the parallel does not hold: but still, it is conceived as existing; and it is only in virtue of being so conceived that the class it is referred to is thought of. The sequence of thought, as it were, follows the channel through which the induction was before reached. In so far as each separate deductive act involves an ascent from the particular to the general, before the descent from the general to the particular; the historic relation between induction and deduction is repeated. In all cases of deduction there is either an induction made on the spur of the moment (which is often the case), or there is a rapid rethinking of the induction before made.
Resuming our more immediate topic—It is to be remarked that the amended, or rather completed, form under which the deductive process is above represented, remains in perfect accordance with the doctrine, developed in foregoing chapters; that reasoning is carried on by comparison of relations. For whether the singular relation is thought of before the plural one; or the plural before the singular; or first one and then the other; it remains throughout manifest, that they are thought of as like (or unlike) relations; and that the possibility of the inference depends on their being so thought of. On the other hand, the syllogistic theory is altogether irreconcilable with the mental processes we have just traced out—irreconcilable as presenting the class, while yet there is nothing to account for its presentation; irreconcilable as predicating of that class a special attribute, while yet there is nothing to account for its being thought of in connexion with that attribute; irreconcilable as embodying in the minor premiss an assertory judgment (this is a man), while the previous reference to the class, men, implies that that judgment had been tacitly formed beforehand; irreconcilable as separating the minor premiss and the conclusion, which ever present themselves to the mind in relation. Whatever merit the syllogism may have as verbally exhibiting the data and conclusion in a succinct form; it wholly misrepresents the mental process by which the conclusion is really reached.
And if the syllogism, considered in the concrete, does not truly display the ratiocinative act; still less do the axiomatic principles reached by analysis of the syllogism, supply anything like a theory of the ratiocinative act. It may be said that it does not fall within the province of Logic to formulate the workings of the intellect—that it is concerned with the objective aspect of reasoning, and not with its subjective aspect, which pertains to Psychology—that all which Logic can do is to reduce overt inductions and deductions to their simplest elements, and to systematic arrangement. And this is true. But there seems to be an undefined yet general impression, that a certain abstract truth said to be involved in every syllogism, is that which the mind recognizes in going through every syllogism; and that the recognition of this abstract truth under any particular embodiment, is the real ratiocinative act. Nevertheless, neither the dictum de omni et nullo—“that whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class;” nor the axiom which Mr. Mill evolves—“that whatever possesses any mark possesses that which it is a mark of;” nor indeed any axiom which it is possible to frame; can express the ratiocinative act. Saying nothing of the special objections to be urged against these or kindred propositions, they are all, in so far as they profess to embody laws of logical thinking, open to the fundamental objection that they are substantive truths perceived by reason; not the mode of rational perception. Each of them describes a piece of knowledge; not a process of knowing. Each of them generalizes a large class of cognitions; but does not by so doing approach any nearer to the nature of the cognitive act. Contemplate all the axioms—“Things that are equal to the same thing are equal to each other;” “Things that coexist with the same thing coexist with each other;” and so forth. Each of these is a rational cognition; and if any supposed logical axiom be added to the number, it, also, must be a rational cognition. But these axioms are manifestly of one family; become known by similar intellectual acts; and no addition of a new one to the list can answer the question—What is the common nature of these intellectual acts? what is the process of thought by which axioms become known? Axioms can belong only to the subject-matter about which we reason; and not to reason itself. They imply cases in which an objective uniformity determines a subjective uniformity; and all these subjective uniformities can no more be reduced to one, than the objective ones can. The utmost that any analysis of reason can effect, is to disclose the form of intuition through which these and all other mediately known truths are discerned: and this we have in the inward perception of likeness or unlikeness of relations. This it is which constitutes, as it were, the common type of rational cognitions, axiomatic or other: and it is manifestly incapable of axiomatic expression; not only because it varies with every variation in the subject-matter of thought; but because the universal process of rational intelligence, cannot become solidified into any single product of rational intelligence.
§ 39. And now, that the truth of the several doctrines enunciated in foregoing chapters may be still more clearly seen, let us glance at the series of special results that have been reached; and observe how harmoniously they unite as parts of one consistent whole.
We noticed that perfect quantitative reasoning, by which alone complete previsions are reached, involves intuitions of coextension, coexistence, and connature in the things reasoned about; besides connature in the compared relations, and cointension in the degree of those relations—equality among the entities in Space, Time, Quality; and among their relations in kind and measure: that thus in the highest reasoning, not only does the idea of likeness rise to its greatest perfection (equality), but it appears under the greatest variety of applications; and that in imperfect quantitative reasoning where non-coextension is predicated, either indefinitely (these magnitudes are unequal) or definitely (this magnitude is greater than that), the idea of exact likeness is no longer so variously involved. We next noticed that in perfect qualitative reasoning, the intuition of coextension ceases to appear; but that there is still coexistence and connature amongst the terms, along with connature and cointension amongst the relations subsisting between those terms: that thus there is a further diminution in the number of implied intuitions of equality; and that in parti-perfect qualitative reasoning, where non-coexistence is predicated either indefinitely (these things do not exist at the same time) or definitely (this follows that), the number of such implied intuitions is still further reduced: though there yet remains equality in the natures of the things dealt with, and in the natures of the compared relations. We have now to notice, what was not noticed in passing, that in imperfect qualitative reasoning we descend still lower; for in it, we have no longer complete equality of nature in the terms of the compared relations. Unlike lines, angles, forces, areas, times, &c., the things with which ordinary class reasoning deals, are not altogether homogeneous. The objects grouped together in an induction are never exactly alike in every one of their attributes; nor is the individual thing respecting which a deduction is made, ever quite indistinguishable in character from the things with which it is classed. No two men, or trees, or stones, have the same absolute homogeneity of nature that two circles have. Similarly with the relations between these terms: though they remain connatural, do not remain cointense. And thus, in our contingent every-day reasoning, we have only likeness of nature in the entities and attributes involved; equality of nature in the relations between them; and more or less of likeness in the degree of those relations. The subjects must be like; the things predicated of them must be like; and the relations must be homogeneous, if nothing more. Even when we come to the most imperfect reasoning of all—reasoning by analogy—it is still to be observed that, though the subjects and predicates have severally become so different that not even likeness of nature can be safely asserted of them; there still remains likeness of nature between the compared relations. If the premised relation is a sequence, the inferred one must be a sequence; or they must be both coexistences. If one is a space-relation and the other a time-relation, reasoning becomes impossible. As a weight cannot be compared with a sound; so, neither can there be any comparison between relations of different orders. And hence, whatever else may disappear, the compared relations must continue to be of like nature. Without this there can be no predication of any other likeness or unlikeness; and therefore no reasoning. This fact, that, as we descend from the highest to the lowest kinds of reasoning, the intuitions of likeness among the elements involved, become both less perfect and less numerous, but never wholly disappear, will hereafter be seen to have great significance.
Passing from the elements of the rational intuitions to their forms, we find that these are divisible into two genera: in the one of which the compared relations, having a common term, are conjoined; and in the other of which the compared relations, having no common term, are disjoined. Let us glance at the several species comprehended under the first of these genera. Having necessarily but three terms, these have for their types the forms If, in the first of these forms, A, B, and C represent magnitudes of any order; then, if they are severally equal, we have the axiom—“Things that are equal to the same thing are equal to each other;” and if they are severally unequal, we have a case of mean proportionals. In the second form, if A, B, and C are magnitudes, we have the converse of the above axiom; whilst the thing determined is the inequality of A and C. And in the third form, the thing determined is the superiority or inferiority of A to C. Again, if A, B, and C instead of being magnitudes are times, either at which certain things continuously exist or at which certain events occur, then the first form represents the axioms—“Things that coexist with the same thing coexist with each other,” and “Events which are simultaneous with the same event are simultaneous with each other.” The second form stands for the converse axioms; and predicates the non-coexistence or non-simultaneity of A and C. While the third symbolizes cases in which A is concluded to be before or after C. To make these facts clear, let us formulate each variety.
It must not be supposed, however, that Time and Space relations are the only ones that can enter into these forms. Relations of Force under its various manifestations, may be similarly dealt with. To use Sir William Hamilton's nomenclature, there is Extensive quantity (in Space); Protensive quantity (in Time); and Intensive quantity (in the degree of the Actions that occur in space and time). It is true, as before shown, (§ 25) that intensive quantities, as those of weight, temperature, &c. cannot be accurately reasoned about without reducing them to equivalent quantities of extension; as by the scales and the thermometer: but it is none the less true that there is a simple order of inferences respecting intensive quantities, exactly parallel to those above given. If, for example, a ribbon matched in colour some fabric left at home; and matches some other fabric at the draper's; it is rightly inferred that these fabrics will match each other: or if, on different occasions, a piece of music had its key note pitched by the same tuning fork; it is to be concluded that the pitch was alike on both occasions. And similarly in various other cases, which it is needless to specify. In all of them, as well as in the various ones above given, the intuition, both in its positive and negative forms, is represented by the symbol
The only further fact of importance to be remarked of them, is, that not only are the two relations homogeneous in nature, but all the three terms are so likewise. Whence, in part, arises the extremely-limited range of conjunctive reasonings.∗
The other genus of rational intuitions, distinguished by having four terms, and therefore two separate or disjoined relations, is represented by the typical forms—
To which must be added the two modified forms which result when the reasoning is imperfect—
If, in the first of these five, the letters represent homogeneous magnitudes; then, when A equals B, and C equals D, we have represented the group of axioms—If equals are added to, subtracted from, multiplied by, &c., equals, the results are equal; as well as all the ordinary algebraic reasonings into which these axioms enter: and when each of the two ratios is not one of equality, we have an ordinary proportion. Supposing that the four terms are not homogeneous throughout, but only in pairs; then the formula stands for common geometrical reasoning: and when the things represented are not magnitudes, but simply entities and attributes that are alternately homogeneous; we have that order of reasoning by which necessary coexistences and sequences are recognized. Again, in the second and third forms—if all the terms are homogeneous magnitudes, then inequations and certain axioms antithetical to the above are symbolized: if the magnitudes are but alternately homogeneous, there is typified that imperfect geometrical reasoning by which certain things are proved always greater or less than certain others: and when the letters stand not for magnitudes but simply for entities, properties, or changes, we have that species of necessary qualitative reasoning which gives negative predications. Lastly, by the fourth and fifth forms are signified all orders of common class-reasoning: from that which is next to necessary to that which is in the highest degree problematical: inclusive alike of Induction, Deduction, Analogy, and Hypothesis. All these sub-genera and species of Disjunctive Reasoning are representable by the one symbol—
And the several varieties may be classified in three distinct modes; according as the basis of classification is—(1) the degree of resemblance between the two relations; (2) the nature of the compared relations; and (3) the comparative number of the premised and inferred relations. Under the first of these classifications, we have the divisions—Positive and Negative; Perfect, Parti-perfect, and Imperfect; Necessary and Contingent; Analogical. Under the second, we have the two great divisions—Quantitative and Qualitative: of which the one may be Proportional, Algebraic, or Geometrical, according as the terms of each relation are or are not homogeneous, and are or are not equal; and of which the other may refer to either coexistences or sequences, whether between attributes, things, or events. Under the third, we have reasoning divided into Inductive, Deductive, Hypothetical; which are classifiable according to the numerical ratio between the premised and inferred relations. Thus, if the inference is
The only further fact to be noted respecting the disjunctive form of reasoning, is, that it includes certain inferences which can be classed neither with the inductive, the deductive, the process from particulars to particulars, nor any of their modifications: inferences namely, that are at once drawn, and correctly drawn, in cases that have not been before paralleled in experience. Thus, if A be but a hundredth part less than B; it is at once inferable that a half of A is greater than a third of B. Neither a general principle nor a particular experience, can be quoted as the premiss for this conclusion. It is reached directly and independently by a comparison of the two relations named; and is satisfactorily explicable neither on the hypothesis of forms of thought, nor on the experience-hypothesis as ordinarily interpreted. We may aptly term it a latent inference; and its genesis, like that of many others, is to be properly understood only from that point of view, whence, as already hinted, these antagonist hypotheses are seen to express opposite sides of the same truth. Of this more in the sequel. Meanwhile let it be observed that while the species of reasoning thus exemplified is obviously effected, like all others, by comparison of relations; it cannot be conformed to any of the current theories.
Respecting those most complex forms of reasoning analyzed in the first chapter, which deal not with the quantitative or qualitative relations of things, but with the quantitative relations of quantitative relations; it is needless now to do more than remind the reader that they arise by duplication of the forms above given; and that in their highest complications they follow the same law. Perceiving as he thus will that the doctrine enunciated applies alike to all orders of reasoning, from the most simple to the most complex—from the necessary to the remotely contingent; from the axiomatic to the analogical; from the most premature induction to the most rigorous deduction—he will see that it fulfils the character of a true generalization: that, namely, of explaining all the phenomena.
§ 40. One other group of confirmatory evidences may with advantage be noticed: those which are supplied by our ordinary forms of speech. Already one or two of them have been incidentally pointed out. They are so numerous and so significant, that even standing alone they would go far to establish the theory that has been developed. Thus we have the Latin ratio, meaning reason; and ratiocinor, to reason. This word ratio we apply to each of the two quantitative relations forming a proportion; and the word ratiocination, which is defined as “the act of deducing consequences from premisses,” is applicable alike to numerical and to other inferences. Conversely, the French use raison in the same sense that ratio is used by us. Throughout, therefore, the implication is that reasoning and ratio-ing are fundamentally identical. Further be it remarked that ratiocination, or reasoning, is defined as “the comparison of propositions or facts, and the deduction of inferences from the comparison.” Now every proposition or asserted fact, involving as it does a subject and a something predicated of it, necessarily expresses a relation: hence the definition may be properly transformed into, “the comparison of relations” &c.: and as the only thing effected by comparison is a recognition of the likeness or unlikeness of the compared things; it follows that inferences said to be deduced from the comparison, must result from the recognition of the likeness or unlikeness of relations. Again, we have the word analogy applied alike to proportional reasoning in mathematics, and to the presumptive reasoning of daily life. The meaning of analogy is, “an agreement or likeness between things in some circumstances or effects, when the things are otherwise entirely different:” and in mathematics, an analogy is “an agreement or likeness between” two ratios in respect of the quantitative contrast between each antecedent and its consequent; though their constituent magnitudes are unlike in amount, or in nature, or in both. So that in either case, to “deny the analogy,” is to deny the assumed likeness of relations. Then we have the common expressions—“by parity of reasoning,” and “the cases are not upon a par.” Parity means equality; and being upon a par means being upon a level; so that here, too, the essential idea is that of likeness or unlikeness. Note also, the familiar qualifications,—”cæteris paribus,” “other things equal;” which are used with the implication that when all the remaining elements of the compared cases stand in like relations, the particular elements in question will stand in like relations. Further, there is the notion of parallelism. It is an habitual practice in argument to draw a parallel, with the view of assuming in the one case what is shown in the other. But parallel lines are those that are always equi-distant—that are like in direction: and thus the fundamental idea is still the same. Once more: not only do men reason by similes of all orders, from the parable down to the mere illustration; but similarity is constantly the alleged ground of inference, alike in necessary and in contingent reasoning. When geometrical figures are known to be similar, and the ratio of any two homologous sides is given; the values of all the remaining sides in the one, may be inferred from their known values in the other: and when the lawyer has established his precedent he goes on to argue, that similarly, &c. Now as, in geometry, the definition of similarity is, equality of ratios amongst the answering parts of the compared figures; it is clear that the similarity on the strength of which ordinary inferences are drawn, means—likeness of relations. Various other phrases, such as, “The comparison is not fair;” “What is true in this case will be true in that;” “Like causes will produce like results;” may be mentioned as having the same implication. Nay more: not only is the process of thought by which both our simplest and our most complex inferences are drawn, fundamentally one with that by which proportional inferences are drawn; but its verbal expression often simulates the same form. Just as in mathematics we say—As A is to B, so is C to D; so in non-quantitative reasoning we say—As a muscle is to be strengthened by exercise, so is the rational faculty to be strengthened by thinking. And indeed, this sentence supplies a double illustration; for not only does each of the two inferences it compares exhibit the proportional form; but the comparison itself exhibits that form. Thus it is throughout manifest, that our habitual modes of expression bear witness to the truth of the foregoing analysis.
§ 41. And now, as an appropriate finish to this somewhat too lengthened exposition, I would briefly point out that the conclusion reached may be established even à priori. When towards the close of this Special Analysis we come to consider the ultimate elements of consciousness; it will be abundantly manifest that the phenomena of reasoning cannot, in the nature of things, be truly generalized in any other way. But without waiting for this simplest and most conclusive proof eventually to be arrived at; it may, even from our present stand-point, be demonstrated by two separate methods, that every inference of necessity involves an intuition of the likeness or unlikeness of relations. Already, incidental reference has been made to these à priori arguments; but they claim a more definite statement than they have hitherto received.
Both of them are based immediately upon the very definition of reason, considered under its universal aspect. What is the content of every rational proposition? Invariably a predication—an assertion that something is, was, or will be, conditioned (or not) in a specified manner—that certain objects, forces, attributes, stand to each other thus or thus, in Time or Space. In other words—the content of every rational proposition is, some relation. But what is the condition under which alone a relation is thinkable? It is thinkable only as of a certain order—as belonging, or not belonging, to some class of before-known relations. It must be with relations as with the terms between which they subsist; which can be thought of as such, or such, only by being thought of as members of this or that class. To say—“This is an animal;” or “This is a stone;” or “This is the colour red;” of necessity implies that animals, stones, and colours have been previously presented to consciousness. And the assertion that this is an animal, a stone, or a colour, is, in such case, a grouping of the new object of perception, with the similar objects before perceived. In like manner the inferences—“That berry is poisonous;” “This solution will crystallize;” are impossible even as conceptions, unless a knowledge of the relations between poison and death, between solution and crystallization, have been previously put into the mind; either immediately by experience, or mediately by description. And if a knowledge of such relations pre-exists in the mind, then the predications—“That berry is poisonous;” “This solution will crystallize;” imply that certain new relations are thought of as belonging to certain classes of relations—as being severally of the same order as one or more relations previously known. It follows, then, that contemplated from this point of view, reasoning is a classification of relations. But what does classification mean? It means the grouping together those that are like—the separation of the like from the unlike. Hence, therefore, in inferring any relation we are necessitated to think of it as one (or not one) of some class of relations; and thus to think of it, is to think of it as like or unlike certain other relations. Inference is impossible on any other condition.
Again, passing to the second à priori argument, let us consider what is the more specific definition of reasoning. Not only does the proposition embodied in every inference, assert a relation; but every proposition, whether expressing mediate or immediate knowledge, asserts a relation. In what, then, does the knowing a relation by reason, essentially differ from the knowing it by perception? It differs by its indirectness. Every cognitive act, consisting as it does in the consciousness of a definite relation between two things, (in contradistinction to that indefinite relation which is already known to obtain between them as severally existing in Space and Time), the process of cognition is distinguishable into two separate kinds; according as the relation is disclosed to the mind directly or indirectly. If the two things are so presented that the relation between them is immediately cognized—if their coexistence, or succession, or juxtaposition, is knowable through the senses; we have a perception: but if their coexistence, or sequence, or juxtaposition, is not knowable through the senses—if the relation between them is mediately cognized; we have a ratiocinative act. Reasoning, then, is definable as the indirect establishment of a definite relation between two things. But now the question arises—By what process can the indirect establishment of a definite relation be effected? There is but one answer. If a relation between two things is not directly knowable; it can be disclosed to the mind only through the intermediation of relations that are directly knowable, or are already known. Two mountains not admitting of a side by side comparison, can have their relative heights determined only by reference to some common datum line; as the level of the sea. The relation between a certain distant sound and the blowing of a horn, can be established in consciousness, only by means of a before-perceived relation between such a sound and such an action. Observe, however, that in neither case can any progress be made so long as the relations are separately contemplated. Knowledge of the altitude of each mountain above the sea, will give no knowledge of their relative altitudes, until their two relations to the sea are thought of together, as having a certain relation. The remembrance that a special kind of sound is simultaneous with the blowing of a horn, will be of no service unless this general relation is thought of in connection with the particular relation to be inferred. Hence, then, every ratiocinative act is the establishment of a definite relation between two definite relations.
These two general truths—That reasoning, whether exhibited in a simple inference, or in a long chain of such inferences, is the indirect establishment of a definite relation between two things; and that the achievement of this, is by one or many steps, each of which consists in the establishment of a definite relation between two definite relations; embody, under the most abstract form, the various results arrived at in previous chapters.∗
[∗]I ought here to mention that some year and a half since, in the course of a conversation in which the axiom—“Things that coexist with the same thing coexist with each other,” was referred to; it was remarked by a distinguished lady—the translator of Strauss and Feuerbach—that perhaps a better axiom would be—“Things that have a constant relation to the same thing have a constant relation to each other.” Not having at that time reached the conclusion that a formula having but three terms could not express our ordinary ratiocinations, which involve four; I was greatly inclined to think this the most general truth to which the propositions known by reason are reducible: the more so as, being expressed in terms of relations, it assimilated with many results at which I had already arrived in the course of analyzing the lower intellectual processes. As will appear, however, from the preceding chapters, subsequent inquiry led me to other conclusions. Nevertheless, this suggestion was of much service in directing my thoughts into a track which they might not else have followed. Respecting this axiom itself, it may be remarked that as the word constant, implies time and uniformity, the application of the axiom is limited to necessary time-relations of the conjunctive class. But if, changing the word constant for a more general one, we say—Things which have a definite relation to the same thing have a definite relation to each other; we get an axiom which expresses the most general truth known by conjunctive reasoning—positive and negative, quantitative and qualitative.
[∗]A brief statement of the theory of Reasoning here elaborated in detail, will be found in an essay on “The Genesis of Science,” published in the British Quarterly Review, for July, 1854. In that essay I have sought to show, that scientific progress conforms to the laws of thought disclosed by the foregoing analysis. It contains accumulated illustrations of the fact, that the discoveries of exact science, from the earliest to the latest, severally consist in the establishment of the equality of certain relations whose equality had not been before perceived. That the progress of human reason, as viewed in its concrete results, should throughout exemplify this generalization, as it does in the clearest manner, affords further confirmation of the foregoing analysis: if further confirmation be needed.