ANALYSIS OF MR. MILL'S SYSTEM OF LOGIC.
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ANALYSIS OF MR. MILL'S SYSTEM OF LOGIC.
W. STEBBING, M.A.
FELLOW OF WORCESTER COLLEGE, OXFORD.
LONDON: LONGMANS, GREEN, AND CO. 1867.
PRINTED BY SPOTTISWOODE AND CO.
PREFACE TO THE SECOND EDITION.
The author's aim has been to produce such a condensation of the original work as may recall its contents to those who have read it, and may serve those who are now reading it in the place of a full body of marginal notes. Mr. Mill's conclusions on the true province and method of Logic have a high substantive value, independent even of the arguments and illustrations by which they are supported; and these conclusions may be adequately, and, it is believed, with much practical utility, embodied in an epitome. The processes of reasoning on which they depend, can, on the other hand, be represented in outline only. But it is hoped that the substance of every paragraph, necessary for the due comprehension of the several steps by which the results have been reached, will be here found at all events suggested.
The author may be allowed to add, that Mr. Mill, before publication, expressed a favourable opinion of the manner in which the work had been executed. Without such commendation the volume would hardly have been offered to the public.
LONDON: Dec. 21, 1865.
NAMES AND PROPOSITIONS.
I. On the Necessity of commencing with an Analysis of Language in Logic 3
II. Names 3
III. The Things denoted by Names 7
IV. Propositions 17
V. The Import of Propositions 19
VI. Propositions merely Verbal 24
VII. The Nature of Classification, and the Five Predicables 26
VIII. Definition 30
I. Inference, or Reasoning in General 35
II. Ratiocination, or Syllogism 36
III. The Functions and Logical Value of the Syllogism 39
IV. Trains of Reasoning, and Deductive Sciences 43
V. & VI. Demonstration and Necessary Truths 46
I. Preliminary Observations on Induction in general 53
II. Inductions improperly so called 54
III. The ground of Induction 57
IV. Laws of Nature 58
V. The Law of Universal Causation 60
VI. The Composition of Causes 66
VII. Observation and Experiment 67
VIII. & Note to IX. The Four Methods of Experimental Enquiry 69
X. Plurality of Causes, and intermixture of Effects 73
XI. The Deductive Method 76
XII. & XIII. The Explanation and Examples of the Explanation of Laws of Nature 77
XIV. The Limits to the Explanation of Laws of Nature; and Hypotheses 79
XV. Progressive Effects, and continued Action of Causes 81
XVI. Empirical Laws 83
XVII. Chance, and its Elimination 85
XVIII. The Calculation of Chances 87
XIX. The Extension of Derivative Laws to Adjacent Cases 89
XX. Analogy 91
XXI. The Evidence of the Law of Universal Causation 92
XXII. Uniformities of Coexistence not dependent on Causation 94
XXIII. Approximate Generalisations, and Probable Evidence 96
XXIV. The remaining Laws of Nature 99
XXV. The grounds of Disbelief 103
OPERATIONS SUBSIDIARY TO INDUCTION.
I. Observation and Description 107
II. Abstraction, or the Formation of Conceptions 108
III. Naming as Subsidiary to Induction 111
IV. The Requisites of a Philosophical Language, and the Principles of Definition 112
V. The Natural History of the Variation in the Meaning of Terms 115
VI. Terminology and Nomenclature 117
VII. Classification, as Subsidiary to Induction 121
VIII. Classification by Series 124
I. Fallacies in general 127
II. Classification of Fallacies 128
III. Fallacies of Simple Inspection; or, a priori Fallacies 130
IV. Fallacies of Observation 134
V. Fallacies of Generalisation 137
VI. Fallacies of Ratiocination 141
VII. Fallacies of Confusion 143
ON THE LOGIC OF THE MORAL SCIENCES.
I. Introductory Remarks 148
II. Liberty and Necessity 148
III. There is, or may be, a Science of Human Nature 150
IV. The Laws of Mind 151
V. Ethology, or the Science of the Formation of Character 153
VI. General Considerations on the Social Science 155
VII. The Chemical, or Experimental, Method in the Social Science 156
VIII. The Geometrical, or Abstract Method 157
IX. The Physical, or Concrete Deductive Method 158
X. The Inverse Deductive, or Historical Method 161
XI. The Logic of Practice, or Art; including Morality and Policy 165
ANALYSIS OF MILL'S LOGIC.
No adequate definition is possible till the properties of the thing to be defined are known. Previously we can define only the scope of the inquiry. Now, Logic has been considered as both the science of reasoning, i.e. the analysis of the mental process when we reason, and the art of reasoning, i.e. the rules for the process. The term reasoning, however, is not wide enough. Reasoning means either syllogising, or (and this is its truer sense) the drawing inferences from assertions already admitted. But the Aristotelian or Scholastic logicians included in Logic terms and propositions, and the Port Royal logicians spoke of it as equivalent to the art of thinking. Even popularly, accuracy of classification, and the extent of command over premisses, are thought clearer signs of logical powers than accuracy of deduction. On the other hand, the definition of logic as a 'science treating of the operations of the understanding in the search of truth,' though wide enough, would err through including truths known from intuition; for, though doubtless many seeming intuitions are processes of inference, questions as to what facts are real intuitions belong to Metaphysics, not to Logic.
Logic is the science, not of Belief, but of Proof, or Evidence. Almost all knowledge being matter of inference, the fields of Logic and of Knowledge coincide; but the two differ in so far that Logic does not find evidence, but only judges of it. All science is composed of data, and conclusions thence: Logic shows what relations must subsist between them. All inferential knowledge is true or not, according as the laws of Logic have been obeyed or not. Logic is Bacon's Ars Artium, the science of sciences. Genius sometimes employs laws unconsciously; but only genius: as a rule, the advances of a science have been ever found to be preceded by a fuller knowledge of the laws of Logic applicable to it. Logic, then, may be described as the science of the operations of the understanding which aid in the estimation of evidence. It includes not only the process of proceeding from the known to the unknown, but, as auxiliary thereto, Naming, Definition, and Classification. Conception, Memory, and other like faculties, are not treated by it; but it presupposes them. Our object, therefore, must be to analyse the process of inference and the subsidiary operations, besides framing canons to test any given evidence. We need not, however, carry the analysis beyond what is necessary for the practical uses of Logic; for one step in analysis is good without a second, and our purpose is simply to see the difference between good and ill processes of inference. Minuter analysis befits Metaphysics; though even that science, when stepping beyond the interrogation of our consciousness, or rather of our memory, is, as all other sciences, amenable to Logic.
NAMES AND PROPOSITIONS.
ON THE NECESSITY OF COMMENCING WITH AN ANALYSIS OF LANGUAGE IN LOGIC.
The fact of Logic being a portion of the art of thinking, and of thought's chief instrument being words, is one reason why we must first inquire into the right use of words. But further, the import of propositions cannot really be examined apart from that of words; and (since whatever can be an object of belief assumes the form of a proposition, and in propositions all truth and error lie) this is a paramount reason why we must, as a preliminary, consider the import of names, the neglecting which, and confining ourselves to things, would indeed be to discard all past experience. The right method is, to take men's classifications of things as shown by names, correcting them as we proceed.
Hobbes's assertion that a name is a sign, not of a thing, but of our conception of it, is untrue (unless he merely mean that the conception, and not the thing itself, is imparted to the hearer); for we intend by a name, not only to make men conceive what we conceive, but to inform them what we believe as to the things themselves.
Names may be divided according to five principles of classification. The first way of dividing them is into General (not as equivalent to Collective) and Individual names; the second, into Concrete, i.e. the names of objects, and Abstract, i.e. the names of attributes (though Locke improperly extends the term to all names gained by abstraction, that is, to all general names). An abstract name is sometimes general, e.g. colour, and sometimes singular, e.g. milk-whiteness. It may be objected to calling attributes abstract, that also concrete adjectives, e.g. white, are attributes. But a word is the name of the things of which it can be predicated. Hence, white is the name of all things so coloured, given indeed because of the quality, but really the name of the thing, and no more the name of the quality than are names generally, since every one of them, if it signifies anything at all, must imply an attribute.
The third division is into Connotative and Non-connotative (the latter being wrongly called Absolute). By connotative are meant, not (as Mr. James Mill explains it) words which, pointing directly to one thing, tacitly refer to another, but words which denote a subject and imply an attribute; while non-connotatives signify a subject only, or attribute only. All concrete general names are connotative. They are also called denominative, because the subject denoted receives a common name (e.g. snow is named white) from the attribute connoted. Even some abstracts are connotative, for attributes may have attributes ascribed to them, and a word which denotes attributes may connote an attribute of them; e.g. fault connotes hurtfulness. Proper names, on the other hand, though concrete, are not connotative. They are merely distinguishing marks, given perhaps originally for a reason, but, when once given, independent of it, since the reason is proved to be no part of the sense of the word by the fact that the name is still used when the reason is forgotten. But other individual names are connotative. Some of these, viz. those connoting some attribute or some set of attributes possessed by one object only, e.g. Sun, God, are really general names, though happening to be predicable only of a single object. But there are also real connotative individual names, part of whose meaning is, that there exists only one individual with the connoted attribute, e.g. The first Emperor, The father of Socrates; and it is so with many-worded names, made up of a general name limited by other words, e.g. The present Prime Minister of England. In short, the meaning of all names, which have any meaning, resides, not in what they denote, but in what they connote. There perpetually, however, arises a difficulty of deciding how much they do connote, that is, what difference in the object would make a difference in the name. This vagueness comes from our learning the connotation, through a rude generalisation and analysis, from the objects denoted. Thus, men use a name without any precise reference to a definite set of attributes, applying it to new objects on account of superficial resemblance, so that at length all common meaning disappears. Even scientific writers, from ignorance, or from the aversion which men at large feel to the use of new names, often force old terms to express an ever-growing number of distinctions. But every concrete general name should be given a definite connotation with the least possible change in the denotation; and this is what is aimed at in every definition of a general name already in use. But we must not confound the use of names of indeterminate connotation, which is so great an evil, with the employment, necessitated by the paucity of names as compared with the demand, of the same words with different connotations in different relations.
A fourth division of names is into Positive and Negative. When the positive is connotative, so is the corresponding negative, for the non-possession of an attribute is itself an attribute. Names negative in form, e.g. unpleasant, are often really positive; and others, e.g. idle, sober, though seemingly positive, are really negative. Privatives are names which are equivalent each to a positive and a negative name taken together. They connote both the absence of certain attributes, and the presence of others, whence the presence of the defaulting ones might have been expected. Thus, blind would be applied only to a non-seeing member of a seeing class.
The fifth division is into Relative and (that we may economise the term Absolute for an occasion when none other is available) Non-relative names. Correlatives, when concrete, are of course connotative. A relation arises from two individuals being concerned in the same series of facts, so that the signification of neither name can be explained except by mentioning another: and any two correlatives connote, not the same attribute indeed, but just this series of facts, which is exactly the same in both cases.
Some make a sixth division, viz. Univocals, i.e. names predicated of different individuals in the same sense, and AEquivocals, i.e. names predicated of different individuals in different senses. But these are not two kinds of names, but only two modes of using them; for an aequivocal name is two names accidentally coinciding in sound. An intermediate case is that of a name used analogically or metaphorically, that is, in two senses, one its primary, the other its secondary sense. The not perceiving that such a word is really two has produced many fallacies.
THE THINGS DENOTED BY NAMES.
Logic is the theory of Proof, and everything provable can be exhibited as a proposition, propositions alone being objects of belief. Therefore, the import of propositions, that is, the import of predication, must be ascertained. But, as to make a proposition, i.e. to predicate, is to assert one thing of another thing, the way to learn the import of predication is, by discovering what are the things signified by names which are capable of being subject or predicate. It was with this object that Aristotle formed his Categories, i.e. an attempted enumeration of all nameable things by the summa genera or highest predicates, one or other of which must, he asserted, be predicable of everything. His, however, is a rude catalogue, without philosophical analysis of the rationale even of familiar distinctions. For instance, his Relation properly includes Action, Passivity, and Local Situation, and also the two categories of Position [Greek: pote] and [Greek: pou], while the difference between [Greek: pou] and [Greek: keisthai] is only verbal, and [Greek: echein] is not a summum genus at all. Besides—only substantives and attributes being there considered—there is no category for sensation and other mental states, since, though these may rightly be placed, so far as they express their relation, if active, to their objects, if passive to their causes, in the Categories of Actio and Passio, the things, viz., the mental states, do not belong there.
The absence of a well-defined concrete name answering to the abstract existence, is one great obstacle to renewing Aristotle's attempt. The words used for the purpose commonly denote substances only, though attributes and feelings are equally existences. Even being is inadequate, since it denotes only some existences, being used by custom as synonymous with substance, both material and spiritual. That is, it is applied to what excites feelings and has attributes, but not to feelings and attributes themselves; and if we called extension, virtue, &c., beings, we should be accused of believing in the Platonic self-existing ideas, or Epicurus's sensible forms—in short, of deeming attributes substances. To fill this gap, the abstract, entity, was made into a concrete, equivalent to being. Yet even entity implies, though not so much as being, the notion of substance. In fact, every word originally connoting simply existence, gradually enlarges its connotation to mean separate existence, i.e. existence freed from the condition of belonging to a substance, so as to exclude attributes and feelings. Since, then, all the terms are ambiguous, that among them (and the same principle applies to terms generally) will be employed here which seems on each occasion to be least ambiguous: and terms will be used even in improper senses, when these by familiar association convey the proper meaning.
Nameable things are—I. Feelings or States of Consciousness.—A feeling, being anything of which the mind is conscious, is synonymous with state of consciousness. It is commonly confined to the sensations and emotions, or to the emotions alone; but it is properly a genus, having for species, Sensation, Emotion, Thought, and Volition. By thought is meant all that we are internally conscious of when we think; e.g. the idea of the sun, and not the sun itself, is a thought; and so, not even an imaginary thing like a ghost, but only the idea of it, is a thought. In like manner, a sensation differs both from the object causing it, and the attribute ascribed to the object. Yet language (except in the case of the sensations of hearing) has seldom provided the sensations with separate names; so that we have to name the sensation from the object or the attribute exciting it, though we might conceive the sensation to exist, though it never actually does, without an exciting cause. Again, another distinction has to be attended to, viz. the difference between the sensation and the state of the bodily organs, which is the physical agency producing it. This distinction escapes notice partly by reason of the division of the feelings into bodily and mental. But really there is no such division, even sensations being states of the sentient mind, and not of the body. The difference, in fact, between sensations, thoughts, and emotions, is only in the different agency producing the feeling; it being, in the case of the sensations, a bodily, and, for the other two, a mental state. Some suppose, after the sensation, in which, they say, the mind is passive, a distinct active process called perception, which is the direct recognition of an external object, as the cause of the sensation. Probably, perceptions are simply cases of belief claiming to be intuitive, i.e. free of external evidence. But, at any rate, any question as to their nature is irrelevant to an inquiry like the present, viz. how we get the non-original part of our knowledge. And so also is the distinction in German metaphysics, between the mind's acts and its passive states. Enough for us now that they are all states of the mind.
II. Substances.—Logicians think they have defined substance and attribute, when they have shown merely what difference the use of them respectively makes in the grammar of a sentence. They say an attribute must be an attribute of something, but that a substance is self-existent (being followed, if a relative, by of, not qua substance, but qua the relation). But this of, as distinguishing attributes, itself needs explanation: besides, we can no more conceive a substance independent of attributes, than an attribute independent of a substance. Metaphysicians go deeper into the distinction than logicians. Substances, most of them say, are either bodies or minds; and, of these, a body is the external cause to which we ascribe sensations. Berkeley and the Idealists, however, deny that there exists any cause of sensations (except, indeed, a First Cause). They argue that the whole of our notion of a body consists of a number of our own or others' sensations occurring together habitually (so that, the thought of one being associated with the thought of the others, we get what Hartley and Locke call a complex idea). They deny that a residuum would remain if all the attributes were pared off; for that, though the sensations are bound together by a law, the existence of a substratum is but one of many forms of mentally realising the connection. And they ask how it is,—since so long as the sensations occurred in the old order, we should not miss such a substratum, supposing it to have once existed and to have perished—that we can know it exists even now? Their opponents used formerly to reply, that the uniform order of sensations implies an external cause determining the law of the order; and that the attributes inhere in this external cause or substratum, viz. matter. But at last it was seen that the existence of matter could not be proved by extrinsic evidence; consequently, now the answer to the idealist argument simply is, that the belief in an external cause of sensations is universal, and as intuitive as our knowledge of sensations themselves. Even Kant allows this (notwithstanding his belief in the existence of a universe of things in themselves, i.e. Nouemena, as contrasted with the mental representation of them, where the sensations, he thinks, furnish the matter, and the laws of the mind, the form). Brown even traced up to the sensations of touch, combined with the sensations seated in the muscular frame, those very properties, viz., extension and figure, which Reid referred to as proving that some qualities must exist, not in the sensations, but in the things themselves, since they cannot possibly be copies of any impression on the senses. We have, in truth, no right to consider a thing's sensible qualities akin to its nature, unless we suppose an absurdity, viz. that a cause must, as such, resemble its effects. In any case, the question whether Ontology be a possible science, concerns, not Logic, but the nature and laws of intuitive knowledge. And the question as to the nature of Mind is as out of place here as that about Body. As body is the unknown exciting cause of sensations, so mind, the other kind of substance, is the unknown recipient both of the sensations and of all the other feelings. Though I call a something myself, as distinct from the series of feelings, the 'thread of consciousness,' yet this self shows itself only through its capacity of feeling or being conscious; and I can, with my present faculties, conceive the gaining no new information but about as yet unknown faculties of feeling. In short, as body is the unsentient cause of all feelings, so mind is the sentient subject (in the German sense) of them, viz. that which feels them. About this inner nature we know nothing, and Logic cares nothing.
III. Attributes.—Qualities are the first class of attributes. Now, if we know nothing about bodies but the sensations they excite, we can mean nothing by the attributes of bodies but sensations. Against this it has been urged that, though we know nothing of sensible objects except the sensations, the quality which we ascribe on the ground of the sensation may yet be a real hidden power or quality in the object, of which the sensation is only the evidence. Seemingly, this doctrine arises only from the tendency to suppose that there must be two different things to answer to two names when not quite synonymous. Quality and sensation are probably names for the same thing viewed in different lights. The doctrine of an entity per se, called quality, is a relic of the scholastic occult causes; the only intelligible cause of sensation being the presence of the assemblage of phenomena, called the object. Why the presence of the object causes the sensation, we know not; and, granting an occult cause, we are still in the dark as to how that produces the effect. However, the question belongs to metaphysics; and it suits this doctrine, as well as the opposed one, to say that a quality has for its foundation a sensation.
Relations form the second class of attributes. In all cases of relation there exists some fact into which the relatives enter as parties concerned; and this is the fundamentum relationis. Whenever two things are involved in some one fact, we may ascribe to them a relation grounded on it, however general the fact may be. As, then, a quality is an attribute based on the fact of a sensation, so a relation is an attribute based on a fact into which two objects enter jointly. This fact in both is always composed entirely of states of consciousness; and this, whether it be complicated, as in many legal relations, or simple, as in the relations expressed by antecedent and consequent and by simultaneous, where the fact consists merely of the two things so related, since the consciousness either of the succession or of the simultaneousness of the two sensations which represent the things, is a feeling not added to, but involved in them, being a condition under which we must suppose things. And so, likewise, with the relations of likeness and unlikeness. The feeling of these sometimes cannot be analysed, when the fundamentum relationis is, as in the case of two simple sensations, e.g. two sensations of white, only the two sensations themselves, the consequent feeling of their resemblance being, like that of their succession or simultaneousness, apparently involved in the sensations themselves. Sometimes, again, the likeness or unlikeness is complex, and therefore can be analysed into simpler cases. In any case, likeness or unlikeness must resolve itself into likeness or unlikeness between states of our own or some other mind; and this, whether the feeling of the resemblance or dissimilarity relate to bodies or to attributes, since the former we know only through the sensations they are supposed to excite, and the latter through the sensations on which they are grounded. And so, again, when we say that two relations are alike (one of the many senses of analogy), we simply assert resemblance between the facts constituting the two fundamenta relationis. Several relations, called by different names, are really cases of resemblance. Thus, equality, i.e. the exact resemblance existing between things in respect of their quantity, is often called identity.
The third species of attributes is Quantity. The assertion of likeness or unlikeness in quantity, as in quality, is always founded on a likeness or unlikeness in the sensations excited. What the difference is all who have had the sensations know, but it cannot be explained to those who never had them.
In fine, all the attributes classed under Quality and Quantity are the powers bodies have of exciting certain sensations. So, Relation generally is but the power which an object has of joining its correlative in producing the series of sensations, which is the only sign of the existence of the fact on which they both are grounded. The relations of succession and simultaneousness, indeed, are not based on any fact (i.e. any feeling) distinct from the related objects. But these relations are themselves states of consciousness; resemblance, for example, being nothing but our feeling of resemblance: at least, we ascribe these relations to objects or attributes simply because they hold between the feelings which the objects excite and on which the attributes are grounded. And as with the attributes of bodies, so also those of minds are grounded on states of consciousness. Considered in itself, we can predicate of a mind only the series of its own feelings: e.g. by devout we mean that the feelings implied in that word form an oft-recurring part of the series of feelings filling up the sentient existence of that mind. Again, attributes may be ascribed to a mind as to a body, as grounded on the thoughts or emotions (not the sensations, for only bodies excite them) which it excites in others: e.g. when we call a character admirable, we mean that it causes feelings in us of admiration. Sometimes, under one word really two attributes are predicated, one a state of the mind, the other of other minds affected by thinking of it: e.g. He is generous. Sometimes, even bodies have the attribute of producing an emotion: e.g. That statue is beautiful.
The general result is, that there are three chief kinds of nameable things:—1. Feelings distinct from the objects exciting and the organs supposed to convey them, and divisible into four classes, perceptions being only a particular case of belief, which is itself a sort of thought, while actions are only volitions followed by an effect. 2. Substances, i.e. the unknown cause and the unknown recipient of our sensations. 3. Attributes, subdivisible into Quality, Relation, Quantity. Of these ([Greek: a]) qualities, like substances, are known only by the states of consciousness which they excite, and on which they are based, and by which alone, though they are treated as a distinct class, they can be described. ([Greek: b]) Relations also, with four exceptions, are based on some fact, i.e. a series of states of consciousness. ([Greek: g]) Quantity is, in the same way, based on our sensations. In short, all attributes are only our sensations and other feelings, or something involved in them. We may, then, classify nameable things thus:—1, Feelings; 2, Minds; 3, Bodies, together with the properties whereby they are popularly (though the evidence is very deficient) supposed to excite sensations; 4, the relations of Succession and Coexistence, Likeness and Unlikeness, which subsist really only between states of consciousness.
These four classes are a substitute for Aristotle's abortive Categories. As they comprise all nameable things, every fact is made up of them or some of them; those that are called subjective facts being composed wholly of feelings as such, and the objective facts, though composed wholly or partly of substances and attributes, being grounded on corresponding subjective facts.
The copula is a mere sign of predication, though it is often confounded with to be, the verb of existence (and that not merely by Greeks, but even by moderns, whose larger experience how one word in one language often answers to several in another, should have saved them from thinking that things with a common name must have a common nature). The first division of propositions is into Affirmative and Negative, the copula in the latter being is not. Hobbes and others, by joining the not to the predicate, made the latter what they call a negative name. But as a negative name is one expressing the absence of an attribute, we thus in fact merely deny its presence, and therefore the affirmative guise these thinkers give to negative propositions is only a fiction. Again, modal propositions cannot be reduced to the common form by joining the modality to the predicate, and turning, e.g. The sun did rise, into, The sun is a thing having risen; for the past time is not a particular kind of rising, and it affects not the predicate, but the predication, i.e. the applicability of the predicate to the subject. There are, however, certain cases in which the qualification may be detached from the copula; e.g. in such expressions as, may be, is perhaps; for, then we really do not mean to assert anything about the fact, but only about the state of our mind about it, so that it is not the predication which is affected: e.g. Caesar may be dead, may properly be rendered, I am not sure that he is alive.
The second division is into Simple and Complex. Several propositions joined by a conjunction do not make a complex proposition. The conjunction, so far from making the two one, adds another, as being an abbreviation generally of an additional proposition: e.g. and is an abbreviation of one additional proposition, viz. We must think of the two together; while but is an abbreviation of two additional propositions, viz. We must think of them together, and we must recollect there is a contrast between them. But hypothetical propositions, i.e. both disjunctives and conditionals, are true complex propositions, since with several terms they contain but a single assertion. Thus, in, If the Koran comes from God, Mahomet is God's prophet, we do not assert the truth of either of the simple propositions therein contained (viz. the Koran comes from God, and Mahomet is God's prophet), but only the inferribility of one from the other. The only difference, then, between a hypothetical and a categorical proposition, is that the former is always an assertion about an assertion (though some categoricals are so likewise; e.g. That the whole is greater than its parts, is an axiom). Their conspicuous place in treatises on Logic arises from this attribute which they predicate of a proposition (for a proposition, like other things, has attributes), viz. its being an inference from something else, being, with reference to Logic, its chief attribute.
The third common division is into Universal, Particular, Indefinite, and Singular. A proposition whose subject is an individual name, even if not a proper name, is singular, e.g. The founder of Rome was killed. In particular propositions, if the part of the class meant by the some were specified, the proposition would become either singular, or universal with a different subject including all the part. Indefinite in Logic is a solecism like doubtful gender in grammar, for the speaker must mean to make either a particular or a universal assertion.
THE IMPORT OF PROPOSITIONS.
The object of an inquiry into the nature of propositions must be to analyse, either, 1, the state of mind called belief, or 2, what is believed. Philosophers have usually, but wrongly, thought the former, i.e. an analysis of the act of judgment, the chief duty of Logic, considering a proposition to consist in the denying or affirming one idea of another. True, we must have the two ideas in the mind together, in order to believe the assertion about the two things; but so we must also in order to disbelieve it. True also, that besides the putting the ideas together, there may be a mental process; but this has nothing to do with the import of propositions, since they are assertions about things, i.e. facts of external nature, not about the ideas of them, i.e. facts in our mental history. Logic has suffered from stress being laid on the relation between the ideas rather than the phenomena, nature thus coming to be studied by logicians second-hand, that is to say, as represented in our minds. Our present object, therefore, must be to investigate judgments, not judgment, and to inquire what it is which we assert when we make a proposition.
Hobbes (though he certainly often shows his belief that all propositions are not merely about the meaning of words, and that general names are given to things on account of their attributes) declares that what we assert, is our belief that the subject and predicate are names of the same thing. This is, indeed, a property of all true propositions, and the only one true of all. But it is not the scientific definition of propositions; for though the mere collocation which makes a proposition a proposition, signifies only this, yet that form, combined with other matter, conveys much more meaning. Hobbes's principle accounts fully only for propositions where both terms are proper names. He applied it to others, through attending, like all nominalists, to the denotation, and not the connotation of words, holding them to be, like proper names, mere marks put upon individuals. But when saying that, e.g. Socrates is wise, is a true proposition, because of the conformity of import between the terms, he should have asked himself why Socrates and wise are names of the same person. He ought to have seen that they are given to the same person, not because of the intention of the maker of each word, but from the resemblance of their connotation, since a word means properly certain attributes, and, only secondarily, objects denoted by it. What we really assert, therefore, in a proposition, is, that where we find certain attributes, we shall find a certain other one, which is a question not of the meaning of names, but of the laws of nature.
Another theory virtually identical with Hobbes's, is that commonly received, which makes predication consist in referring things to a class; that is (since a class is only an indefinite number of individuals denoted by a general name), in viewing them as some of those to be called by that general name. This view is the basis of the dictum de omni et nullo, on which is supposed to rest the validity of all reasoning. Such a theory is an example of [Greek: hysteron proteron]: it explains the cause by the effect, since the predicate cannot be known for a class name which includes the subject, till several propositions having it for predicate have been first assented to. This doctrine seems to suppose all individuals to have been made into parcels, with the common name outside; so that, to know if a general name can be predicated correctly of the subject, we need only search the roll so entitled. But the truth is, that general names are marks put, not upon definite objects, but upon collections of objects ever fluctuating. We may frame a class without knowing a single individual belonging to it: the individual is placed in the class because the proposition is true; the proposition is not made true by the individual being placed there.
Analysis of different propositions shows what is the real import of propositions not simply verbal. Thus, we find that even a proposition with a proper name for subject, means to assert that an individual thing has the attributes connoted by the predicate, the name being thought of only as means for giving information of a physical fact. This is still more the case in propositions with connotative subjects. In these the denoted objects are indicated by some of their attributes, and the assertion really is, that the predicate's set of attributes constantly accompanies the subject's set. But as every attribute is grounded on some fact or phenomenon, a proposition, when asserting the attendance of one or some attributes on others, really asserts simply the attendance of one phenomenon on another; e.g. When we say Man is mortal, we mean that where certain physical and moral facts called humanity are found, there also will be found the physical and moral facts called death. But analysis shows that propositions assert other things besides (although this is indeed their ordinary import) this coexistence or sequence of two phenomena, viz. two states of consciousness. Assertions in propositions about those unknowable entities (nouemena) which are the hidden causes of phenomena, are made, indeed, only in virtue of the knowable phenomena. Still, such propositions do, besides asserting the sequence or coexistence of the phenomena, assert further the existence of the nouemena; and, moreover, in affirming the existence of a nouemenon, which is an unknowable cause, they assert causation also. Lastly, propositions sometimes assert resemblance between two phenomena. It is not true that, as some contend, every proposition whose predicate is a general name affirms resemblance to the other members of the class; for such propositions generally assert only the possession by the subject of certain common peculiarities; and the assertion would be true though there were no members of the class besides those denoted by the subject. Nevertheless, resemblance alone is sometimes predicated. Thus, when individuals are put into a class as belonging to it, not absolutely, but rather than to any other, the assertion is, not that they have the attributes connoted, but that they resemble those having them more than they do other objects. So, again, only resemblance is predicated, when, though the predicate is a class name, the class is based on general unanalysable resemblance. The classes in question are those of the simple feelings; the names of feelings being, like all concrete general names, connotative, but only of a mere resemblance.
In short, one of five things, viz. Existence, Coexistence (or, to be more particular, Order in Place), Sequence (or, more particularly, Order in Time, which comprises also the mere fact of Coexistence), Causation, and Resemblance, is asserted or denied in every proposition. This division is an exhaustive classification with respect to all things that can be believed. Although only propositions with concrete terms have been spoken of, it is equally the fact that, in propositions with an abstract term or terms, we predicate one of these same five things. There cannot be any difference in the import of these two classes of propositions, since there is none in the import of their terms, for the real signification of a concrete term resides in its connotation (so that in a concrete proposition we really predicate an attribute), and what the concrete term connotes forms the whole sense of the abstract. Thus, all propositions with abstract terms can be turned into equivalent ones with concrete, the new terms being either the names which connote the attributes, or names of the facts which are the fundamenta of the attributes: e.g. Thoughtlessness is danger, is equivalent to, Thoughtless actions (the fundamentum) are dangerous.
Finally, as these five are the only things affirmable, so are they the only things deniable.
PROPOSITIONS MERELY VERBAL.
The object of Logic is to find how propositions are to be proved. As preliminary to this, it has been already shown that the Conceptualist view of propositions, viz. that they assert a relation between two ideas, and the Nominalist, that they assert agreement or disagreement between the meanings of two names, are both wrong as general theories: for that generally the import of propositions is, to affirm or deny respecting a phenomenon, or its hidden source, one of five kinds of facts. There is, however, a class of propositions which relate not to matter of fact, but to the meaning of names, and which, therefore, as names and their meanings are arbitrary, admit not of truth or falsity, but only of agreement or disagreement with usage. These verbal propositions are not only those in which both terms are proper names, but also some, viz. essential propositions, thought to be more closely related to things than any others. The Aristotelians' belief that objects are made what they are called by the inherence of a certain general substance in the individuals which get from it all their essential properties, prevented even Porphyry (though more reasonable than the mediaeval Realists) from seeing that the only difference between altering a non-essential (or accidental) property, which, he says, makes the thing [Greek: alloion], and altering an essential one, which makes it [Greek: allo] (i.e. a different thing), is, that the latter change makes the object change its name. But even when it was no longer believed that there are real entities answering to general terms, the doctrine based upon it, viz. that a thing's essence is that without which the thing could neither be, nor be conceived to be, was still generally held, till Locke convinced most thinkers that the supposed essences of classes are simply the significations of their names. Yet even Locke supposed that, though the essences of classes are nominal, individuals have real essences, which, though unknown, are the causes of their sensible properties.
An accidental proposition (i.e. in which a property not connoted by the subject is predicated of it) tacitly asserts the existence of a thing corresponding to the subject; otherwise, such a proposition, as it does not explain the name, would assert nothing at all. But an essential proposition (i.e. in which a property connoted by the subject is predicated of it) is identical. The only use of such propositions is to define words by unfolding the meaning involved in a name. When, as in mathematics, important consequences seem to follow from them, such really follow from the tacit assumption, through the ambiguity of the copula, of the real existence of the object named.
Accidental propositions include, 1, those with a proper name for subject, since an individual has no essence (although the schoolmen, and rightly, according to their view of genera and species as entities inhering in the individuals, attributed to the individual the essence of his class); and, 2, all general or particular propositions in which the predicate connotes any attribute not connoted by the subject. Accidental propositions may be called real; they add to our knowledge. Their import may be expressed (according as the attention is directed mainly, either to what the proposition means, or to the way in which it is to be used), either, by the formula: The attributes of the subject are always (or never) accompanied by those signified by the predicate; or, by the formula: The attributes of the subject are evidence, or a mark, of the presence of those of the predicate. For the purposes of reasoning, since propositions enter into that, not as ultimate results, but as means for establishing other propositions, the latter formula is preferable.
THE NATURE OF CLASSIFICATION, AND THE FIVE PREDICABLES.
It is merely an accident when general names are names of classes of real objects: e.g. The unity of God, in the Christian sense, and the non-existence of the things called dragons, do not prevent those names being general names. The using a name to connote attributes, turns the things, whether real or imaginary, into a class. But, in predicating the name, we predicate only the attributes; and even when a name (as, e.g. those in Cuvier's system) is introduced as a means of grouping certain objects together, and not, as usually, as a means of predication, it still signifies nothing but the possession of certain attributes.
Classification (as resulting from the use of general language) is the subject of the Aristotelians' Five Predicables, viz. Genus, Species, Differentia, Proprium, Accidens. These are a division of general names, not based on a distinction in their meaning, i.e. in the attributes connoted, but on a distinction in the class denoted. They express, not the meaning of the predicate itself, but its relation (a varying one) to the subject. Commonly, the names of any two classes (or, popularly, the classes themselves), one of which includes all the other and more, are called respectively genus and species. But the Aristotelians, i.e. the schoolmen, meant by differences in kind (genere or specie) something which was in its nature (and not merely with reference to the connotation of the name) distinct from differences in the accidents. Now, it is the fact that, though a fresh class may be founded on the smallest distinction in attributes, yet that some classes have, to separate them from other classes, no common attributes except those connoted by the name, while others have innumerable common qualities (from which we have to select a few samples for connotation) not referrible to a common source. The ends of language and of classification would be subverted if the latter (not if the former) sorts of difference were disregarded. Now, it was these only that the Aristotelians called kinds (genera or species), holding differences made up of certain and definite properties to be differences in the accidents of things. In conformity with this distinction—and it is a true one—any class, e.g. negro as opposed to white man, may, according as physiology shall show the differences to be infinite or finite, be discovered to be a distinct kind or species (though not according to the naturalist's construction of species, as including all descended from the same stock), or merely a subdivision of the kind or species, Man. Among kinds, a genus is a class divisible into other kinds, though it may be itself a species in reference to higher genera; that which is not so divisible, is an individual's proximate kind or infima species (species praedicabilis and also subjicibilis), whose common properties must include all the common properties of every other real kind to which the individual can be referred.
The Aristotelians said that the differentia must be of the essence of the subject. They vaguely understood, indeed, by the essence of a thing, that which makes it the kind of thing that it is. But, as a kind is such from innumerable qualities not flowing from a common source, logicians selected the qualities which make the thing be what it is called, and termed these the essence, not merely of the species, but, in the case of the infima species, of the individual also. Hence, the distinction between the predicables, Differentia, Proprium, and Accidens, is founded, not on the nature of things, but on the connotation of names. The specific difference is that which must be added to the connotation of the genus to complete the connotation of the species. A species may have various differences, according to the principle of the particular classification. A kind, and not merely a class, may be founded on any one of these, if there be a host of properties behind, of which this one is the index, and not the source. Sometimes a name has a technical as well as an ordinary connotation (e.g. the name Man, in the Linnaean system, connotes a certain number of incisor and canine teeth, instead of its usual connotation of rationality and a certain general form); and then the word is in fact ambiguous, i.e. two names. Genus and Differentia are said to be of the essence; that is, the properties signified by them are connoted by the name denoting the species. But both proprium and accidens are said to be predicated of the species accidentally. A proprium of the species, however, is predicated of the species necessarily being an attribute, not indeed connoted by the name, but following from an attribute connoted by it. It follows, either by way of demonstration as a conclusion from premisses, or by way of causation as effect from cause; but, in either case, necessarily. Inseparable accidents, on the other hand, are attributes universal, so far as we know, to the species (e.g. blackness to crows), but not necessary; i.e. neither involved in the meaning of the name of the species, nor following from attributes which are. Separable accidents do not belong to all, or if to all, not at all times (e.g. the fact of being born, to man), and sometimes are not constant even in the same individual (e.g. to be hot or cold).
A definition is a proposition declaring either the special or the ordinary meaning, i.e. in the case of connotative names, the connotation, of a word. This may be effected by stating directly the attributes connoted; but it is more usual to predicate of the subject of definition one name of synonymous, or several which, when combined, are of equivalent, connotation. So that, a definition of a name being thus generally the sum total of the essential propositions which could be framed with that name for subject, is really, as Condillac says, an analysis. Even when a name connotes only a single attribute, it (and also the corresponding abstract name itself) can yet be defined (in this sense of being analysed or resolved into its elements) by declaring the connotation of that attribute, whether, if it be a union of several attributes (e.g. Humanity), by enumerating them, or, if only one (e.g. Eloquence), by dissecting the fact which is its foundation. Even when the fact which is the foundation of the attribute is a simple feeling, and therefore incapable of analysis, still, if the simple feeling have a name, the attribute and the object possessing it may be defined by reference to the fact: e.g. a white object is definable as one exciting the sensation of white; and whiteness, as the power of exciting that sensation. The only names, abstract or concrete, incapable of analysis, and therefore of definition, are proper names, as having no meaning, and also the names of the simple feelings themselves, since these can be explained only by the resemblance of the feelings to former feelings called by the same or by an exactly synonymous name, which consequently equally needs definition.
Though the only accurate definition is one declaring all the facts involved in the name, i.e. its connotation, men are usually satisfied with anything which will serve as an index to its denotation, so as to guard them from applying it inconsistently. This was the object of logicians when they laid down that a species must be defined per genus et differentiam, meaning by the differentia one attribute included in the essence, i.e. in the connotation. And, in fact, one attribute, e.g. in defining man, Rationality (Swift's Houyhnhms having not been as yet discovered) often does sufficiently mark out the objects denoted. But, besides that a definition of this kind ought, in order to be complete, to be per genus et differentias, i.e. by all the connoted attributes not implied in the name of the genus, still, even if all were given, a summum genus could not be so defined, since it has no superior genus. And for merely marking out the objects denoted, Description, in which none of the connoted attributes are given, answers as well as logicians' so-called essential definition. In Description, any one or a combination of attributes may be given, the object being to make it exactly coextensive with the name, so as to be predicable of the same things. Such a description may be turned into an essential definition by a change of the connotation (not the denotation) of the name; and, in fact, thus are manufactured almost all scientific definitions, which, being landmarks of classification, and not meant to declare the meaning of the name (though, in fact, they do declare it in its new use), are ever being modified (as is the definition of a science itself) with the advance of knowledge. Thus, a technical definition helps to expound the artificial classification from which it grows; but ordinary definition cannot expound, as the Aristotelians fancied it could, the natural classification of things, i.e. explain their division into kinds, and the relations among the kinds: for the properties of every kind are innumerable, and all that definition can do is to state the connotation of the name.
Both these two modes, viz. the essential but incomplete Definition, and the accidental, or Description, are imperfect; but the Realists' distinction between definition of names and of things is quite erroneous. Their doctrine is now exploded; but many propositions consistent with it alone (e.g. that the science of geometry is deduced from definitions) have been retained by Nominalists, such as Hobbes. Really a definition, as such, cannot explain a thing's nature, being merely an identical proposition explaining the meaning of a word. But definitions of names known to be names of really existing objects, as in geometry, include two propositions, one a definition and another a postulate. The latter affirms the existence of a thing answering to the name. The science is based on the postulates (whether they rest on intuition or proof), for the demonstration appeals to them alone, and not on the definitions, which indeed might, though at some cost of brevity, be dispensed with entirely. It has been argued that, at any rate, definitions are premisses of science, provided they give such meanings to terms as suit existing things: but even so, the inference would obviously be from the existence, not of the name which means, but of the thing which has the properties.
One reason for the belief that demonstrative truths follow from the definitions, not from the postulates, was because the postulates are never quite true (though in reality so much of them is true as is true of the conclusions). Philosophers, therefore, searching for something more accurately true, surmised that definitions must be statements and analyses, neither of words nor of things, as such, but of ideas; and they supposed the subject-matter of all demonstrative sciences to be abstractions of the mind. But even allowing this (though, in fact, the mind cannot so abstract one property, e.g. length, from all others; it only attends to the one exclusively), yet the conclusions would still follow, not from the mere definitions, but from the postulates of the real existence of the ideas.
Definitions, in short, are of names, not things: yet they are not therefore arbitrary; and to determine what should be the meaning of a term, it is often necessary to look at the objects. The obscurity as to the connotation arises through the objects being named before the attributes (though it is from the latter that the concrete general terms get their meaning), and through the same name being popularly applied to different objects on the ground of general resemblance, without any distinct perception of their common qualities, especially when these are complex. The philosopher, indeed, uses general names with a definite connotation; but philosophers do not make language—it grows: so that, by degrees, the same name often ceases to connote even general resemblance. The object in remodelling language is to discover if the things denoted have common qualities, i.e. if they form a class; and, if they do not, to form one artificially for them. A language's rude classifications often serve, when retouched, for philosophy. The transitions in signification, which often go on till the different members of the group seem to connote nought in common, indicate, at any rate, a striking resemblance among the objects denoted, and are frequently an index to a real connection; so that arguments turning apparently on the double meaning of a term, may perhaps depend on the connection of two ideas. To ascertain the link of connection, and to procure for the name a distinct connotation, the resemblances of things must be considered. Till the name has got a distinct connotation, it cannot be defined. The philosopher chooses for his connotation of the name the attributes most important, either directly, or as the differentiae leading to the most interesting propria. The enquiry into the more hidden agreement on which these obvious agreements depend, often itself arises under the guise of enquiries into the definition of a name.
INFERENCE, OR REASONING IN GENERAL.
The preceding book treated, not of the proper subject of logic, viz. the nature of proof, but of assertion. Assertions (as, e.g. definitions) which relate to the meaning of words, are, since that is arbitrary, incapable of truth or falsehood, and therefore of proof or disproof. But there are assertions which are subjects for proof or disproof, viz. the propositions (the real, and not the verbal) whose subject is some fact of consciousness, or its hidden cause, about which is predicated, in the affirmative or negative, one of five things, viz. existence, order in place, order in time, causation, resemblance: in which, in short, it is asserted, that some given subject does or does not possess some attribute, or that two attributes, or sets of attributes, do or do not (constantly or occasionally) coexist.
A proposition not believed on its own evidence, but inferred from another, is said to be proved; and this process of inferring, whether syllogistically or not, is reasoning. But whenever, as in the deduction of a particular from a universal, or, in Conversion, the assertion in the new proposition is the same as the whole or part of the assertion in the original proposition, the inference is only apparent; and such processes, however useful for cultivating a habit of detecting quickly the concealed identity of assertions, are not reasoning.
Reasoning, or Inference, properly so called, is, 1, Induction, when a proposition is inferred from another, which, whether particular or general, is less general than itself; 2, Ratiocination, or Syllogism, when a proposition is inferred from others equally or more general; 3, a kind which falls under neither of these descriptions, yet is the basis of both.
RATIOCINATION, OR SYLLOGISM.
The syllogistic figures are determined by the position of the middle term. There are four, or, if the fourth be classed under the first, three. But syllogisms in the other figures can be reduced to the first by conversion. Such reduction may not indeed be necessary, for different arguments are suited to different figures; the first figure, says Lambert, being best adapted to the discovery or proof of the properties of things; the second, of the distinctions between things; the third, of instances and exceptions; the fourth, to the discovery or exclusion of the different species of a genus. Still, as the premisses of the first figure, got by reduction, are really the same as the original ones, and as the only arguments of great scientific importance, viz. those in which the conclusion is a universal affirmative, can be proved in the first figure alone, it is best to hold that the two elementary forms of the first figure are the universal types, the one in affirmatives, the other in negatives, of all correct ratiocination.
The dictum de omni et nullo, viz. that whatever can be affirmed or denied of a class can be affirmed or denied of everything included in the class, which is a true account generalised of the constituent parts of the syllogism in the first figure, was thought the basis of the syllogistic theory. The fact is, that when universals were supposed to have an independent objective existence, this dictum stated a supposed law, viz. that the substantia secunda formed part of the properties of each individual substance bearing the name. But, now that we know that a class or universal is nothing but the individuals in the class, the dictum is nothing but the identical proposition, that whatever is true of certain objects is true of each of them, and, to mean anything, must be considered, not as an axiom, but as a circuitous definition of the word class.
It was the attempt to combine the nominalist view of the signification of general terms with the retention of the dictum as the basis of all reasoning, that led to the self-contradictory theories disguised under the ultra-nominalism of Hobbes and Condillac, the ontology of the later Kantians, and (in a less degree) the abstract ideas of Locke. It was fancied that the process of inferring new truths was only the substitution of one arbitrary sign for another; and Condillac even described science as une langue bien faite. But language merely enables us to remember and impart our thoughts; it strengthens, like an artificial memory, our power of thought, and is thought's powerful instrument, but not its exclusive subject. If, indeed, propositions in a syllogism did nothing but refer something to or exclude it from a class, then certainly syllogisms might have the dictum for their basis, and import only that the classification is consistent with itself. But such is not the primary object of propositions (and it is on this account, as well as because men will never be persuaded in common discourse to quantify the predicate, that Mr. De Morgan's or Sir William Hamilton's quantification of the predicate is a device of little value). What is asserted in every proposition which conveys real knowledge, is a fact dependent, not on artificial classification, but on the laws of nature; and as ratiocination is a mode of gaining real knowledge, the principle or law of all syllogisms, with propositions not purely verbal, must be, for affirmative syllogisms, that; Things coexisting with the same thing coexist with one another; and for negative, that; A thing coexisting with another, with which a third thing does not coexist, does not coexist with that third thing. But if (see supra, p. 26) propositions (and, of course, all combinations of them) be regarded, not speculatively, as portions of our knowledge of nature, but as memoranda for practical guidance, to enable us, when we know that a thing has one of two attributes, to infer it has the other, these two axioms may be translated into one, viz. Whatever has any mark has that which it is a mark of; or, if both premisses are universal, Whatever is a mark of any mark, is a mark of that of which this last is a mark.
THE FUNCTIONS AND LOGICAL VALUE OF THE SYLLOGISM.
The question is, whether the syllogistic process is one of inference, i.e. a process from the known to the unknown. Its assailants say, and truly, that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii; and Dr. Whately's defence of it, that its object is to unfold assertions wrapped up and implied (i.e. in fact, asserted unconsciously) in those with which we set out, represents it as a sort of trap. Yet, though no reasoning from generals to particulars can, as such, prove anything, the conclusion is a bona fide inference, though not an inference from the general proposition. The general proposition (i.e. in the first figure, the major premiss) contains not only a record of many particular facts which we have observed or inferred, but also instructions for making inferences in unforeseen cases. Thus the inference is completed in the major premiss; and the rest of the syllogism serves only to decipher, as it were, our own notes.
Dr. Whately fails to make out that syllogising, i.e. reasoning from generals to particulars, is the only mode of reasoning. No additional evidence is gained by interpolating a general proposition, and therefore we may, if we please, reason directly from the individual cases, since it is on these alone that the general proposition, if made, would rest. Indeed, thus are in fact drawn, as well the inferences of children and savages, and of animals (which latter having no signs, can frame no general propositions), as even those drawn by grown men generally, from personal experience, and particularly the inferences of men of high practical genius, who, not having been trained to generalise, can apply, but not state, their principles of action. Even when we have general propositions we need not use them. Thus Dugald Stewart showed that the axioms need not be expressly adverted to in order to make good the demonstrations in Euclid; though he held, inconsistently, that the definitions must be. All general propositions, whether called axioms, or definitions, or laws of nature, are merely abridged statements of the particular facts, which, as occasion arises, we either think we may proceed on as proved, or intend to assume.
In short, all inference is from particulars to particulars; and general propositions are both registers or memoranda of such former inferences, and also short formulae for making more. The major premiss is such a formula; and the conclusion is an inference drawn, not from, but according to that formula. The actual premisses are the particular facts whence the general proposition was collected inductively; and the syllogistic rules are to guide us in reading the register, so as to ascertain what it was that we formerly thought might be inferred from those facts. Even where ratiocination is independent of induction, as, when we accept from a man of science the doctrine that all A is B; or from a legislator, the law that all men shall do this or that, the operation of drawing thence any particular conclusion is a process, not of inference, but of interpretation. In fact, whether the premisses are given by authority, or derived from our own (or predecessors') observation, the object is always simply to interpret, by reference to certain marks, an intention, whether that of the propounder of the principle or enactment, or that which we or our predecessors had when we framed the general proposition, so that we may draw no inferences that were not intended to be drawn. We assent to the conclusion in a syllogism on account of its consistency with what we interpret to have been the intention of the framer of the major premiss, and not, as Dr. Whately held, because the supposition of a false conclusion from the premisses involves a contradiction, since, in fact, the denial, e.g. that an individual now living will die, is not in terms contradictory to the assertion that his ancestors and their contemporaries (to which the general proposition, as a record of facts, really amounts) have all died.
But the syllogistic form, though the process of inference, which there always is when a syllogism is used, lies not in this form, but in the act of generalisation, is yet a great collateral security for the correctness of that generalisation. When all possible inferences from a given set of particulars are thrown into one general expression (and, if the particulars support one inference, they always will support an indefinite number), we are more likely both to feel the need of weighing carefully the sufficiency of the experience, and also, through seeing that the general proposition would equally support some conclusion which we know to be false, to detect any defect in the evidence, which, from bias or negligence, we might otherwise have overlooked. But the syllogistic form, besides being useful (and, when the validity of the reasoning is doubtful, even indispensable) for verifying arguments, has the acknowledged merit of all general language, that it enables us to make an induction once for all. We can, indeed, and in simple cases habitually do, reason straight from particulars; but in cases at all complicated, all but the most sagacious of men, and they also, unless their experience readily supplied them with parallel instances, would be as helpless as the brutes. The only counterbalancing danger is, that general inferences from insufficient premisses may become hardened into general maxims, and escape being confronted with the particulars.
The major premiss is not really part of the argument. Brown saw that there would be a petitio principii if it were. He, therefore, contended that the conclusion in reasoning follows from the minor premiss alone, thus suppressing the appeal to experience. He argued, that to reason is merely to analyse our general notions or abstract ideas, and that, provided that the relation between the two ideas, e.g. of man and of mortal, has been first perceived, we can evolve the one directly from the other. But (to waive the error that a proposition relates to ideas instead of things), besides that this proviso is itself a surrender of the doctrine that an argument consists simply of the minor and the conclusion, the perception of the relation between two ideas, one of which is not implied in the name of the other, must obviously be the result, not of analysis, but of experience. In fact, both the minor premiss, and also the expression of our former experience, must both be present in our reasonings, or the conclusion will not follow. Thus, it appears that the universal type of the reasoning process is: Certain individuals possess (as I or others have observed) a given attribute; An individual resembles the former in certain other attributes: Therefore (the conclusion, however, not being conclusive from its form, as is the conclusion in a syllogism, but requiring to be sanctioned by the canons of induction) he resembles them also in the given attribute. But, though this, and not the syllogistic, is the universal type of reasoning, yet the syllogistic process is a useful test of inferences. It is expedient, first, to ascertain generally what attributes are marks of a certain other attribute, so as, subsequently, to have to consider, secondly, only whether any given individuals have those former marks. Every process, then, by which anything is inferred respecting an unobserved case, we will consider to consist of both these last-mentioned processes. Both are equally induction; but the name may be conveniently confined to the process of establishing the general formula, while the interpretation of this will be called 'Deduction.'
TRAINS OF REASONING, AND DEDUCTIVE SCIENCES.
The minor premiss always asserts a resemblance between a new case and cases previously known. When this resemblance is not obvious to the senses, or ascertainable at once by direct observation, but is itself matter of inference, the conclusion is the result of a train of reasoning. However, even then the conclusion is really the result of induction, the only difference being that there are two or more inductions instead of one. The inference is still from particulars to particulars, though drawn in conformity, not to one, but to several formulae. This need of several formulae arises merely from the fact that the marks by which we perceive that an inference can be drawn (and of which marks the formulae are records) happen to be recognisable, not directly, but only through the medium of other marks, which were, by a previous induction, collected to be marks of them.
All reasoning, then, is induction: but the difficulties in sciences often lie (as, e.g. in geometry, where the inductions are the simple ones of which the axioms and a few definitions are the formulae) not at all in the inductions, but only in the formation of trains of reasoning to prove the minors; that is, in so combining a few simple inductions as to bring a new case, by means of one induction within which it evidently falls, within others in which it cannot be directly seen to be included. In proportion as this is more or less completely effected (that is, in proportion as we are able to discover marks of marks), a science, though always remaining inductive, tends to become also deductive, and, to the same extent, to cease to be one of the experimental sciences, in which, as still in chemistry, though no longer in mechanics, optics, hydrostatics, acoustics, thermology, and astronomy, each generalisation rests on a special induction, and the reasonings consist but of one step each.
An experimental science may become deductive by the mere progress of experiment. The mere connecting together of a few detached generalisations, or even the discovery of a great generalisation working only in a limited sphere, as, e.g. the doctrine of chemical equivalents, does not make a science deductive as a whole; but a science is thus transformed when some comprehensive induction is discovered connecting hosts of formerly isolated inductions, as, e.g. when Newton showed that the motions of all the bodies in the solar system (though each motion had been separately inferred and from separate marks) are all marks of one like movement. Sciences have become deductive usually through its being shown, either by deduction or by direct experiment, that the varieties of some phenomenon in them uniformly attend upon those of a better known phenomenon, e.g. every variety of sound, on a distinct variety of oscillatory motion. The science of number has been the grand agent in thus making sciences deductive. The truths of numbers are, indeed, affirmable of all things only in respect of their quantity; but since the variations of quality in various classes of phenomena have (e.g. in mechanics and in astronomy) been found to correspond regularly to variations of quantity in the same or some other phenomena, every mathematical formula applicable to quantities so varying becomes a mark of a corresponding general truth respecting the accompanying variations in quality; and as the science of quantity is, so far as a science can be, quite deductive, the theory of that special kind of qualities becomes so likewise. It was thus that Descartes and Clairaut made geometry, which was already partially deductive, still more so, by pointing out the correspondence between geometrical and algebraical properties.
CHAPTERS V. AND VI.
DEMONSTRATION AND NECESSARY TRUTHS.
All sciences are based on induction; yet some, e.g. mathematics, and commonly also those branches of natural philosophy which have been made deductive through mathematics, are called Exact Sciences, and systems of Necessary Truth. Now, their necessity, and even their alleged certainty, are illusions. For the conclusions, e.g. of geometry, flow only seemingly from the definitions (since from definitions, as such, only propositions about the meaning of words can be deduced): really, they flow from an implied assumption of the existence of real things corresponding to the definitions. But, besides that the existence of such things is not actual or possible consistently with the constitution of the earth, neither can they even be conceived as existing. In fact, geometrical points, lines, circles, and squares, are simply copies of those in nature, to a part alone of which we choose to attend; and the definitions are merely some of our first generalisations about these natural objects, which being, though equally true of all, not exactly true of any one, must, actually, when extended to cases where the error would be appreciable (e.g. to lines of perceptible breadth), be corrected by the joining to them of new propositions about the aberration. The exact correspondence, then, between the facts and those first principles of geometry which are involved in the so-called definitions, is a fiction, and is merely supposed. Geometry has, indeed (what Dugald Stewart did not perceive), some first principles which are true without any mixture of hypothesis, viz. the axioms, as well those which are indemonstrable (e.g. Two straight lines cannot enclose a space) as also the demonstrable ones; and so have all sciences some exactly true general propositions: e.g. Mechanics has the first law of motion. But, generally, the necessity of the conclusions in geometry consists only in their following necessarily from certain hypotheses, for which same reason the ancients styled the conclusions of all deductive sciences necessary. That the hypotheses, which form part of the premisses of geometry, must, as Dr. Whewell says, not be arbitrary—that is, that in their positive part they are observed facts, and only in their negative part hypothetical—happens simply because our aim in geometry is to deduce conclusions which may be true of real objects: for, when our object in reasoning is not to investigate, but to illustrate truths, arbitrary hypotheses (e.g. the operation of British political principles in Utopia) are quite legitimate.
The ground of our belief in axioms is a disputed point, and one which, through the belief arising too early to be traced by the believer's own recollection, or by other persons' observation, cannot be settled by reference to actual dates. The axioms are really only generalisations from experience. Dr. Whewell, however, and others think that, though suggested, they are not proved by experience, and that their truth is recognised a priori by the constitution of the mind as soon as the meaning of the proposition is understood. But this assumption of an a priori recognition is gratuitous. It has never been shown that there is anything in the facts inconsistent with the view that the recognition of the truth of the axioms, however exceptionally complete and instant, originates simply in experience, equally with the recognition of ordinary physical generalisations. Thus, that we see a property of geometrical forms to be true, without inspection of the material forms, is fully explained by the capacity of geometrical forms of being painted in the imagination with a distinctness equal to reality, and by the fact that experience has informed us of that capacity; so that a conclusion on the faith of the imaginary forms is really an induction from observation. Then, again, there is nothing inconsistent with the theory that we learn by experience the truth of the axioms, in the fact that they are conceived by the mind as universally and necessarily true, that is, that we cannot figure them to ourselves as being false. Our capacity or incapacity of conceiving depends on our associations. Educated minds can break up their associations more easily than the uneducated; but even the former not entirely at will, even when, as is proved later, they are erroneous. The Greeks, from ignorance of foreign languages, believed in an inherent connection between names and things. Even Newton imagined the existence of a subtle ether between the sun and bodies on which it acts, because, like his rivals the Cartesians, he could not conceive a body acting where it is not. Indeed, inconceivableness depends so completely on the accident of our mental habits, that it is the essence of scientific triumphs to make the contraries of once inconceivable views themselves appear inconceivable. For instance, suppositions opposed even to laws so recently discovered as those of chemical composition appear to Dr. Whewell himself to be inconceivable. What wonder, then, that an acquired incapacity should be mistaken for a natural one, when not merely (as in the attempt to conceive space or time as finite) does experience afford no model on which to shape an opposed conception, but when, as in geometry, we are unable even to call up the geometrical ideas (which, being impressions of form, exactly resemble, as has been already remarked, their prototypes), e.g. of two straight lines, in order to try to conceive them inclosing a space, without, by the very act, repeating the scientific experiment which establishes the contrary.
Since, then, the axioms and the misnamed definitions are but inductions from experience, and since the definitions are only hypothetically true, the deductive or demonstrative sciences—of which these axioms and definitions form together the first principles—must really be themselves inductive and hypothetical. Indeed, it is to the fact that the results are thus only conditionally true, that the necessity and certainty ascribed to demonstration are due.
It is so even with the Science of Number, i.e. arithmetic and algebra. But here the truth has been hidden through the errors of two opposite schools; for while many held the truths in this science to be a priori, others paradoxically considered them to be merely verbal, and every process to be simply a succession of changes in terminology, by which equivalent expressions are substituted one for another. The excuse for such a theory as this latter was, that in arithmetic and algebra we carry no ideas with us (not even, as in a geometrical demonstration, a mental diagram) from the beginning, when the premisses are translated into signs, till the end, when the conclusion is translated back into things. But, though this is so, yet in every step of the calculation, there is a real inference of facts from facts: but it is disguised by the comprehensive nature of the induction, and the consequent generality of the language. For numbers, though they must be numbers of something, may be numbers of anything; and therefore, as we need not, when using an algebraical symbol (which represents all numbers without distinction), or an arithmetical number, picture to ourselves all that it stands for, we may picture to ourselves (and this not as a sign of things, but as being itself a thing) the number or symbol itself as conveniently as any other single thing. That we are conscious of the numbers or symbols, in their character of things, and not of mere signs, is shown by the fact that our whole process of reasoning is carried on by predicating of them the properties of things.
Another reason why the propositions in arithmetic and algebra have been thought merely verbal, is that they seem to be identical propositions. But in 'Two pebbles and one pebble are equal to three pebbles,' equality but not identity is affirmed; the subject and predicate, though names of the same objects, being names of them in different states, that is, as producing different impressions on the senses. It is on such inductive truths, resting on the evidence of sense, that the Science of Number is based; and it is, therefore, like the other deductive sciences, an inductive science. It is also, like them, hypothetical. Its inductions are the definitions (which, as in geometry, assert a fact as well as explain a name) of the numbers, and two axioms, viz. The sums of equals are equal; the differences of equals are equal. These axioms, and so-called definitions are themselves exactly, and not merely hypothetically, true. Yet the conclusions are true only on the assumption that, 1 = 1, i.e. that all the numbers are numbers of the same or equal units. Otherwise, the certainty in arithmetical processes, as in those of geometry or mechanics, is not mathematical, i.e. unconditional certainty, but only certainty of inference. It is the enquiry (which can be gone through once for all) into the inferences which can be drawn from assumptions, which properly constitutes all demonstrative science.
New conclusions may be got as well from fictitious as from real inductions; and this is even consciously done, viz. in the reductio ad absurdum, in order to show the falsity of an assumption. It has even been argued that all ratiocination rests, in the last resort, on this process. But as this is itself syllogistic, it is useless, as a proof of a syllogism, against a man who denies the validity of this kind of reasoning process itself. Such a man cannot in fact be forced to a contradiction in terms, but only to a contradiction, or rather an infringement, of the fundamental maxim of ratiocination, viz. 'Whatever has a mark, has what it is a mark of;' and, since it is only by admitting premisses, and yet rejecting a conclusion from them, that this axiom is infringed, consequently nothing is necessary except the connection between a conclusion and premisses.
PRELIMINARY OBSERVATIONS ON INDUCTION IN GENERAL.
As all knowledge not intuitive comes exclusively from inductions, induction is the main topic of Logic; and yet neither have metaphysicians analysed this operation with a view to practice, nor, on the other hand, have discoverers in physics cared to generalise the methods they employed.
Inferences are equally inductive, whether, as in science, which needs its conclusions for record, not for instant use, they pass through the intermediate stage of a general proposition (to which class Dr. Whewell, without sanction from facts, or from the usage of Reid and Stewart, the founders of modern English metaphysical terminology, limits the term induction), or are drawn direct from particulars to a supposed parallel case. Neither does it make any difference in the character of the induction, whether the process be experiment or ratiocination, and whether the object be to infer a general proposition or an individual fact. That, in the latter case, the difficulty of the practical enquiries, e.g. of a judge or an advocate, lies chiefly in selecting from among all approved general propositions those inductions which suit his case (just as, even in deductive sciences, the ascertaining of the inductions is easy, their combination to solve a problem hard) is not to the point: the legitimacy of the inductions so selected must at all events be tried by the same test as a new general truth in science. Induction, then, may be treated here as though it were the operation of discovering and proving general propositions; but this is so only because the evidence which justifies an inference respecting one unknown case, would justify a like inference about a whole class, and is really only another form of the same process: because, in short, the logic of science is the universal logic applicable to all human enquiries.