Article on Logic, Encyclopedia Britannica, 1911
The article on Logic from the Encyclopedia Britannica, 1911 contains little that is directly interesting to us today. Both the Aristotelian logic that held undisputed sway over two millennia, and the point of view of it represented by the article are completely outmoded and almost forgotten. The article was written after some of the most exciting developments of modern logic. Russell had published his discovery of what he called "the Contradiction" (which we now call "Russell's paradox) in The Principles of Mathematics of 1903. Zermelo had published his seminal paper formalising Cantor's set theory in 1908. Most of Frege's work had been published for over twenty years. Yet none of these works are mentioned in the article (curiously, the only work from 1908 to be cited is Joyce's Principles of Logic (of which a copy is available online here), a neo-scholastic seminary manual which was archaic even in 1908. This is despite Frege's opus being exactly contemporary with the works mentioned in the bibliography.
It is interesting for that reason. It provides insight into the intellectual world of the time, in which Russell, Frege, Brentano, Venn and others were working. The article provides a thorough conspectus of received opinion on logical matters in 1911. Many of the writers mentioned in the article or in the bibliography (such as Benno Erdmann, Wundt, Theodore Lipps and others) were those whose theories were engaged by Frege.
It also mentions many ideas that were essential to the development of modern logic (e.g. the existential nature of the categorical judgment, the equivalence of universal categorical and hypothetical judgments). Some of these were advanced by logicians of the older school. Everyone today knows about Frege, the inventor of modern logic. Most know about the most fundamental idea underlying his logic: that existence is not a property of individuals, and that every categorical judgment A is B is therefore existential. Yet this idea was published by Brentano 1874, while Frege was still at university, and it is Brentano whose theory is discussed in the article. Two whole sections (here and here) are devoted to analysing (and disparaging) Brentano's theory of judgment, and the article states that the view originates with him. The fact that Brentano was influential in 1911, and Frege was unheard of, is interesting in itself.
I have altered the original article as little as possible, except: to make formatting changes consistent with modern information technology. I have not attempted to include parenthesised expressions in Greek (which the scanner did not understand). Some material was lost entirely – I have indicated these using the dots of ellipsis in brackets, thus […].
The name given to one of the four main departments of philosophy, though its sphere is very variously delimited.
Introduction. Logic is the science of the processes of inference. What, then, is inference? It is that mental operation which proceeds by combining two premises so as to cause a consequent conclusion. Some suppose that we may infer from one premise by a so-called immediate inference. But one premise can only reproduce itself in another form, e.g. all men are some animals; therefore some animals are men. It requires the combination of at least two premises to infer a conclusion different from both. There are as many kinds of inference as there are different ways of combining premises, and in the main three types:
1. Analogical Inference, from particular to particular: e.g.
border-war between Thebes and Phocis is evil
border-war between Thebes and Athens is similar to
that between Thebes and
therefore, border-war between Thebes and Athens is evil.
2. Inductive Inference, from particular to universal: e.g.
border-war between Thebes and Phocis is evil
all border-war is like that between Thebes and
therefore, all border-war is evil.
3. Deductive or Syllogistic Inference, from universal to particular, e.g.
all border-war is evil
border-war between Thebes and Athens is border-war
therefore border-war between Thebes and Athens is evil.
In each of these kinds of inference there are three mental judgments capable of being expressed as above in three linguistic propositions; and the two first are the premises which are combined, while the third is the conclusion which is consequent on their combination. Each proposition consists of two terms, the subject and its predicate, united by the copula. Each inference contains three terms. In syllogistic inference the subject of the conclusion is the minor term, and its predicate the major term, while between these two extremes the term common to the two premises is the middle term, and the premise containing the middle and major terms is the major premise, the premise containing the middle and minor term the minor premise. Thus in the example of syllogism given above, border-war between Thebes and Athens is the minor term, evil the major term, and border-war the middle term. Using S for minor, P for major and M for middle, and preserving these signs for corresponding terms in analogical and inductive inferences, we obtain the following formula of the three inferences:
Analogical. Inductive. Deductive or Syllogistic.
S1 is P S is P Every M is P
S1 is similar to S2 Every M is similar to S S is M
S2 is P Every M is P S is P.
The love of unity has often made logicians attempt to resolve these three processes into one. But each process has a peculiarity of its own; they are similar, not the same. Analogical and inductive inference alike begin with a particular premise containing one or more instances; but the former adds a particular premise to draw a particular conclusion, the latter requires a universal premise to draw a universal conclusion. A citizen of Athens, who had known the evils of the border-war between Thebes and Phocis, would readily perceive the analogy of a similar war between Thebes and Athens, and conclude analogously that it would be evil; but he would have to generalize the similarity of all border-wars in order to draw the inductive conclusion that all alike are evil. Induction and deduction differ still more, and are in fact opposed, as one makes a particular premise the evidence of a universal conclusion, the other makes a universal premise evidence of a particular conclusion. Yet they are alike in requiring the generalization of the universal and the belief that there are classes which are whole numbers of similars. On this point both differ from inference by analogy, which proceeds entirely from particular premises to a particular conclusion. Hence we may re-divide inference into particular inference by analogy and universal inference by induction and deduction. Universal inference is what we call reasoning; and its two species are very closely connected, because universal conclusions of induction become universal premises of deduction. Indeed, we often induce in order to deduce, ascending from particular to universal and descending from universal to particular in one act as it were; so that we may proceed either directly from particular to particular by analogical inference, or indirectly from particular through universal to particular by an inductive deductive inference which might be called perduction. On the whole, then, analogical, inductive and deductive inferences are not the same but three similar and closely connected processes.
The three processes of inference, though different from one another, rest on a common principle of similarity of which each is a different application. Analogical inference requires that one particular is similar to another, induction that a whole number or class is similar to its particular instances, deduction that each particular is similar to the whole number or class. Not that these inferences require us to believe, or assume, or premise or formulate this principle either in general, or in its applied forms: the premises are all that any inference needs the mind to assume. The principle of similarity is used, not assumed by the inferring mind, which in accordance with the similarity of things and the parity of inference spontaneously concludes in the form that similars are similarly determined (similia similibus convenire ). In applying this principle of similarity, each of the three processes in its own way has to premise both that something is somehow determined and that something is similar, and by combining these premises to conclude that this is similarly determined to that. Thus the very principle of inference by similarity requires it to be a combination of premises in order to draw a conclusion.
The three processes, as different applications of the principle of similarity, consisting of different combinations of premises, cause different degrees of cogency in their several conclusions. Analogy hardly requires as much evidence as induction. Men speculate about the analogy between Mars and the earth, and infer that it is inhabited, without troubling about all the planets. Induction has to consider more instances, and the similarity of a whole number or class. Even so, however, it starts from a particular premise which only contains many instances, and leaves room to doubt the universality of its conclusions. But deduction, starting from a premise about all the members of a class, compels a conclusion about every and each of necessity. One border-war may be similar to another, and the whole number may be similar, without being similarly evil; but if all alike are evil, each is evil of necessity. Deduction or syllogism is superior to analogy and induction in combining premises so as to involve or contain the conclusion. For this reason it has been elevated by some logicians above all other inferences, and for this very same reason attacked by others as no inference at all. The truth is that, though the premises contain the conclusion, neither premise alone contains it, and a man who knows both but does not combine them does not draw the conclusion; it is the synthesis of the two premises which at once contains the conclusion and advances our knowledge; and as syllogism consists, not indeed in the discovery, but essentially in the synthesis of two premises, it is an inference and an advance on each premise and on both taken separately. As again the synthesis contains or involves the conclusion, syllogism has the advantage of compelling assent to the consequences of the premises. Inference in general is a combination of premises to cause a conclusion; deduction is such a combination as to compel a conclusion involved in the combination, and following from the premises of necessity.
Nevertheless, deduction or syllogism is not independent of the other processes of inference. It is not the primary inference of its own premises, but constantly converts analogical and inductive conclusions into its particular and universal premises. Of itself it causes a necessity of consequence, but only a hypothetical necessity; if these premises are true, then this conclusion necessarily follows. To eliminate this if ultimately requires other inferences before deduction. Especially, induction to universals is the warrant and measure of deduction from universals. So far as it is inductively true that all border-war is evil, it is deductively true that a given border-war is therefore evil. Now, as an inductive combination of premises does not necessarily involve the inductive conclusion, induction normally leads, not to a necessary, but to a probable conclusion; and whenever its probable conclusions become deductive premises, the deduction only involves a probable conclusion. Can we then infer any certainty at all? In order to answer this question we must remember that there are many degrees of probability, and that induction, and therefore deduction, draw conclusions more or less probable, and rise to the point at which probability becomes moral certainty, or that high degree of probability which is sufficient to guide our lives, and even condemn murderers to death. But can we rise still higher and infer real necessity? This is a difficult question, which has received many answers. Some noologists suppose a mental power of forming necessary principles of deduction a priori; but fail to show how we can apply principles of mind to things beyond mind. Some empiricists, on the other hand, suppose that induction only infers probable conclusions which are premises of probable deductions; but they give up all exact science. Between these extremes there is room for a third theory, empirical yet providing a knowledge of the really necessary. In some cases of induction concerned with objects capable of abstraction and simplification, we have a power of identification, by which, not a priori but in the act of inducing a conclusion, we apprehend that the things signified by its subject and predicate are one and the same thing which cannot exist apart from itself. Thus by combined induction and identification we apprehend that one and one are the same as two, that there is no difference between a triangle and a three-sided rectilineal figure, that a whole must be greater than its part by being the whole, that inter-resisting bodies necessarily force one another apart, otherwise they would not be inter-resisting but occupy the same place at the same moment. Necessary principles, discovered by this process of induction and identification, become premises of deductive demonstration to conclusions which are not only necessary consequents on the premises, but also equally necessary in reality. Induction thus is the source of deduction, of its truth, of its probability, of its moral certainty; and induction, combined with identification, is the origin of the necessary principles of demonstration or deduction to necessary conclusions.
Analogical inference in its turn is as closely allied with induction. Like induction, it starts from a particular premise, containing one or more examples or instances; but, as it is easier to infer a particular than a universal conclusion, it supplies particular conclusions which in their turn become further particular premises of induction. Its second premise is indeed merely a particular apprehension that one particular is similar to another, whereas the second premise of induction is a universal apprehension that a whole number of particulars is similar to those from which the inference starts; but at bottom these two apprehensions of similarity are so alike as to suggest that the universal premise of induction has arisen as a generalized analogy. It seems likely that man has arrived at the apprehension of a whole individual, e.g. a whole animal including all its parts, and thence has inferred by analogy a whole number, or class, e.g. of animals including all individual animals; and accordingly that the particular analogy of one individual to another has given rise to the general analogy of every to each individual in a class, or whole number of individuals, contained in the second premise of induction. In this case, analogical inference has led to induction, as induction to deduction. Further, analogical inference from particular to particular suggests inductive deductive inference from particular through universal to particular.
Newton, according to Dr Pemberton, thought in 1666 that the moon moves so like a falling body that it has a similar centripetal force to the earth, 20 years before he demonstrated this conclusion from the laws of motion in the Principia. In fact, analogical, inductive and deductive inferences, though different processes of combining premises to cause different conclusions, are so similar and related, so united in principle and interdependent, so consolidated into a system of inference, that they cannot be completely investigated apart, but together constitute a single subject of science. This science of inference in general is logic.
Logic, however, did not begin as a science of all inference. Rather it began as a science of reasoning (Myos), of syllogism, of deductive inference. Aristotle was its founder. He was anticipated of course by many generations of spontaneous thinking (logica naturalis). Many of the higher animals infer by analogy: otherwise we cannot explain their thinking. Man so infers at first: otherwise we cannot explain the actions of young children, who before they begin to speak give no evidence of universal thinking. It is likely that man began with particular inference and with particular language; and that, gradually generalizing thought and language, he learnt at last to think and say all, to infer universally, to induce and deduce, to reason, in short, and raise himself above other animals. In ancient times, and especially in Egypt, Babylon and Greece, he went on to develop reason into science or the systematic investigation of definite subjects, e.g. arithmetic of number, geometry of magnitude, astronomy of stars, politics of government, ethics of goods. In Greece he became more and more reflective and conscious of himself, of his body and soul, his manners and morals, his mental operations and especially his reason. One of the characteristics of Greek philosophers is their growing tendency, in investigating any subject, to turn round and ask themselves what should be the method of investigation. In this way the Presocratics and Sophists, and still more Socrates and Plato, threw out hints on sense and reason, on inferential processes and scientific methods which may be called anticipations of logic. But Aristotle was the first to conceive of reasoning itself as a definite subject of a special science, which he called analytics or analytic science, specially designed to analyse syllogism and especially demonstrative syllogism, or science, and to be in fact a science of sciences. He was therefore the founder of the science of logic.
Among the Aristotelian treatises we have the following, which together constitute this new science of reasoning
1. The Categories, or names signifying things which can become predicates
2. The De Interpretatione, or the enumeration of conceptions and their combinations by
(1) nouns and verbs (names),
(2) enunciations (propositions)
3. The Prior Analytics, on syllogism
4. The Posterior Analytics, on demonstrative syllogism, or science
5. The Topics, on dialectical syllogism; or argument
6. The Sophistica Elenchi, on sophistical or contentious syllogism, or sophistical fallacies.
So far as we know, Aristotle had no one name for all these investigations. Analytics is only applied to the Prior and Posterior Analytics, and logical, which he opposed to analytical, only suits the Topics and at most the Sophistical Elenchi; secondly, while he analyzed syllogism into premises, major and minor, and premises into terms, subject and predicate, he attempted no division of the whole science; thirdly, he attempted no order and arrangement of the treatises into a system of logic, but only of the Analytics, Topics and Sophistical Elenchi into a system of syllogisms. Nevertheless, when his followers had arranged the treatises into the Organon, as they called it to express that it is an instrument of science, then there gradually emerged a system of syllogistic logic, arranged in the triple division terms, propositions and syllogisms which has survived to this day as technical logic, and has been the foundation of all other logics, even of those which aim at its destruction.
The main problem which Aristotle set before him was the analysis of syllogism, which he defined as reasoning in which certain things having been posited something different from them of necessity follows by their being those things (Prior Analytics, i. I). What then did he mean by reasoning, or rather by the Greek word [..] of which reasoning is an approximate rendering? It was meant (cf. Post. An. i. 10) to be both internal, in the soul, and external, in language: hence after Aristotle the Stoics distinguished […]. It meant, then, both reason and discourse of reason (cf. Shakespeare, Hamlet, i. 2). On its mental side, as reason it meant combination of thoughts. On its linguistic side, as discourse it was used for any combination of names to form a phrase, such as the definition rational animal, or a book, such as the Iliad. It had also the mathematical meaning of ratio; and in its use for definition it is sometimes transferred to essence as the object of definition, and has a mixed meaning, which may be expressed by account. In all its uses, however, the common meaning is combination. When Aristotle called syllogism […], he meant that it is a combination of premises involving a conclusion of necessity. Moreover, he tended to confine the term […] to syllogistic inference. Not that he omitted other inferences. On the contrary, to him (cf. Prior Analytics, ii. 24) we owe the triple distinction into inference from particular to particular (or what we call analogy ), inference from particular to universal (induction), and inference from universal to particular (syllogism, or deduction). But he thought that inferences other than syllogism are imperfect; that analogical inference is rhetorical induction; and that induction, through the necessary preliminary of syllogism and the sole process of ascent from sense, memory and experience to the principles of science, is itself neither reasoning nor science. To be perfect he thought that all inference must be reduced to syllogism of the first figure, which he regarded as the specially scientific inference. Accordingly, the syllogism appeared to him to be the rational process, and the demonstrative syllogism from inductively discovered principles to be science. Hence, without his saying it in so many words, Aristotle's .logic perforce became a logic of deductive reasoning, or syllogism. As it happened this deductive tendency helped the development of logic. The obscurer premises of analogy and induction, together with the paucity of experience and the backward state of physical science in Aristotle's time would have baffled even his analytical genius. On the other hand, the demonstrations of mathematical sciences of his time, and the logical forms of deduction evinced in Plato's dialogues,- provided him with admirable examples of deduction, which is also the inference most capable of analysis. Aristotle's analysis of the syllogism showed man how to advance by combining his thoughts in trains of deductive reasoning. Nevertheless, the wider question remained for logic: what is the nature of all inference, and the special form of each of its three main processes?
As then the reasoning of the syllogism was the main problem of Aristotle's logic, what was his analysis of, it? In distinguishing inner and outer reason, or reasoning and discourse, he added that it is not to outer reason but to inner reason in the soul that demonstration and syllogism are directed (Post. An. i. ro). One would expect, then, an analysis of mental reasoning into mental judgments as premises and conclusion. In point of fact, he analysed it into premises, but then analysed a premise into terms, which he divided into subject and predicate, with the addition of the copula is or is not. This analysis, regarded as a whole and as it is applied in the Analytics and in the other logical treatises, was evidently intended as a linguistic analysis. So in the Categories, lie first divided things said into uncombined and combined, or names and propositions, and then divided the former into categories; and in the De Interpretatione he expressly excluded mental conceptions and their combinations, and confined himself to nouns and verbs and enunciations, or, as we should say, to names and propositions. Aristotle apparently intended, or at all events has given logicians in general the impression, that he intended to analyse syllogism into propositions as premises, and premise into names as terms. His logic therefore exhibits the curious paradox of being an analysis of mental reasoning into linguistic elements. The explanation is that outer speech is more obvious than inner thought, and that grammar and poetic criticism, rhetoric and dialectic preceded logic, and that out of those arts of language arose the science of reasoning. The sophist Protagoras had distinguished various kinds of sentences, and Plato had divided the sentence into noun and verb, signifying a thing and the action of a thing. Rhetoricians had enumerated various means of persuasion, some of which are logical forms, e.g. probability and sign, example and enthymeme. Among the dialecticians, Socrates had used inductive arguments to obtain definitions as data of deductive arguments against his opponents, and Plato had insisted on the processes of ascending to and descending from an unconditional principle by the power of giving and receiving argument. All these points about speech, eloquence and argument between man and man were absorbed into Aristotle's theory of reasoning, and in particular the grammar of the sentence consisting of noun and verb caused the logic of the proposition consisting of subject and predicate.. At the same time, Aristotle was well aware that the science of reasoning is no art of language and must take up a different position towards speech as the expression of thought. In the Categories he classified names, not, however, as a grammarian by their structure, but as a logician by their signification. In the De Interpretatione, having distinguished the enunciation, or proposition, from other sentences as that in which there is truth or falsity, he relegated the rest to rhetoric or poetry, and founded the logic of the proposition, in which, however, he retained the grammatical analysis into noun and verb. In the Analytics he took the final step of originating the logical analysis of the proposition as premise into subject and predicate as terms mediated by the copula, and analysed the syllogism into these elements. Thus did he become the founder of the logical but linguistic analysis of reasoning as discourse into propositions and terms. Nevertheless, the deeper question remained, what is the logical but mental analysis of reasoning itself into its mental premises and conclusion?
Aristotle thus was the founder of logic as a science. But he laid too much stress on reasoning as syllogism or deduction, and on deductive science; and he laid too much stress on the linguistic analysis of rational discourse into proposition and terms. These two defects remain ingrained in technical logic to this day. But in the course of the development of the science, logicians have endeavoured to correct those defects, and have diverged into two schools. Some have devoted themselves to induction from sense and experience and widened logic till it has become a general science of inference and scientific method. Others have devoted themselves to the mental analysis of reasoning, and have narrowed logic into a science of conception, judgment and reasoning. The former belong to the school of empirical logic, the latter to the school of conceptual and formal logic. Both have started from points which Aristotle indicated without developing them. But we shall find that his true descendants are the empirical logicians.
Aristotle was the first of the empiricists. He consistently maintained that sense is knowledge; of particulars and the origin of scientific knowledge of universals. In his view, sense is a congenital form of judgment (Post. An. ii. 19); a sensation of each of the five senses is always true of its proper object; without sense there is no science; sense is the origin of induction, which is the origin of deduction and science. The Analytics end (Post. An. ii. 19) with a detailed system of empiricism, according to which sense is the primary knowledge of particulars, memory is the retention of a sensation, experience is the sum of many memories, induction infers universals, and intelligence is the true apprehension of the universal principles of science, which is rational, deductive, demonstrative, from empirical principles.
This empirical groundwork of Aristotle's logic was accepted by the Epicureans, who enunciated most distinctly the fundamental doctrine that all sensations are true of their immediate objects, and falsity begins with subsequent opinions, or what the moderns call interpretation. Beneath deductive logic, in the logic of Aristotle and the canonic of the Epicureans, there already lay the basis of empirical logic: sensory experience is the origin of all inference and science. It remained for Francis Bacon to develop these beginnings into a new logic of induction. He did not indeed accept the infallibility of sense or of any other operation unaided. He thought, rather, that every operation becomes infallible by method. Following Aristotle, in this order sense, memory, intellect he resolved the whole process of induction into three ministrations:
1. The ministration to sense, aided by observation and experiment.
2. The ministration to memory, aided by registering and arranging the data, of observation and experiment in tables of instances of agreement, difference and concomitant variations.
3. The ministration to intellect or reason, aided by the negative elimination by means of contradictory instances of whatever in the instances is not always present, absent and varying with the given subject investigated, and finally by the positive inference that whatever in the instances is always present, absent and varying with the subject is its essential cause.
Bacon, like Aristotle, was anticipated in this or that point; but, as Aristotle was the first to construct a system of deduction in the syllogism and its three figures, so Bacon was the first to construct a system of induction in three ministrations, in which the requisites of induction, hitherto recognized only in sporadic hints, were combined for the first time in one logic of induction. Bacon taught men to labour in inferring from particular to universal, to lay as much stress on induction as on deduction, and to think and speak of inductive reasoning, inductive science, inductive logic. Moreover, while Aristotle had the merit of discerning the triplicity of inference, to Bacon we owe the merit of distinguishing the three processes without reduction:
1. Inference from particular to particular by Experientia Literata, in plano;
2. Inference from particular to universal by Inductio, ascendendo
3. Inference from universal to particular by Syllogism, descendendo.
In short, the comprehensive genius of Bacon widened logic into a general science of inference.
On the other hand, as Aristotle over-emphasized deduction so Bacon over-emphasized induction by contending that it is the only process of discovering universals (axiomata), which deduction only applies to particulars. J. S. Mill in his Logic pointed out this defect, and without departing from Baconian principles remedied, it by quoting scientific examples, in which deduction, starting from inductive principles, applies more general to less general universals, e.g. when the more general law of gravitation is shown to include the less general laws of planetary gravitation. Mill's logic has the great merit of copiously exemplifying the principles of the variety of method according to subject-matter. It teaches us that scientific method is sometimes induction, sometimes deduction, and sometimes the consilience of both, either by the inductive verification of previous deductions, or by the deductive explanation of previous inductions.
It is also most interesting to notice that Aristotle saw further than Bacon in this direction. The founder of logic anticipated the latest logic of science, when he recognized, not only the deduction of mathematics, but. also the experience of facts followed by dc duct we explanations of their causes in physics.
The consilience of empirical and deductive processes was an Aristotelian discovery, elaborated by Mill against Bacon. Or the whole, however, Aristotle, Bacon and Mill, purged from their errors, form one empirical school, gradually growing by adapting itself to the advance of science; a school in which Aristotle was most influenced by Greek deductive Mathematics, Bacon by the rise of empirical physics at the Renaissance, and Mill by the Newtonian combination of empirical facts and mathematical principles in the Principia. From studying this succession of empirical logicians, we cannot doubt that sense, memory and experience are the real origin of inference, analogical, inductive and deductive. The deepest problem of logic is the relation of sense and inference. But we must first consider the mental analysis of inference, and this brings us to conceptual and formal logic.
Aristotle's logic has often been called formal logic; it was really a technical logic of syllogism analysed into linguistic elements, and of science rested on an empirical basis. At the same time his psychology, though maintaining his empiricism, contained some seeds of conceptual logic, and indirectly of formal logic. Intellectual development, which according to the logic of the Analytics consists of sense, memory, experience, induction and intellect, according to the psychology of the De Anima consists of sense, imagination and intellect, and one division of intellect is into conception of the undivided and combination of conceptions as one (De An. iii. 6). The De Interpretatione opens with a reference to this psychological distinction, implying that names represent conceptions, propositions represent combinations of conceptions. But the same passage relegates conceptions am their combinations to the De Anima, and confines the De Interpretatione to names and propositions in conformity with the linguistic analysis which pervades the logical treatises of Aristotle, who neither brought his psychological distinction between conceptions and their combinations into his logic, nor advanced the combinations of conceptions as a definition of judgment, nor employed the mental distinction between conceptions and judgments as an analysis of inference, or reasoning, or syllogism: he was no conceptual logician. The history of logic shows that the linguistic distinction between terms and propositions was the sole analysis of reasoning in the logical treatises of Aristotle; that the mental distinction between conceptions and judgments was imported into logic by the Stoics; and that this mental distinction became the logical analysis of reasoning under the authority of St Thomas Aquinas. In his commentary on the De Interpretatione, St Thomas, after citing from the De Anima Aristotle's duplex operatio intellectus, said, Additur autem et tertia operatio, scilicet ratiocinandi, and concluded that, since logic is a rational science (rationalis scientia), its consideration must be directed to all these operations of reason. Hence arose conceptual logic; according to which conception is a simple apprehension of an idea without belief in being or not being, e.g. the idea of man or of running; judgment is a combination of conceptions, adding being or not being, e.g. man is running or not running; and reasoning is a combination of judgments: conversely, there is a mental analysis of reasoning into judgments, and judgment into conceptions, beneath the linguistic analysis of rational discourse into propositions, and propositions into terms. Logic, according to this new school, which has by our time become an old school, has to co-ordinate these three operations, direct them, and, beginning with conceptions, combine conceptions into judgments, and judgments into inference, which thus becomes a complex combination of conceptions, or, in modern parlance, an extension of our ideas. Conceptual logicians were, indeed, from the first aware that sense supplies the data, and that judgment and therefore inference contains belief that things are or are not.
But they held, and still hold that sensation and conception are alike mere apprehensions, and that the belief that things are or are not arises somehow after sensation and conception in judgment, from which it passes into inference. At first, they were more sanguine of extracting from these unpromising beginnings some knowledge of things beyond ideas. But at length many of them became formal logicians, who held that logic is the investigation of formal thinking, or consistent. conception, judgment and reasoning; that it shows how we infer formal truths of consistency without material truth of signifying things; that, as the science of the form or process, it must entirely abstract from the matter, Or objects, of thought; and that it does not tell us how we infer from experience. Thus has logic drifted further and further from the real and empirical logic of Aristotle the founder and Bacon the reformer of the science.
The great merit of conceptual logic was the demand for a mental analysis of mental reasoning, and the direct analysis of reasoning into judgments which are the sole premises and conclusions of reasoning and of all mental inferences. Aristotle had fallen into the paradox of resolving a mental act into verbal elements. The Schoolmen, however, gradually came to realize that the result to their logic was to make it a sermocionalis scientia, and to their metaphysics the danger of nominalism. St Thomas made a great advance by making logic throughout a rationalis scientia; and logicians are now agreed that reasoning consists of judgments, discourse of propositions. This distinction is, moreover, vital to the whole logic of inference, because we always think all the judgments of which our inference consists, but seldom state all, the propositions by which it is expressed. We omit propositions, curtail them, and even express a judgment by a single term, e.g. Good, Fire! . Hence the linguistic expression is not a true measure of inference; and to say that an inference consists of two propositions causing a third is not strictly true. But to say that it is two judgments causing a third is always true, and the very essence of inference, because we must think the two to conclude the third in the sessions of sweet silent thought., Inference, in short, consists of actual judgments capable of being expressed in propositions.
Inference always consists of judgments. But judgment does not always consist of conceptions. It is not a combination of conceptions; it does not arise from conceptions, nor even at first require conception. Sense is the origin of judgment. One who feels pained or pleased, who feels hot or cold or resisting in touch, who tastes the flavoured, who smells the odorous, who hears the sounding, who sees the coloured, or is conscious, already believes that something sensible exists before conception, before inference, and before language; and his belief is true of the immediate object of sense, the sensible thing, e.g. the hot felt in touch. But a belief in the existence of something is a judgment and a categorical judgment of existence. Sense, then, outer and inner, or sensation and consciousness, is the origin of sensory judgments which are true categorical beliefs in the existence of sensible things; and primary judgments are such true categorical sensory beliefs that things exist, and neither require conception nor are combinations of conceptions. Again, since sense is the origin of memory and experience, memorial and experiential judgments are categorical and existential judgments, which so far as they report sensory judgments are always true. Finally, since sense, memory and experience are the origin of inference, primary inference is categorical and existential, starting from sensory, memorial and experiential judgments as premises, and proceeding to inferential judgments as conclusions, which are categorical and existential, and are true, so far as they depend on sense, memory and experience.
Sense, then, is the origin of judgment; and the consequence is that primary judgments are true, categorical and existential judgments of sense, and primary inferences are inferences from categorical and existential premises to categorical and existential conclusions, which are true so far as they arise from outer and inner sense, and proceed to things similar to sensible things. All other judgments and inferences about existing things, or ideas, or names, whether categorical or hypothetical, are afterthoughts, partly true and partly false.
Sense, then, because it involves a true belief in existence is fitted to be the origin of judgment. Conception on the other hand is the simple apprehension of an idea, particular or universal, but without belief that anything is or is not, and therefore is unfitted to beget judgment. Nor could a combination of conceptions make a difference so fundamental as that between conceiving and believing. The most that it could do would be to cause an ideal judgment, e.g. that the idea of a centaur is the idea of a man-horse; and even here some further origin is needed for the addition of the copula is.
So far from being a cause, conception is not even a condition of all judgments; a sensation of hot is sufficient evidence that hot exists, before the idea of hot is either present or wanted. Conception is, however, a condition of a memorial judgment: in order to remember being hot, we require an idea of hot. Memory, however, is not that idea, but involves a judgment that there previously existed the hot now represented by the idea, which is about the sensible thing beyond the conceived idea; and the cause of this memorial judgment is past sense and present memory. So sense, memory and experience, the sum of sense and memory, though requiring conception, are the causes of the experiential judgment that there exist and have existed many similar, sensible things, and these sensory, memorial and experiential judgments about ,the existence of past and present sensible things beyond conceived ideas become the particular premises of primary inference. Starting from them, inference is enabled to draw conclusions which are inferential judgments about the existence of things similar to sensible things beyond conceived ideas. In rising, however, from particular to universal inference, induction, as we have seen, adds to its particular premise, S is P, a universal premise, every M is similar to S, in order to infer the universal conclusion, every M is P. This universal premise requires a universal conception of a class or whole number of similar particulars, as a condition. But the premise is not that conception; it is a belief that there is a whole number of particulars similar to those already experienced. The generalization of a class is not, as the conceptual logic assumes, the abstraction of a genera) idea, but an inference from the analogy of a whole individual thing, e.g. a whole man, to a whole number of similar individuals, e.g. the whole of men. The general idea of all men or the combination that the idea of all men is similar to the idea of particular men would not be enough; the universal premise that all men in fact are similar to those who have died is required to induce the universal conclusion that all men in fact die. Universal inference thus requires particular and universal conceptions as its condition; but, so far as it arises from sense, memory, experience, and involves generalization, it consists of judgments which do not Consist of conceptions, but are beliefs in things existing beyond conception. Inference then, so far as it starts from categorical and existential premises, causes conclusions, or inferential judgments, which require conceptions, but are categorical and existential judgments beyond conception. Moreover, as it becomes more deductive, and causes conclusions further from sensory experience, these inferential judgments become causes of inferential conceptions. For example, from the evidence of molar changes due to the obvious parts of bodies, science first comes to believe in molecular changes due to imperceptible particles, and then tries to conceive the ideas of particles, molecules, atoms, electrons. The conceptual logic supposes that conception always precedes judgment; but the truth is that sensory judgment begins and inferential judgment ends by preceding conception. The supposed triple order conception, judgment, reasoning is defective and false. The real order is sensation and sensory judgment, conception, memory and memorial judgment, experience and experiential judgment, inference, inferential judgment, inferential conception. This is not all: inferential conceptions are, inadequate, and finally fail. They are often symbolical; that is, we conceive one thing only by another like it, e.g. atoms by minute bodies not nearly small enough. Often the symbol is not like. What idea can the physicist form of intra spatial ether? What believer in God pretends to conceive Him as lie really is? We believe many things that we cannot conceive; as Mill said, the inconceivable is not the incredible; and the point of science is not what we can conceive but what we should believe on evidence. Conception is the weakest, judgment the strongest power of mans mind. Sense before conception is the original cause of judgment; and inference from sense enables judgment to continue after conception ceases. Finally, as there is judgment without conception, so there is conception without judgment. We often say 1 understand, but do not decide. But this suspension of judgment is a highly refined act, unfitted to the beginning of thought. Conception begins as a condition of memory, and after a long continuous process of inference ends in mere ideation. The conceptual logic has made the mistake of making ideation a stage in thought prior to judgment.
It was natural enough that the originators of conceptual logic, seeing that judgments can be expressed by propositions, and conceptions by terms, should fall into the error of supposing that, all propositions consist of terms, so judgments consist of conceptions, and that there is a triple mental order conception, judgment, reasoning parallel to the triple linguistic order term, proposition, discourse. They overlooked the fact that man thinks long before he speaks, makes judgments which he does not express at all, or expresses them by interjections, names and phrases, before he uses regular propositions, and that he does not begin by conceiving and naming, and then proceed to believing and proposing. Feeling and sensation, involving believing or judging, come before conception and language. As conceptions are not always present in judgment as they are only occasional conditions, and as they are unfitted cause beliefs or judgments, and especially judgments of existence and as judgments both precede conceptions in sense and continue after them in inference, it follows that conceptions are not the constituents of judgment, and judgment is not a combination of conceptions. Is there then any analysis of judgment? Paradoxical as it may sound, the truth seems to be that primary judgment beginning as it does with the simplest feeling and sensation, is no a combination of two mental elements into one, but is a divisor of one sensible thing into the thing itself and its existence and the belief that it is determined as existing, e.g. that hot exists, […………….] a cause, namely sense, but no mental elements. Afterwards come judgments of complex sense, e.g. that the existing hot is burning or becoming more or less hot, &c. Thus there is a, combination of sensations causing the judgment; but the judgment is still a division of the sensible thing into itself and its being, and a belief that it is so determined. Afterwards follow judgments arising from more complex causes, e.g. memory, experience, inference. But however complicated these mental causes, there still remain these points common to all judgment:
(1) The mental causes of judgment are sense, memory, experience and inference; while conception is a condition of some judgments.
(2) A judgment is not a combination either of its causes or of its conditions, e.g. it is not a combination of sensations any more than of ideas.
(3) A judgment is a unitary mental act, dividing not itself but its object into the object itself and itself as determined, and signifying that it is so determined.
(4) A primary judgment is a judgment that a sensible thing is determined as existing; but later judgments are concerned with either existing things, or with ideas, or with words, and signify that they are determined in all sorts of ways.
(5) When a judgment is expressed by a proposition, the proposition expresses the results of the division by two terms, subject and predicate, and by the copula that what is signified by the subject is what is signified by the predicate; and the proposition is a combination of the two terms; e.g. border-war is evil.
(6) A complex judgment is a combination of two judgments, and may be copulative, e.g. you and I are men, or hypothetical, or disjunctive, &c.
Empirical logic, the logic of Aristotle and Bacon, is on the right way. It is the business of the logician to find the causes of the judgments which form the premises and the conclusions of inference, reasoning and science. What knowledge do we get by sense, memory and experience, the first mental causes of judgment? What is judgment, and what its various kinds? What is inference, how does it proceed by combining judgments as premises to cause judgments as conclusions, and what are its various kinds? How does inference draw conclusions more or less probable up to moral certainty? How does it by the aid of identification convert probable into necessary conclusions, which become necessary principles of de1ilonstration? How is categorical succeeded by conditional inference? What is scientific method as a system of inferences about definite subjects? How does inference become the source of error and fallacy? How does the whole process from sense to inference discover the real truth of judgments, which are true so far as they signify things known by sense, memory, experience and inference? These are the fundamental questions of the science of inference. Conceptual logic, on the other hand, is false from the start. It is not the first business of logic to direct us to form conceptions signified by terms, because sense is a prior cause of judgment and inference. It is not the second business of logic to direct us how out of conceptions to form judgments signified by propositions, because the real causes of judgments are sense, memory, experience and inference. It is, however, the main business of logic to direct us how out of judgments to form inferences signified by discourse; and this is the one point which conceptual logic has contributed to the science of inference. But why spoil the further mental analysis of inference by supposing that conceptions are constituents of judgment and therefore of inference, which thus becomes merely a complex combination of conceptions, an extension of ideas? The mistake has been to convert three operations of mind into three processes in a fixed order conception, judgment, inference. Conception and judgment are decisions: inference alone is a process from decisions to decision, from judgments to judgment. Sense not conception, is the origin of judgment. Inference is the process which from judgments about sensible things proceeds t judgments about things similar to sensible things. Though some conceptions are its conditions and some judgments in causes, inference itself in its conclusions causes many more judgments and conceptions. Finally, inference is an extension not of ideas, but of beliefs, at first about existing things, afterwards about ideas, and even about words; about anything - in short about which we think, in what is too fancifully called the universe of discourse.
Formal logic has arisen out of the narrowness of conceptual logic. The science of inference no doubt has to deal primarily with formal truth or the consistency of premises and conclusion […. ]of consistency become real rules of truth, when the premises are true and the consistent conclusion is therefore true. The science of inference again rightly emphasizes the formal thinking of the syllogism in which the combination of premises involves the conclusion. But the combinations of premises in analogical and inductive inference, although the combination does not involve the conclusion, yet causes us to infer it, and in so similar a way that the science of inference is not complete without investigating all the combinations which characterize different kinds of inference. The question of logic is how we infer in fact, as well as perfectly; and we cannot understand inference unless we consider inferences of probability of all kinds. Moreover, the study of analogical and inductive inference is necessary to that of the syllogism itself, because they discover the premises of syllogism. The formal thinking of syllogism alone is merely necessary consequence; but when its premises are necessary principles, its conclusions are not only necessary consequents but also necessary truths. Hence the manner in which induction aided by identification discovers necessary principles must be studied by the logician in order to decide when the syllogism can really arrive at necessary conclusions. Again, the science of inference has for its subject the form, or processes, of thought, but not its matter or objects. But it does not follow that it can investigate the former without the latter. Formal logicians say that, if they had to consider the matter, they must either consider all things, which would be impossible, or select some, which would be arbitrary. But there is an intermediate alternative, which is neither impossible nor arbitrary; namely, to consider the general distinctions and principles of all things; and without this general consideration of the matter the logician cannot know the form of thought, which consists in drawing inferences about things on these general principles. Lastly, the science of inference is not indeed the science of sensation, memory and experience, but at the same time it is the science of using those mental operations as data of inference; and, if logic does not show how analogical and inductive inferences directly, and deductive inferences indirectly, arise from experience, it becomes a science of mere thinking without knowledge.
Logic is related to all the sciences, because it considers the common inferences and varying methods used in investigating different subjects; But it is most closely related to the sciences of metaphysics and psychology, which form with it a triad of sciences. Metaphysics is the science of being in general, and therefore of the things which become objects apprehended by our minds. Psychology is the science of mind in general, and therefore of the mental operations, of which inference is one. Logic is the science of the processes of inference. These three sciences, of the objects of mind, of the operations of mind, of the processes used in the inferences of mind, are differently, but closely related, so that they are constantly confused. The real point is their interdependence, which is so intimate that one sign of great philosophy is a consistent metaphysics, psychology and logic. If the world of things is known to be partly material and partly mental, then the mind must have powers of sense and inference enabling it to know these things, and there must be processes of inference carrying us from and beyond the sensible to the insensible world of matter and mind. If the whole world of things is matter, operations and processes of mind are themselves material. If the whole world of things is mind, operations and processes of mind have only to recognize their like all the world over. It is clear then that a mans metaphysics and psychology must colour his logic. It is accordingly necessary to the logician to know beforehand the general distinctions and principles of things in metaphysics, and the mental operations of sense, conception, memory and experience in psychology, so as to discover the processes of inference from experience about things in logic.
The interdependence of this triad of sciences has sometimes led to their confusion. Hegel, having identified being with thought, merged metaphysics in logic. But he divided logic into objective and subjective, and thus practically confessed that there is one science of the objects and another of the processes of thought. Psychologists, seeing that inference is a mental operation, often extemporize a theory of inference to the neglect of logic. But we have a double consciousness of inference. We are conscious of it as one operation among many, and of its omnipresence, so to speak, to all the rest. But we are also conscious of the processes of the operation of inference. To a certain extent this second consciousness applies to other operations: for example, we are conscious of the process of association by which various mental causes recall ideas in the imagination. But how little does the psychologist know about the association of ideas, compared with what the logician has discovered about the processes of inference! The fact is that our primary consciousness of all mental operations is hardly equal to our secondary consciousness of the processes of the one operation of inference from premises to conclusions permeating long trains and pervading whole sciences. This elaborate consciousness of inferential process is the justification of logic as a distinct science, and is the first step in its method. But it is not the whole method of logic, which also and rightly considers the mental process necessary to language, without substituting linguistic for mental distinctions.
Nor are consciousness and linguistic analysis all the instruments of the logician. Logic has to consider the things we know, the, minds by which we know them from sense, memory and experience to inference, and the sciences which systematize and extend our knowledge of things; and having considered these facts, the logician must make such a science of inference as will explain the power and the poverty of human knowledge.
General Tendencies Of Modern Logic
There are several grounds for hope in the logic of our day. In the first place, it tends to take up an intermediate position between the extremes of Kant and Hegel. It does not, with the former, regard logic as purely formal in the sense of abstracting thought from being, nor does it follow the latter in amalgamating - metaphysics with logic by identifying being with thought. Secondly, it does not content itself with the mere formulae of thinking, but pushes forward to theories of method, knowledge and science; and it is a hopeful sign to find this epistemological spirit, to which England was accustomed by Mill, animating German logicians such as Lotze, Duhring, Schuppe, Sigwart and Wundt. Thirdly, there is a determination to reveal the psychological basis of logical processes, and not merely to describe them as they are in adult reasoning, but to explain also how they arise from simpler mental operations and primarily from sense. This attempt is connected with the psychological turn given to recent philosophy by Wundt and others, and is dangerous only so far as psychology itself is hypothetical. Unfortunately, however, these merits are usually connected with a less admirable characteristic contempt for tradition. Writing his preface to his second edition in 1888, Sigwart says:
Important works have appeared by Lotze, Schuppe, Wundt and Bradley, to name only the most eminent; and all start from the conception which has guided this attempt. That is, logic is grounded by them, not upon an effete tradition but upon a new investigation of thought as it actually is in its psychological foundations, in its significance for knowledge, and its actual operation in scientific methods. How strange! The spirit of every one of the three reforms above enumerated is an unconscious return to Aristotle's Organon. Aristotle's was a logic which steered, as Trendelenburg has shown, between Kantian formalism and Hegelian metaphysics; it was a logic which in the Analytics investigated the syllogism as a means to understanding knowledge and science: it was a logic which, starting from the psychological foundations of sense, memory and experience, built up the logical structure of induction and deduction on the profoundly Aristotelian principle that there is no process from universals without induction, and none by induction without sense. Wundt's comprehensive view that logic looks backwards to psychology and forward to epistemology was hundreds of years ago one of the many discoveries of Aristotle.
I. Judgment and Conception. The emphasis now laid on judgment, the recovery from Hume's confusion of beliefs with ideas and the association of ideas, and the distinction of the mental act of judging from its verbal expression in a proposition, are all healthy signs in recent logic. The most fundamental question, before proceeding to the investigation of inference, is not what we say but what we think in making the judgments which, whether we express them in propositions or not, are both the premises and the conclusion of inference; and, as this question has been diligently studied of late, but has been variously answered, it will be well to give a list of the more important theories of judgment as follows:
a. It expresses a relation between the content of two ideas, not a relation of these ideas (Lotze).
b. It is consciousness concerning the objective validity of a subjective combination of ideas, i.e. whether between the corresponding objective elements an analogous combination exists (Ueberweg).
c. It is the synthesis of ideas into unity and consciousness of their objective validity, not in the sense of agreement with external reality but in the sense of the logical necessity of their synthesis (Sigwart).
d. It is the analysis of an aggregate idea (Gesamvorstellung) into subject and predicate; based on a previous association of ideas, on relating and comparing, and on the apperceptive synthesis of an aggregate idea in consequence; but itself consisting in an apperceptive analysis of that aggregate idea; and requiring will in the form of apperception or attention (Wundt).
e. It requires an idea, because every object is conceived as well as recognized or denied; but it is itself an assertion of actual fact, every perception counts for a judgment, and every categorical is changeable into an existential judgment without change of sense (Brentano, who derives his theory from Mill except that, he denies the necessity of a combination of ideas, and reduces a categorical to an existential judgment).
f. It is a decision of the validity of an idea requiring will (Bergmann, following Brentano).
g. Judgment (Urteil) expresses that two ideas belong together: by-judgment (Beurtheilung) is the reaction of will expressing the validity or invalidity of the combination of ideas (Windelband, following Bergmann, but distinguishing the decision of validity from the judgment).
h. Judgment is consciousness of the identity or difference and of the causal relations of the given; naming the actual combinations of the data, but also requiring a priori categories of the understanding, the notions of identity, difference and causality, as principles of thought or laws, to combine the plurality of the given into a unity (Schuppe).
i. Judgment is the act which, refers an ideal content recognized as such to a reality beyond the act, predicating an idea of a reality, a what of a that; so that the subject is reality and the predicate the meaning of an idea, while the judgment refers the idea to reality by an identity of content (Bradley and Bosanquet).
k. Judgment is an assertion of reality, requiring comparison and ideas which render it directly expressible in words (Hobhouse, mainly following Bradley).
These theories are of varying value in proportion to their proximity to Aristotle's point that predication is about things, and to Mill's point that judgments and propositions are about things, not about ideas. The essence of judgment is belief that something is (or is not) determined, either as existing (e.g. I am, A centaur is not) or as something in particular (e.g. I am a man, I am not a monkey). Neither Mill, however, nor any of the later logicians whose theories we have quoted, has been able quite to detach judgment from conception. They all suppose that an idea, or ideas, is a condition of all judgment. But judgment starts from sensation (Empfindung) and feeling (Gefutil), and not from idea (Vorstellung). When I feel pleased or pained, or when I use my senses to perceive a pressure, a temperature, a flavour, an odour, a colour, a sound, or when I am conscious of feeling and perceiving, I cannot resist the belief that something sensible is present; and this belief that something exists is already a judgment, a judgment of existence, and, so far as it is limited to sense without inference a true judgment. It is a matter of words whether or not we should call this sensory belief a judgment; but it is no matter of choice to the logician, who regards all the constituents or inference as judgments; for the fundamental constituents are sensory beliefs, which are therefore judgments in the logical sense. Sense is the evidence of inference; directly of analogical and inductive, directly or indirectly of deductive, inference; and therefore, if logic refuses to include sensory beliefs among judgments, it will omit the fundamental constituents of inference, inference will no longer consist of judgments but of sensory beliefs plus judgments, and the second part of logic, the logic of judgment, the purpose of which is to investigate the constituents of inference, will be like Hamlet without the prince of Denmark. If, on the other hand, all the constituents of inference are judgments, there are judgments of sense; and the evidence of the senses means that a judgment of sense is true, while a judgment of inference is true so far as it is directly or indirectly concluded from judgments of sense.
Now a sensory judgment, e.g. that a sensible pressure is existing, is explained by none of the foregoing theories, because it requires nothing but sensation and belief. It requires no will, but is usually involuntary, for the stimulus forces ones attention, which is not always voluntary; not all judgment then requires will, as Wundt supposes. It requires no reference to reality beyond the sensible pressure, because it is merely a belief that this exists without inference of the external stimulus or any inference at all: not all judgment then requires the reference of subjective to objective supposed by Ueberweg, or the consciousness of logical necessity supposed by Sigwart. It requires in addition to the belief that something exists, no consideration as to whether the belief itself be true, because a man who feels pressure believes in the thing without further question about the belief: not all judgment then requires a decision of validity, as Bergmann supposes. It requires nothing beyond the sensation and belief in the given existence of the given pressure: not all judgment then requires categories of understanding, or notions of identity, difference and causality, or even of existence, such as Schuppe supposes. It requires no comparison in order to express it in words, for a judgment need not be expressed, and a sensory ,judgment of pressure is an irresistible belief that a real pressure exists, without waiting for words, or for a comparison which is wanted not to make a sensation a judgment, but to turn a judgment into language: not all judgment then requires comparison with a view to its expression, as supposed by Hobhouse.
Lastly, all the authors of the above-quoted theories err in supposing that all judgment requires conception; for even Mill thinks a combination of ideas necessary, and Brentano, who comes still nearer to the nature of sensory judgment when he says, Every perception counts for a judgment, yet thinks that an idea is necessary at the same time in order to understand the thing judged. In reality, the sensation and the belief are sufficient; when I feel a sensible pressure, I cannot help believing in its reality, and therefore judging that it is real, without any tertium quid - an idea of pressure, or of existence or of pressure existing - intervening between the sensation and the belief. Only after sensation has ceased does an idea, or representation of what is not presented, become necessary as a substitute for a sensation and as a condition not of the first judgment that there is, but of a second judgment that there was, something sensible. Otherwise there would be no judgment of sensible fact, for the first sensation would not give it, and the idea following the sensation would be still farther off. The sensory judgment then, which is nothing but a belief that at the moment of sense something sensible exists, is a proof that not all judgment requires conception, or synthesis or analysis of ideas, or decision about the content, or about the validity, of ideas, or reference of an ideal content to reality, as commonly, though variously, supposed in the logic of our day.
Not, however, that all judgment is sensory: after the, first judgments of sense follow judgments of memory, and memory requires ideas. Yet memory is not mere conception, as Aristotle and Mill after him, have perceived. To remember, we must have a present idea; but we must also have a belief that the thing, of which the idea is a representation. was (or was not) determined; and this belief is the memorial judgment. Origin ally such judgments arise from sensory judgments followed by ideas, and are judgments of memory after sense that something sensible existed, e.g. pressure existed: afterwards come judgments of memory after inference, e.g. Caesar was murdered. Finally, most judgments are inferential. These are conclusions which primarily are inferred from sensory and memorial judgments; and so far as inference starts from sense of something sensible in the present, and from memory after sense of something sensible in the past, and concludes similar things, inferential judgments are indirect beliefs in being and in existence beyond ideas. When from the sensible pressures between the parts of my mouth, which I feel and remember and judge that they exist and have existed, I infer another similar pressure (e.g. of the food which presses and is pressed by my mouth in eating), the inferential judgment with which I conclude is a belief that the latter exists as well as the former (e.g. the pressure of food without as well as the sensible pressures within).
Inference, no doubt, is closely involved with conception. So far as it depends on memory, an inferential judgment presupposes memorial ideas in its data; and so far as it infers universal classes and laws, it produces general ideas. But even so the part played by conception is quite subordinate to that of belief. in the first place, the remembered datum, from which an inference of pressure starts, is not the conceived idea, but the belief that the sensible pressure existed. Secondly, the conclusion in which it ends is not the general idea of a class, but the belief that a class, represented by a general idea, exists, and is (or is not) otherwise determined (e.g. that things pressing and pressed exist and move). Two things are certain about inferential judgment: one, that when inference is based on sense and memory, inferential judgment starts from a combination of sensory and memorial judgment, both of which are beliefs that things exist; the other, that in consequence inferential judgment is a belief that similar things exist. There are thus three primary judgments: judgments of sense, of memory after sense and of inference from sense. All these are beliefs in being and existence, and this existential belief is first in sense, and afterwards transferred to memory and inference. Moreover, it is transferred in the same irresistible way : frequently we cannot help either feeling pressure, or remembering it, or inferring it; and as there are involuntary sensation and attention, so there are involuntary memory and inference.
Again, in a primary judgment existence need not be expressed; but if expressed, it may be expressed either by the predicate, e.g. I exist, or by the subject, e.g. I who exist think. There are indeed differences between primary judgments, in that the sensory is a belief in present, the memorial in past, and the inferential in present, past and future existence. But these differences in detail do not alter the main point that all these are beliefs in the existing, in the real as opposed to the ideal, in actual things which are not ideas. In short, a primary judgment is a belief in something existing apart from our idea of it; and not because we have an idea of it, or by comparing an idea with, or referring an idea to, reality; but because we have a sensation of it, or a memory of it or an inference of it. Sensation, not conception, is the origin of judgment.
2. Different Significations of Being in different Kinds of Judgment. As Aristotle remarked both in the De Interpretatione and in the Sophistici Elenchi, not-being is thinkable does not mean not-being exists. In the latter treatise he added that it is a fallacia a dicto secundum quid ad dictum simpliciter to argue from the former to the latter; for, as he says, it is not the same thing to be something and to exist absolutely. Without realizing their debt to tradition, Herbart, Mill and recently Sigwart, have repeated Aristotle's separation of the copula from the verb of existence, as if it were a modern discovery that it is not the same as "exists". It may be added that they do not quite realize what the copula exactly signifies: it does not signify existence, but it does signify a fact, namely, that something is (or is not) determined, either absolutely in a categorical judgment, or conditionally in a conditional judgment.
Now we have seen that all primary judgments signify more than this fact; they are also beliefs in the existence of the thing signified by the subject. But, in the first place, primary judgments signify this existence never by the copula, but sometimes by the predicate, and sometimes by the subject; and, secondly, it does not follow that all judgments whatever signify existence. Besides inference of existence there is inference of non-existence, of things inconsistent with the objects of primary judgments. Hence secondary judgments, which no longer contain a belief that the thing exists, e.g. the judgment, not-being is thinkable, cited by Aristotle; the judgment, A square circle is impossible, cited by Herbart; the judgment, A centaur is a fiction of the poets, cited by Mill. These secondary judgments of non-existence are partly like and partly unlike primary judgments of existence. They resemble them in that they are beliefs in being signified by the copula. They are beliefs in things of a sort; for, after all, ideas and names are things; their objects, even though non-existent, are at all events things conceivable or nameable; and therefore we are able to make judgments that things, non-existent but conceivable or nameable, are (or are not) determined in a particular manner.
Thus the judgment about a centaur is the belief, A conceivable centaur is a fiction of the poets, and the judgment about a square circle is the belief, A so-called square circle is an impossibility. But, though beliefs that things of some sort are (or are not) determined, these secondary judgments fall short of primary judgments of existence. Whereas in a primary judgment there is a further belief, signified by subject or predicate, that the thing is an existing thing in the sense of being a real thing (e.g. a man), different from the idea of it as well as from the name for it; in a secondary judgment there is no further belief that the thing has any existence beyond the idea (e.g. a centaur), or even beyond the name (e.g. a square circle): though the idea or name exists, there is no belief that anything represented by idea or name exists. Starting, then, from this fundamental distinction between judgments of existence and judgments of non-existence, we may hope to steer our way between two extreme views which emanate from two important thinkers, each of whom has produced a flourishing school of psychological logic.
On the one hand, early in the 19th century Herbart started the view that a categorical judgment is never a judgment of existence, but always hypothetical; on the other hand, in the latter part of the century Brentano started the view that all categorical judgments are existential. The truth lies between these contraries. The view of Herbart and his school is contradicted by our primary judgments of and from sense, in which we cannot help believing existence; and it gives an inadequate account even of our secondary judgments in which we no longer indeed believe existence, but do frequently believe that a nonexistent thing is (or is not) somehow determined unconditionally. It is true, as Herbart says, that the judgment, A square circle is an impossibility, does not contain the belief, A square circle is existent ; but when he goes on to argue that it means, If a square circle is thought, the conception of impossibility must be added in thought, he falls into a non-sequitur. To be categorical, a judgment does not require a belief in existence, but only that something, existent or not, is (or is not) determined; and there are two quite different attitudes of mind even to a non-existent thing, such as a square circle, namely, unconditional and conditional belief. The judgment, A non-existent but so-called square circle is an impossibility, is an unconditional, or categorical judgment of non-existence, quite different from any hypothetical judgment, which depends on the conditions if it is thought, or if it exists, or any other if. On the other hand, the view of Brentano and his school is contradicted by these very categorical judgments of non-existence; and while it applies only to categorical judgments of existence, it does so inadequately. To begin with the latter objection, Brentano proposed to change the four Aristotelian forms of judgment, A, E, I, O, into the following existential forms:
A. There is not an immortal man.
E. There is not a live stone.
I. There is a sick man.
O. There is an unlearned man.
This reconstruction, which merges subject and predicate in one expression, in order to combine it with the verb of existence, is repeated in similar proposals of recent English logicians. Venn, in his Symbolic Logic, proposes the four forms, x~=o, xyo, xy>o, x3>o (where 5~ means not-y ), but only as alternative to the ordinary forms. Bradley says that SP is real attributes SP, directly or indirectly, to the ultimate reality, and agrees with Brentano that is never stands for anything but exists; while Bosanquet, who follows Bradley, goes so far as to define a categorical judgment as that which affirms the existence of its subject, or, in other words, asserts a fact. Now it is true that our primary judgments do contain a belief in existence; but they do not all contain it in the same way, but are beliefs sometimes that something is determined as existing, and sometimes that something existing is particularly determined. Brentano's forms do not express such a judgment of existence, as All existing men are mortal : nor does Bradley's form, Reality includes SP. Metaphysically, all realities are parts of one ultimate reality; but logically, even philosophers think more often only of finite realities, existing men, dogs, horses, &c.; and children know that their parents exist long before they apprehend ultimate reality. The normal form, then, of a judgment of existence is either S is a real P, or A real S is P. Hence the reconstruction of all categorical judgments by merging subject and predicate, either on Brentano's or on Bradley's plan, is a misrepresentation even of normal categorical judgments of existence.
Secondly, it is much more a misrepresentation of categorical judgments of non-existence. No existential form suits a judgment such as A centaur is a fiction, when we do not believe that there is a centaur, or that reality includes a centaur. As Mill pointed out, it cannot be implied that a centaur exists, since the very thing asserted is that the thing has no real existence. In a correspondence with Mill, Brentano rejoined that the centaur exists in imagination; Bradley says, inside our heads. According to one, then, the judgment becomes There is an imaginary centaur ; according to the other Reality includes an imaginary centaur. The rejoinder, however, though partly true, is not to the point. The idea of the centaur does exist in our imagination, and inside our heads, and the name of it in our mouths. But the point is that the centaur conceived and named does not exist beyond the idea of it and the name for it; it is not, like a man, a real thing which is neither the idea of it nor the name for it. No amount of subtlety will remove the difference between a categorical judgment of existence, e.g. An existing man is mortal, and a categorical judgment of non-existence, e.g. A conceivable centaur is a fiction, because in the former we believe and mean that the thing exists beyond the idea, and in the latter we do not. If, contrary to usage, we choose to call the latter a judgment of existence, there is no use in quarrelling about words; but we must insist that new terms must in that case be invented to express so fundamental a difference as that between judgments about real men and judgments about ideal centaurs.
So long, however, as we use words in the natural sense, and call the former judgments of existence, and the latter judgments of non-existence, then is will not be, as Bradley supposes, the same as exists, for we use is in both judgments, but exists only in the first kind. Bosanquet's definition of a categorical judgment contains a similar confusion. To assert a fact and to affirm the existence of a subject are not, as he makes out, the same thing: a judgment often asserts a fact and denies existence in the same breath, e.g. Jupiter is nonexistent. Here, as usual in logic, tradition is better than innovation. All categorical judgment is an unconditional belief in the fact, signified by the copula, that a thing of some sort is (or is not) determined; but some categorical judgments are also beliefs that the thing is an existing thing, signified either by the subject or by the predicate, while others are not beliefs that the thing exists at all, but are only beliefs in something conceivable, or nameable, or in something or other, without particularizing what. Judgment then always signifies being, but not always existence.
3. Particular and Universal Judgments. Aristotle, by distinguishing affirmative and negative, particular and universal, made the fourfold classification of judgments, A, E, I and O, the foundation both of opposition and of inference. With regard to inference, he remarked that a universal judgment means by all, not every individual we know, but every individual absolutely, so that, when it becomes a major premise, we know therein every individual universally, not individually, and often do not know a given individual individually until we add a minor premise in a syllogism. Whereas, then, a particular judgment is a belief that some, a universal judgment is a belief that all, the individuals of a kind or total of similar individuals, are similarly determined, whether they are known or unknown individuals.
Now, as we have already seen, what is signified by the subject may be existing or not, and in either case a judgment remains categorical so long as it is a belief without conditions. Thus, Some existing men are poets, All existing men are mortal, Some conceivable centaurs are human in their forequarters, All conceivable centaurs are equine in their hindquarters, are all categorical judgments, while the two first are also categorical judgments of existence. Nevertheless these obvious applications of Aristotelian traditions have been recently challenged, especially by Sigwart, who holds in his Logic (secs. 27, 36) that, while a particular is a categorical judgment of existence, a universal is hypothetical, on the ground that it does not refer to a definite number of individuals, or to individuals at all, but rather to general ideas, and that the appropriate form of all M is P is if anything is M it is P. This view, which has influenced not only German but also English logicians, such as Venn, Bradley and Bosanquet, destroys the fabric of inference, and reduces scientific laws to mere hypotheses. In reality, however, particular and universal judgments are too closely connected to have such different imports. In opposition, a categorical particular is the contradictory of a universal, which is also categorical, not hypothetical, e.g., not all M is P is the contradictory of all M is P, not of if anything is M it is P. In inference, a particular is an example of a universal which in its turn may become a particular example of a higher universal. For instance, in the history of mechanics it was first inferred from some that all terrestrial bodies gravitate, and then from these as some that all ponderable bodies, terrestrial and celestial, gravitate. How absurd to suppose that here we pass from a particular categorical to a universal hypothetical, and then treat this very conclusion as a particular categorical to pass to a higher universal hypothetical!
Sigwart, indeed, is deceived both about particulars and universals. On the one hand, some particulars are not judgments of existence, e.g. some imaginary deities are goddesses ; on the other hand, some universals are not judgments of non-existence, e.g. every existing man is mortal. Neither kind is always a judgment of existence, but each is sometimes the one and sometimes the other. In no case is a universal hypothetical, unless we think it under a condition; for in a universal judgment about the non-existing, e.g. about all conceivable centaurs, we do not think, If anything is a centaur, because we do not believe that there are any; and in a universal judgment about the existent, e.g. about all existing men, we do not think, If anything is a man, because we believe that there is a whole class of men existing at different times and places. The cause of Sigwart's error is his misconception of all. So far as he follows Aristotle in saying that all does not mean a definite number of individuals he is right; but when he says that we mean no individuals at all he deserts Aristotle and goes wrong. By all we mean every individual whatever of a kind; and when from the experience of sense and memory we start with particular judgments of existence, and infer universal judgments of existence and scientific laws, we further mean those existing individuals which we have experienced, and every individual whatever of the kind which exists. We mean neither a definite number of individuals, nor yet an infinite number, but an incalculable number, whether experienced or inferred to exist. We do not mean existing here and now, nor yet out of time and place, but at any time and place (semper et ubique), past, present and future being treated as simply existing, by what logicians used to call suppositio naturalis. We mean then by all existing every similar individual whatever, whenever, and wherever existing.
Hence Sigwart is right in saying that All bodies are extended means Whatever is a body is extended, but wrong in identifying this form with If anything is a body it is extended. Whatever is not if anything. For the same reason it is erroneous to confuse all existing with a general idea. Nor does the use of abstract ideas and terms make any difference. When Bosanquet says that in Heat is a mode of motion there is no reference to individual objects, but a pure hypothetical form which absolutely neglects the existence of objects, he falls far short of expressing the nature of this scientific judgment, for in his Theory of Heat Clerk Maxwell describes it as believing heat as it exists in a hot body to be in the form of kinetic energy. As Bacon would say, it is a belief that all individual bodies qua hot are individually but similarly moving in their particles. When, again, Bradley and Bosanquet speak of the universal as if it always meant one ideal content referred to reality, they forget that in universal judgments of existence, such as All men existing are mortal, we believe that every individually existing man dies his own death individually, though similarly to other men; and that we are thinking neither of ideas nor of reality; but of all existent individual men being individually but similarly determined. A universal is indeed one whole; but it is one whole of many similars, which are not the same with one another. This is indeed the very essence of distribution, that a universal is predicable, not singly or collectively, but severally and similarly of each and every individual of a kind, or total of similar individuals. So also the essence of a universal judgment is that every individual of the kind is severally but similarly determined.
Finally, a universal judgment is often existential; but whether it is so or not it remains categorical, so long as it introduces no hypothetical antecedent about the existence of the thing signified by the subject. It is true that even in universal judgments of existence there is often a hypothetical element; for example, All men are mortal contains a doubt whether every man whatever, whenever and wherever existing, must die. But this is only a doubt whether all the things signified by the subject are similarly determined as signified by the predicate, and not a doubt whether there are such things at all. Hence the hypothetical element is not a hypothetical antecedent If anything is a man, but an uncertain conclusion that All existing men are mortal. In other words, a categorical universal is often problematic, but a problematic is not the same as a hypothetical judgment.
4. The Judgment and the Proposition. Judgment in general is the mental act of believing that something is (or is not) determined. A proposition is the consequent verbal expression of such a belief, and consists in asserting that the thing as signified by the subject is (or is not) determined as signified by the predicate. But the expression is not necessary. Sensation irresistibly produces a judgment of existence without needing language. Children think long before they speak; and indeed, as mere vocal sounds are not speech, and as the apprehension that a word signifies a thing is a judgment, judgment is originally not an effect, but a cause of significant language. At any rate, even when we have learnt to speak, we do not express all we think, as we may see not only from the fewness of words known to a child, but also from our own adult consciousness. The principle of thought is to judge enough to conclude. The principle of language is to speak only so far as to understand and be understood.
Hence speech is only a curtailed expression of thought. Sometimes we express a whole judgment by one word, e.g. Fire! or by a phrase, e.g. What a fire! and only usually by a proposition. But even the normal proposition in the syllogistic form tertii adjacentis, with subject, predicate and copula, is seldom a complete expression of the judgment. The consequence is that the proposition, being different from a judgment arising after a judgment, and remaining an imperfect copy of judgment, is only a superficial evidence of its real nature. Fortunately, we have more profound evidences, and at least three evidences in all: the linguistic expression of belief in the proposition; the consciousness of what we mentally believe; and the analysis of reasoning, which shows what we must believe, and have believed, as data for inference. In these ways we find that a judgment is both different from, and more than, a proposition.
But recent logicians, although they perceive the difference, nevertheless tend to make the proposition the measure of the judgment. This makes them omit sensory judgments, and count only those which require ideas, and even general ideas expressed in general terms. Sigwart, for example, gives as instances of our most elementary judgments, This is Socrates, This is snow beliefs in things existing beyond ourselves which require considerable inferences from many previous judgments of sense and memory. Worse still, logicians seem unable to keep the judgment apart from the proposition. Herbart says that the judgment A is B does not contain the usually added thought that A is, because there is no statement of As existence; as if the statement mattered to the thought. So Sigwart, in order to reduce universals to hypotheticals, while admitting that existence is usually thought, argues that it is not stated in the universal judgment; so also Bosanquet. But in the judgment the point is not what we state, but what we think; and so long as the existence of A is added in thought, the judgment in question must contain the thought that A exists as well as that A is B, and therefore is a judgment that something is determined both as existing and in a particular manner. The statement only affects the proposition; and whenever we believe the existence of the thing, the belief in existence is part of the judgment thought, whether it is part of the proposition stated or not.
Here Sir William Hamilton did a real service to logic in pointing out that Logic postulates to be allowed to state explicitly in language all that is implicitly contained in the thought. Not that men should or can carry this logical postulate out in ordinary life; but it is necessary in the logical analysis of judgments, and yet logicians neglect it. This is why they confuse the categorical and the universal with the hypothetical. Taking the carelessly expressed propositions of ordinary life, they do not perceive that similar judgments are often differently expressed, e.g. I, being a man, am mortal, and If I am a man, I am mortal ; and conversely, that different judgments are often similarly expressed. In ordinary life we may say, All men are mortal, All centaurs are figments, All square circles are impossibilities, All candidates arriving five minutes late are fined (the last proposition being an example of the identification of categorical with hypothetical in Keynes's Formal Logic). But of these universal propositions the first imperfectly expresses a categorical belief in existing things, the second in thinkable things, and the third in nameable things, while the fourth is a slipshod categorical expression of the hypothetical belief, If any candidates arrive late they are fined The four judgments are different, and therefore logically the propositions fully expressing them are also different. The judgment, then, is the measure of the proposition, not the proposition the measure of the judgment.
On the other hand, we may go too far in the opposite direction, as Hamilton did in proposing the universal quantification of the predicate. If the quantity of the predicate were always thought, it ought logically to be always stated. But we only sometimes think it. Usually we leave the predicate indefinite, because, as long as the thing in question is (or is not) determined, it does not matter about other things, and it is vain for us to try to think all things at once. It is remarkable that in Barbara, and therefore in many scientific deductions, to think the quantity of the predicate is not to the point either in the premises or in the conclusion; so that to quantify the propositions, as Hamilton proposes, would be to express more than a rational man thinks and judges. In judgments, and therefore in propositions, indefinite predicates are the rule, quantified predicates the exception. Consequently, A E I O are the normal propositions with indefinite predicates; whereas propositions with quantified predicates are only occasional forms, which we should use whenever we require to think the quantity of the predicate, e.g. (1) in conversion, when we must think that all men are some animals, in order to judge that some animals are men; (2) in syllogisms of the 3rd figure, when the predicate of the minor premise must be particularly quantified in thought in order to become the particularly quantified subject of the conclusion; (3) in identical propositions including definitions, where we must think both that 1 + I are 2 and 2 are I + 1. But the normal judgment, and therefore the normal proposition, do not require the quantity of the predicate. It follows also that the normal judgment is not an equation. The symbol of equality (=) is not the same as the copula (is); it means is equal to, where equal to is part of the predicate, leaving is as the copula.
Now, in all judgment we think is, but in few judgments predicate equal to. In quantitative judgments we may think x=y, or, as Boole proposes, x =vy =~y, or, as Jevons proposes, x =xy, or, as Venn proposes, x which is not y~o; and equational symbolic logic is useful whenever we think in this quantitative way. But it is a byway of thought. In most judgments all we believe is that x is (or is not) y, that a thing is (or is not) determined, and that the thing signified by the subject is a thing signified by the predicate, hut not that it is the only thing, or equal to everything signified by the predicate. The symbolic logic, which confuses is with is equal to, having introduced a particular kind of predicate into the copula, falls into the mistake of reducing all predication to the one category of the quantitative; whereas it is more often in the substantial, e.g. I am a man, not I am equal to a man, or in the qualitative, e.g. I am white, not I am equal to white, or in the relative, e.g. I am born in sin, not I am equal to born in sin. Predication, as Aristotle saw, is as various as the categories of being. Finally, the great difficulty of the logic of judgment is to find the mental act behind the linguistic expression, to ascribe to it exactly what is thought, neither more nor less, and to apply the judgment thought to the logical proposition, without expecting to find it in ordinary propositions. Beneath Hamilton's postulate there is a deeper principle of logic. A rational being thinks only to the point, and speaks only to understand and be understood.
The nature and analysis of inference have been so fully treated in the Introduction that here we may content ourselves with some points of detail.
1. False Views of Syllogism arising from False Views of Judgment. The false views of judgment, which we have been examining, have led to false views of inference. On the one hand, having reduced categorical judgments to an existential form, Brentano proposes to reform the syllogism, with the results that it must contain four terms, of which two are opposed and two appear twice; that, when it is negative, both premises are negative; and that, when it is affirmative, one premise, at least, is negative. In order to infer the universal affirmative that every professor is mortal because he is a man, Brentano's existential syllogism would run as follows:
There is not a not-mortal man.
There is not a not-human professor.
There is not a not-mortal professor.
On the other hand, if on the plan of Sigwart categorical universals were reducible to hypotheticals, the same inference would be a pure hypothetical syllogism, thus: If anything is a man it is mortal. If anything is a professor it is a man, if anything is a professor it is mortal.
But both these unnatural forms, which are certainly not analyses of any conscious process of categorical reasoning, break down at once, because they cannot explain those moods in the third figure, e.g. Darapti, which reason from universal premises to a particular conclusion. Thus, in order to infer that some wise men are good from the example of professors, Brentano's syllogism would be the following non-sequitur:
There is not a not-good professor.
There is not a not-wise professor.
There is a wise good (non-sequitur).
So Sigwart's syllogism would be the following non-sequitur:
If anything is a professor, it is good.
If anything is a professor, it is wise.
Something wise is good (non-sequitur).
But as by the admission of both logicians these reconstructions of Darapti are illogical, it follows that their respective reductions of categorical universals to existentials and hypotheticals are false, because they do not explain an actual inference. Sigwart does not indeed shrink from this and greater absurdities; he reduces the first figure to the modus ponens and the second to the modus tollens of the hypothetical syllogism, and then, finding no place for the third figure, denies that it can infer necessity; whereas it really infers the necessary consequence of particular conclusions But the crowning absurdity is that, if all universals were hypothetical, Barbara in the first figure would become a purely) hypothetical syllogism - a consequence which seems innocent enough until we remember that all universal affirmative conclusions in all sciences would with their premises dissolve into mere hypothesis. No logic can be sound which leads to the following analysis:
If anything is a body it is extended.
If anything is a planet it is a body.
If anything is a planet it is extended.
Sigwart, indeed, has missed the essential difference between the categorical and the hypothetical construction of syllogisms. In a categorical syllogism of the first figure, the major premise, Every M whatever is P, is a universal, which we believe - on account of previous evidence without any condition about the thing signified by the subject M, which we simply believe sometimes to be existent (e.g. Every man existent), and sometimes not (e.g., Every centaur conceivable); and the minor premise, S is M, establishes no part of the major, but adds the evidence of a particular not thought of in the major at all But in a hypothetical syllogism of the ordinary mixed type, the first or hypothetical premise is a conditional belief, e.g. If anything is M it is P, containing a hypothetical antecedent, If anything is M, which is sometimes a hypothesis of existence (e.g. If anything is an angel ), and sometimes a hypothesis of fact (e.g. If an existing man is wise); and the second premise or assumption, Something is M, establishes part of the first, namely, the hypothetical antecedent, whether as regards existence (e.g. Something is an angel ), or as regards fact (e.g. This existing man is wise ). These very different relations of premises are obliterated by Sigwart's false reduction of categorical universals to hypotheticals. But even Sigwart's errors are outdone by Lotze, who not only reduces Every M is P so If S is M, S is P, but proceeds to reduce this hypothetical to the disjunctive, If S is M, S is Pi or P1 or P3, and finds fault with the Aristotelian syllogism because it contents itself with inferring S is P without showing what P. Now there are occasions when we want to reason in this disjunctive manner, to consider whether S is Pi or P3, and to conclude that S is a particular P ; but ordinarily all we want to know is that S is P ; e.g. in arithmetic, that 2 + 2 are 4, not any particular 4, and in life that all our contemporaries must die, without enumerating all their particular sorts of deaths. Lotze's mistake is the same as that of Hamilton about the quantification of the predicate, and that of those symbolists who held that reasoning ought always to exhaust all alternatives by equations. It is the mistake of exaggerating exceptional into normal forms of thought, and ignoring the principle that a rational being thinks only to the point.
2. Quasi-syllogisms. Besides reconstructions of the syllogistic fabric, we find in recent logic attempts to extend the figures of the syllogism beyond the syllogistic rules. An old error that we may have a valid syllogism from merely negative premises (ex omnibus negativis), long ago answered by Alexander and Boethius, is now revived by Lotze, Jevons and Bradley, who do not perceive that the supposed second negative is really an affirmative containing a not which can only be carried through the syllogism by separating it from the copula and attaching it to one of the extremes, thus:
The just are not unhappy (negative).
The just are not-recognized (affirmative).
Some not-recognized are not unhappy (negative).
Here the minor being the infinite term not-recognized in the conclusion, must be the same term also in the minor premise. Schuppe, however, who is a fertile creator of quasi-syllogisms has managed to invent some examples from two negative premises of a different kind:
(1) (2) (3)
No M is P. No M is P No P is M.
S is not M. S is not M. S is not M.
Neither S nor M is P S may be P S may be P.
But (1) concludes with a mere repetition, (2) and (3) with contingent may be, which, as Aristotle says, also may not be, and therefore nihil certo colligibur. The same answer applies to Schuppes supposed syllogisms from two particular premises:
Some M is P Some M is P.
Some S is M. Some M is S.
Some S may be P Some S may be P.
The only difference between these and the previous examples (2) and (3) is that, while those break the rule against two negative premises, these break that against undistributed middle. Equally fallacious are two other attempts of Schuppe to produce syllogisms from invalid moods:
(1) 1st Fig. (2) 2nd Fig.
All M is P P is M.
No S is M. S is M.
S may be P S is partially identical with P.
In the first the fallacy is the indifferent contingency of the conclusion caused by the non-sequitur from a negative premise to an affirmative conclusion; while the second is either a mere repetition of the premises if the conclusion means S is like P in being M, or, if it means S is P, a non-sequitur on account of the undistributed middle. It must not be thought that this trifling with logical rules has no effect. The last supposed syllogism, namely, that having two affirmative premises and entailing an undistributed middle in the second figure, is accepted by Wundt under the title Inference by Comparison (Vergleichungsschluss), and is supposed by him to he useful for abstraction and subsidiary to induction, and by Bosanquet to be useful for analogy. Wundt, for example, proposes the following premises:
Gold is a shining, fusible, ductile, simple body.
Metals are shining, fusible, ductile, simple bodies.
But to say from these premises, Gold and metal are similar in what is signified by the middle term, is a mere repetition of the premises; to say, further, that Gold may be a metal is a non-sequitur, because, the middle being undistributed, the logical conclusion is the contingent Gold may or may not be a metal, which leaves the question quite open, and therefore there is no syllogism. Wundt, who is again followed by Bosanquet, also supposes another syllogism in the third figure, under the title of Inference by Connection (Verbindungsschluss), to be useful for induction. He proposes, for example, the following premises : Gold, silver, copper, lead, are fusible.
Gold, silver, copper, lead, are metals.
Here there is no syllogistic fallacy in the premises; but the question is what syllogistic conclusion can be drawn, and there is only one which follows without an illicit process of the minor, namely, Some metals arc fusible. The moment we stir a step further with Wundt in the direction of a more general conclusion (cm allgemeinerer Satz), we cannot infer from the premises the conclusion desired by Wundt, Metals and fusible are connected ; nor can we infer All metals are fusible, nor Metals are fusible, nor Metals may be fusible, nor All metals may be fusible, nor any assertory conclusion, determinate or indeterminate, but the indifferent contingent, All metals may or may not be fusible, which leaves the question undecided, so that there is no syllogism. We do not mean that in Wundt's supposed inferences of relation by comparison and connection the premises are of no further use; but those of the first kind are of no syllogistic use in the second figure, and those of the second kind of no syllogistic use beyond particular conclusions in the third figure. What they really are in the inferences proposed by Wundt is not premises for syllogism, hut data for induction parading as syllogism. We must pass the same sentence on Lotze attempt to extend the second figure of the syllogism for inductive purposes, thus
S is M.
Q is M.
R is M.
Every ~, which is common to S Q, R, is M.
We could not have a more flagrant abuse of the rule Ne eslo plus ?nlnusquc in conclusione quam in praemissis. As we see from Lotze's own defence, the conclusion cannot he drawn without. another premise or premises to the effect that S, Q, R, are ~, and ~ is the one real subject of M. But how is all this to be got into the second figure? Again, Wundt and B. Erdmann propose new moods of syllogism with convertible premises, containing definitions and equations. Wundt's Logic has the following forms:
(1) 1st Fig. (2) 2nd Fig. (3) 3rd Fig.
Only M is P. xy yx.
No S is M. z=y. y=z.
No S is P x=z xz.
Now, there is no doubt that, especially in mathematical equations, universal conclusions are obtainable from convertible premises expressed in these ways. But the question is how the premises must be thought, and they must be thought in the converse way to produce a logical conclusion. Thus, we must think in (I) All P is M to avoid illicit process of the major, in (2) All y is z to avoid undistributed middle, in (3) All x is y to avoid illicit process of the minor. Indeed, it is the very essence of a convertible judgment to think it in both orders, and especially to think it in the order necessary to an inference from it. Accordingly, however expressed, the syllogisms quoted above are, as thought, ordinary syllogisms, (1) being Camestres in the second figure, (2) and (3) Barbara in the first figure. Aristotle, indeed, was as well aware as German logicians of the force of convertible premises; but he was also aware that they require no special syllogisms, and made it a point that, in a syllogism from a definition, the definition is the middle, and the definitum the major in a convertible major premise of Barbara in the first figure, e.g.
The interposition of an opaque body is (essentially) deprivation of light.
The moon suffers the interposition of the opaque earth.
The moon suffers deprivation of light.
It is the same with all the recent attempts to extend the syllogism beyond its rules, which are not liable to exceptions, because they follow from the nature of syllogistic inference from universal to particular. To give the name of syllogism to inferences which infringe the general rules against undistributed middle, illicit process, two negative premises, non -sequitur from negative to affirmative, and the introduction of what is not in the premises into the conclusion, and which consequently infringe the special rules against affirmative conclusions in the second figure, and against universal conclusions in the third figure, is to open the door to fallacy, and at best to confuse the syllogism with other kinds of inference, without enabling us to understand any one kind.
3. Analytic and Synthetic Deduction. Alexander the Commentator defined synthesis as a progress from principles to consequences, analysis as a regress from consequences to principles; and Latin logicians preserved the same distinction between the progressus a principiis ad principiata, and the regressus a principiatis ad principia. No distinction is more vital in the logic of inference in general and of scientific inference in particular; and yet none has been so little understood, because though analysis is the more usual order of discovery, synthesis is that of instruction, and therefore, by becoming more familiar, tends to replace and obscure the previous analysis. The distinction, however, did not escape Aristotle, who saw that a progressive syllogism can be reversed thus:
i. Progression. (1) (2)
All M is P. All P is M. All S is P:
All S is M. All S is P. All M is S.
All S is P All S is M. All M is P.
Proceeding from one order to the other, by converting one of the premises, and substituting the conclusion as premise for the other premise, so as to deduce the latter as conclusion, is what he calls circular inference; and he remarked that the process is fallacious unless it contains propositions which are convertible, as in mathematical equations. Further, he perceived that the difference between the progressive and regressive orders extends from mathematics to physics, and that there are two kinds of syllogism: one progressing a priori from real ground to consequent fact, and the other regressing a posteriori from consequent fact to real ground. For example, as he says, the sphericity of the moon is the real ground of the fact of its light waxing; but we can deduce either from the other, as follows:
I. Progression. 2. Regression.
What is spherical waxes What waxes is spherical.
The moon is spherical. The moon waxes.
.The moon waxes The moon is spherical.
These two kinds of syllogism are synthesis and analysis in the ancient sense. Deduction is analysis when it is regressive from consequence to real ground, as when we start from the proposition that the angles of a triangle are equal to two right angles and deduce analytically that therefore (I) they are equal to equal angles made by a straight line standing on another straight line, and (2) such equal angles are two right angles. Deduction is synthesis when it is progressive from real ground to consequence, as when we start from these two results of analysis as principles and deduce synthetically the proposition that therefore the angles of a triangle are equal to two right angles, in the order familiar to the student of Euclid. But the full value of the ancient theory of these processes cannot be appreciated until we recognize that as Aristotle planned them Newton used them. Much of the Principia consists of synthetical deductions from definitions and axioms. But the discovery of the centripetal force of the planets to the sun is an analytic deduction from the facts of their motion discovered by Kepler to their real ground, and is so stated by Newton in the first regressive order of Aristotle P-M, S-P, S-M. Newton did indeed first show synthetically what kind of motions by mechanical laws have their ground in a centripetal force varying inversely as the square of the distance (all P is M); but his next step was, not to deduce synthetically the planetary motions, but to make a new start from the planetary motions as facts established by Kepler's laws and as examples of the kind of motions in question (all S is P); and then, by combining these two premises, one mechanical and the other astronomical, he analytically deduced that these facts of planetary motion have their ground in a centripetal force varying inversely as the squares of the distances of the planets from the sun (all S is M). (See Principia I. prop. 2-4 coroll. 6; III. Phaenomena, 4-5; prop. 2.) What Newton did, in short, was to prove by analysis that the planets, revolving by Kepler's astronomical laws round the sun, have motions such as by mechanical laws are consequences of a centripetal force to the sun. This done, as the major is convertible, the analytic order P-M, S-P, S-M was easily inverted into the synthetic order M-P, S-M, S-P; and in this progressive order the deduction as now taught begins with the centripetal force of the sun as real ground, and deduces the facts of planetary motion as consequences. Thereupon the Newtonian analysis which preceded this synthesis, became forgotten; until at last Mill in his Logic, neglecting the Principia, had the temerity to distort Newton's discovery, which was really a pure example of analytic deduction, into a mere hypothetical deduction; as if the author of the saying Hypotheses non fingo started from the hypothesis of a centripetal force to the sun, and thence deductively explained the facts of planetary motion, which reciprocally verified the hypothesis. This gross misrepresentation has made hypothesis a kind of logical fashion. Worse still, Jevons proceeded to confuse analytic deduction from consequence to ground with hypothetical deduction from ground to consequence under the common term inverse deduction. Wundt attempts, but in vain, to make a compromise between the old and the new. He re-defines analysis in the very opposite way to the ancients; whereas they defined it as a regressive process from consequence to ground, according to Wundt it is a progressive process of taking for granted a proposition and deducing a consequence, which being true verifies the proposition. He then divides it into two species: one categorical, the other hypothetical. By the categorical he means the ancient analysis from a given proposition to more general propositions. By the hypothetical he means the new-fangled analysis from a given proposition to more particular propositions, i.e. from a hypothesis to consequent facts. But his account of the first is imperfect, because in ancient analysis the more general propositions, with which it concludes, are not mere consequences, but the real grounds of the given proposition; while his addition of the second reduces the nature of analysis to the utmost confusion, because hypothetical deduction is progressive from hypothesis to consequent facts whereas analysis is regressive from consequent facts to real ground. There is indeed a sense in which all inference is from ground to consequence, because it is from logical ground (principium cognoscendi) to logical consequence. But in the sense in which deductive analysis is opposed to deductive synthesis, analysis is deduction from real consequence as logical ground principiatum as principium) cognoscendi) to real ground (principium essendi), e.g. from the consequential facts of planetary motion to their real ground, i.e. centripetal force to the sun. Hence Sigwart is undoubtedly right in distinguishing analysis from hypothetical deduction, for which he proposes the name reduction. We have only further to add that many scientific discoveries about sound, heat, light, colour and so forth, which it is the fashion to represent as hypotheses to explain facts, are really analytical deductions from the facts to their real grounds in accordance with mechanical laws. Recent logic does scant justice to scientific analysis.
4. Induction. As induction is the process from particulars to universals, it might have been thought that it would always have been opposed to syllogism, in which one of the rules is against using particular premises to draw universal conclusions. Yet such is the passion for one type that from Aristotle's time till now constant attempts have been made to reduce induction to syllogism. Aristotle himself invented an inductive syllogism in which the major (P) is to be referred to the middle (M) by means of the minor (5), thus:
A, B, C magnets (S) attract iron (P).
A, B, C magnets (S) are all magnets whatever (M).
All magnets whatever (M) attract iron (P).
As the second premise is supposed to be convertible, he reduced the inductive to a deductive syllogism as follows:
Every S is P.
Every S is P.
Every S is M (convertibly).
Every M is S. .~. Every M is P. :. Every M is P.
I In the reduced form the inductive syllogism was described by Aldrich as Syllogismus in Barbara cujus minor (i.e. every M is S) reticetur. Whately, on the other hand, proposed an inductive syllogism with the major suppressed, that is, instead of the minor premise above, he supposed a major premise, Whatever belongs to A, B, C magnets belongs to all. Mill thereupon supposed a still more general premise, an assumption of the uniformity of nature. Since Mill's time, however, the logic of induction tends to revert towards syllogisms more like that of Aristotle. Jevons supposed induction to be inverse deduction, distinguished from direct deduction as analysis from synthesis, e.g. as division from multiplication; but he really meant that it is a deduction from a hypothesis of the law of a cause to particular effects which, being true, verify the hypothesis. Sigwart declares himself in agreement with Jevons; except that, being aware of the difference between hypothetical deduction and mathematical analysis, and seeing that, whereas analysis (e.g. in division) leads to certain conclusions, hypothetical deduction is not certain of the hypothesis, he arrives at the more definite view that induction is not analysis proper but hypothetical deduction, or reduction, as he proposes to call it. Reduction he defines as the framing of possible premises for given propositions, or the construction of a syllogism when the conclusion and one premise is given. On this view induction becomes a reduction in the form: all M is P (hypothesis), S is M (given),
S is P (given). The views of Jevons and Sigwart are in agreement in two main points. According to both, induction, instead of inferring from A, B, C magnets the conclusion Therefore all magnets attract iron, infers from the hypothesis, Let every magnet attract iron, to A, B, C magnets, whose given attraction verifies the hypothesis. According to both again, the hypothesis of a law with which the process starts contains more than is present in the particular data: according to Jevons, it is the hypothesis of a law of a cause from which induction deduces particular effects; and according to Sigwart, it is a hypothesis of the ground from which the particular data necessarily follow according to universal laws. Lastly, Wundt's view is an interesting piece of eclecticism, for he supposes that induction begins in the form of Aristotle's inductive syllogism, SP, SM, MP, and becomes an inductive method in the form of Jevons's inverse deduction, or hypothetical deduction, or analysis, MP, SM, SP. In detail, he supposes that, while an inference by comparison, which he erroneously calls an affirmative syllogism in the second figure, is preliminary to induction, a second inference by connection, which he erroneously calls a syllogism in the third figure with an indeterminate conclusion, is the inductive syllogism itself. This is like Aristotle's inductive syllogism in the arrangement of terms; but, while on the one hand Aristotle did not, like Wundt, confuse it with the third figure, on the other hand Wundt does not, like Aristotle, suppose it to be practicable to get inductive data so wide as the convertible premise, All S is M, and all M is S, which would at once establish the conclusion, All M is P. Wundt's point is that the conclusion of the inductive syllogism is neither so much as all, nor so little as some, but rather the indeterminate M and P are connected. The question therefore arises, how we are to discover All M is P, and this question Wundt answers by adding an inductive method, which involves inverting the inductive syllogism in the style of Aristotle into a deductive syllogism from a hypothesis in the style of Jevons, thus:
S is P. Every M is P.
S is M. S is M.
- -. M and P are connected. . -. S is P.
He agrees with Jevons in calling this second syllogism analytical deduction, and with Jevons and Sigwart in calling it hypothetical deduction. It is, in fact, a common point of Jevons, Sigwart and Wundt that the universal is not really a conclusion inferred from given particulars, but a hypothetical major premise from which given particulars are inferred, and that this major contains presuppositions of causation not contained in the particulars.
It is noticeable that Wundt quotes Newtons discovery of the centripetal force of the planets to the sun as an instance of this supposed hypothetical, analytic, inductive method; as if Newtons analysis were a hypothesis of the centripetal force to the sun, a deduction of the given facts of planetary motion, and a verification of the hypothesis by the given facts, and as if such a process of hypothetical deduction could be identical with either analysis or induction. The abuse of this instance of Newtonian analysis betrays the whole origin of the current confusion of induction with deduction. One confusion has led to another. Mill confused Newtons analytical deduction with hypothetical deduction; and thereupon Jevons confused induction with both. The result is that both Sigwart and Wundt transform the inductive process of adducing particular examples to induce a universal law into a deductive process of presupposing a universal law as a ground to deduce particular consequences. But we can easily extricate ourselves from these confusions by comparing induction with different kinds of deduction. The point about induction is that it starts from experience, and that, though in most classes we ,can experience only some particulars individually, yet we infer all. Hence induction cannot be reduced to Aristotle's inductive syllogism, because experience cannot give the convertible premise, Every S is M, and every M is S; that All A, B, C are magnets is, but that All magnets are A, B, C is not, a fact of experience. For the same reason induction cannot be reduced to analytical deduction of the second kind in the form, SP, MS, :. MP; because, though both end in a universal conclusion, the limits of experience prevent induction from such inference as:
Every experienced magnet attracts iron.
Every magnet whatever is every experienced magnet.
Every magnet whatever attracts iron.
Still less can induction be reduced to analytical deduction of the first kind in the form P--M, SP, . . SM, of which Newton has left so conspicuous an example in his Principia. As the example shows, that analytic process starts from the scientific knowledge of a universal and convertible law (every M is P, and every P is M), e.g. a mechanical law of all centripetal force, and ends in a particular application, e.g. this centripetal force of planets to the sun. But induction cannot start from a known law. Hence it is that Jevons, followed by Sigwart and Wundt, reduces it to deduction from a hypothesis in the form Let every M be P, S is M, . . S is P. There is a superficial resemblance between induction and this hypothetical deduction. Both in a way use given particulars as evidence. But in induction the given particulars are the evidence by which we discover the universal, e.g. particular magnets attracting iron are the origin of an inference that all do; in hypothetical deduction, the universal is the evidence by which we explain the given particulars, as when we suppose undulating aether to explain the facts of heat and light. In the former process, the given particulars are the data from which we infer the universal; in the latter, they are only the consequent facts by which we verify it. Or rather, there are two uses of induction: inductive discovery before deduction, and inductive verification after deduction. But neither use of induction is the same as the deduction itself:
the former precedes, the latter follows it. Lastly, the theory of Mill, though frequently adopted, e.g. by B. Erdmann, need not detain us long. Most inductions are made without any assumption of the uniformity of nature; for, whether it is itself induced, or a priori or postulated, this like every assumption is a judgment, and most men are incapable of judgment on so universal a scale, when they are quite capable of induction. The fact is that the uniformity of nature stands to induction as the axioms of syllogism do to syllogism; they are not premises, but conditions of inference, which ordinary men use spontaneously, as was pointed out in Physical Realism, and afterwards in Venn's Empirical Logic. The axiom of contradiction is not a major premise of a judgment: the dictum de omni et nullo is not a major premise of a syllogism: the principle of uniformity is not a major premise of an induction. Induction, in fact, is no species of deduction; they are opposite processes, as Aristotle regarded them except in the one passage where he was reducing the former to the latter, and as Bacon always regarded them. But it is easy to confuse them by mistaking examples of deduction for inductions. Thus Whewell mistook Kepler's infertile that Mars moves in an ellipse for an induction, though it required the combination of Tycho's and Kepler's observations, as a minor, with the laws of conic sections discovered by the Greeks, as a major, premise. Jevons, in his Principles of Science, constantly makes the same sort of mistake. For example, the inference from the similarity between solar spectra and the spectra of various gases on the earth to the existence of similar gases in the sun, is called by him an induction; but it really is an analytical deduction from effect to cause, thus:
Such and such spectra are effects of various gases.
Solar spectra are such spectra.
Solar spectra are effects of those gases.
In the same way, to infer a machine from hearing the regular tick of a clock, to infer a player from finding a pack of cards arranged in suits, to infer a human origin of stone implements, and all such inferences from patent effects to liftent [?] causes, though they appear to Jevons to be typical inductions, are really deductions which, besides the minor premise stating the particular effects, require a major premise discovered by a previous induction and stating the general kind of effects of a general kind of cause. B. Erdmann, again, has invented an induction from particular predicates to a totality of predicates which he calls erganzende Induction, giving as an example, This body has the colour, extensibility and specific gravity of magnesium; therefore it is magnesium. But this inference contains the tacit major, What has a given colour, &c., is magnesium, and is a syllogism of recognition. A deduction is often like an induction, in inferring from particulars; the difference is that deduction combines a law in the major with the particulars in the minor premise, and infers syllogistically that the particulars of the minor have the predicate of the major premise, whereas induction uses the particulars simply as instances to generalize a law. An infallible sign of an induction is that the subject and predicate of the universal conclusion are merely those of the particular instances generalized; e.g. These magnets attract iron, .. all do.
This brings us to another source of error. As we have seen, Jevons, Sigwart and Wundt all think that induction contains a belief in causation, in a cause, or ground, which is not present in the particular facts of experience, but is contributed by a hypothesis added as a major premise to the particulars in order to explain them by the cause or ground. Not so; when an induction is causal, the particular instances are already, beliefs in particular causes, e.g. My right hand is exerting pressure reciprocally with my left, A, B, C magnets attract iron ; and the problem is to generalize these causes, not to introduce them. Induction is not introduction. It would make no difference to the form of induction, if, as Kant thought, the notion of causality is a priori; for even Kant thought that it is already contained in experience. But whether Kant be right or wrong, Wundt and his school are decidedly wrong in supposing supplementary notions which are not contained in experience itself, but are gained by a process of logical treatment of this experience ; as if our behalf in causality could be neither a posteriori nor a priori, but beyond experience wake up in a hypothetical major premise of induction. Really, we first experience that particular causes have particular effects; then induce that causes similar to those have effects similar to these; finally, deduce that when a particular cause of the kind occurs it has a particular effect of the kind by synthetic deduction, and that when a particular effect of the kind occurs it has a particular cause of the kind by analytic deduction with a convertible premise, as when Newton from planetary motions, like terrestrial motions, analytically deduced a centripetal force to the sun like centripetal forces to the earth. Moreover, causal induction is itself both synthetic and analytic: according as experiment combines elements into a compound, or resolves a compound into elements, it is the origin of a synthetic or an analytic generalization. Not, however, that all induction is causal; but where it is not, there is still less reason for making it a deduction from hypothesis. When from the fact that the many crows in our experience are black, we induce the probability that all crows whatever are black, the belief in the particulars is quite independent of this universal. How then can this universal be called, as Sigwart, for example, calls it, the ground from which these particulars follow? I do not believe that the crows I have seen are black because all crows are black, but vice versa. Sigwart simply inverts the order of our knowledge. In all induction, as Aristotle said, the particulars are the evidence, or ground Of our knowledge (principium cognoscendi), of the universal. In causal induction, the particulars further contain the cause, or ground of the being (principium essendi), of the effect, as well as the ground of our inducing the law. In all induction the universal is the conclusion, in none a major premise, and in none the ground of either the being or the knowing of the particulars. Induction is generalization. It is not syllogism in the form of Aristotle's or Wundt's inductive syllogism, because, though starting only from some particulars, it concludes with a universal it is not syllogism in the form called inverse deduction by Jevons reduction by Sigwart, inductive method by Wundt, because it often uses particular facts of causation to infer universal law of causation; it is not syllogism in the form of Mills syllogism from a belief in uniformity of nature, because few men have believed in uniformity, but all have induced from particular to universals. Bacon alone was right in altogether opposing induction to syllogism, and in finding inductive rules for the inductive process from particular instances of presence, absence in similar circumstances, and comparison.
5. Inference in General. There are, as we have seen, three types [of inference]: syllogism, induction and analogy. Different as they are, the three kinds have something in common: first, they are all processes from similar to similar; secondly, they all consist in combining two judgments so as to cause a third, whether expressed in so many propositions or not; thirdly, as a judgment is a belief in being, they all proceed from premises which are beliefs in being to a conclusion which is a belief in being. Nevertheless, simple as this account appears, it is opposed in every point to recent logic. In the first place, the point of Bradley's logic is that similarity is not a principle which works. What operates is identity, and that identity is a universal. This view makes inference easy: induction is all over before it begins; for, according to Bradley, every one of the instances is already a universal proposition; and it is not a particular fact or phenomenon at all, so that the moment you observe that this magnet attracts iron, you ipso facto know that every magnet does so, and all that remains for deduction is to identify a second magnet as the same with the first, and conclude that it attracts iron.
In dealing with Bradley's works we feel inclined to repeat what Aristotle says of the discourses of Socrates: they all exhibit excellence, cleverness, novelty and inquiry, but their truth is a difficult matter; and the Socratic paradox that virtue is knowledge is not more difficult than the Bradleian paradox that as two different things are the same, inference is identification. The basis of Bradley's logic is the fallacious dialectic of Hegel's metaphysics, founded on the supposition that two things, which are different, but have something in common, are the same. For example, according to Hegel, being and not-being are both indeterminate and therefore the same. If, says Bradley, A and B, for instance, both have lungs or gills, they are so far the same. The answer to Hegel is that being and not-being are at most similarly indeterminate, and to Bradley that each animal has its own different lungs, whereby they are only similar. If they were the same, then in descending, two things, one of which has healthy and the other diseased lungs, would be the same; and in ascending, two things, one of which has lungs and the other has not, but both of which have life, e.g. plants and animals, would be so far the same. There would be no limit to identity either downwards or upwards; so that a man would be the same as a man-of-war, and all things would be the same thing, and not different parts of one universe. But a thing which has healthy lungs and a thing which has diseased lungs are only similar individuals numerically different. Each individual thing is the same only with itself, although related to other things; and each individual of a class has its own individual, though similar, attributes.
The consequence of this true metaphysics to logic is twofold: on the one hand, one singular or particular judgment, e.g. this magnet attracts iron, is not another, e.g. that magnet attracts iron, and neither is universal; on the other hand, a universal judgment, e.g. every magnet attracts iron, means, distributively, that each individual magnet exerts its individual attraction, though it is similar to other magnets exerting similar attractions. A universal is not one identical point, but one distributive whole. Hence in a syllogism, a middle term, e.g. magnets, is absolutely the same, not in the sense of one identical point making each individual the same as any other, as Bradley supposes, but only in the sense of one whole class, or total of many similar individuals, e.g. magnets each of which is separately though similarly a magnet, not magnet in general, Hence also induction is a real process, because when we know that this individual magnet attracts iron, we are very far from knowing that all alike do so similarly; and the question of inductive logic, how we get from some similars to al similars, remains, as before, a difficulty, but not to be solved b) the fallacy that inference is identification.
Secondly, a subordinate point in Bradley's logic is that there are inferences which are not syllogisms; and this is true. But when he goes-on to propose, as a complete independent inference A is to the right of B, B is to the right of C, therefore A is t the right of C, he confuses two different operations. When A B and C are objects of sense, their relative positions are matters not of inference, but of observation; when they are not, there i an inference, but a syllogistic inference with a major premise induced from previous observations, whenever of three things the first is to the right of the second, and the second to the right of the third, the first is to the right of the third. To reply that this universal judgment is not expressed, or that its expression is cumbrous, is no answer, because, whether expressed or not, it is required for the thought. As Aristotle puts it, the syllogism is directed not to the outer, but to the inner discourse, or as we should say, not to the expression but to the thought, not to the proposition but to the judgment, and to the inference not verbally but mentally. Bradley seems to suppose that the major premise of a syllogism must be explicit, or else is nothing at all. But it is often thought without being expressed, and to judge the syllogism by its mere explicit expression is to commit an ignoratio elenchi; for it has been known all along that we express less than we think, and the very purpose of syllogistic logic is to analyse the whole thought necessary to the conclusion. In this syllogistic analysis two points must always be considered: one, that we usually use premises in thought which we do not express; and the other, that we sometimes use them unconsciously, and therefore infer and reason unconsciously, in the manner excellently described by Zeller in his l?ortrage, iii. pp. 249-255. Inference is a deeper thinking process from judgments to judgment, which only occasionally and partially emerges in the linguistic process from propositions to proposition. We may now then reassert two points about inference against Bradley's logic: the first, that it is a process from similar to similar, and not a process of identification, because two different things are not at all the same thing; the second, that it is the mental process from judgments to judgment rather than the linguistic process from propositions to proposition, because, besides the judgments expressed in propositions, it requires judgments which are not always expressed, and are sometimes even unconscious.
Our third point is that, as a process of judgments, inference is a process of concluding from two beliefs in being to another belief in being, and not an ideal construction, because a judgment does not always require ideas, but is always a belief about things, existing or not. This point is challenged by all the many ideal theories of judgment already quoted. If, for example, judgment were an analysis of an aggregate idea as Wundt supposes, it would certainly be true with him to conclude that as judgment is an immediate, inference is a mediate, reference of the members of an aggregate of ideas to one another. But really a judgment is a belief that something, existing, or thinkable, or nameable or what not, is (or is not) determined; and inference is a process from and to such beliefs in being. Hence the fallacy of those who, like Bosanquet, or like Paulsen in his Einieitung in die Philosophie, represent the realistic theory of inference as if it meant that knowledge starts from ideas and then infers that ideas are copies of things, and who then object, rightly enough, that we could not in that case compare the copy with the original, but only be able to infer from idea to idea.
But there is another realism which holds that inference is a process neither from ideas to ideas, nor from ideas to things, but from beliefs to beliefs, from judgments about things in the premises to judgments about similar things in the conclusion. Logical inference never goes through the impossible process of premising nothing but ideas, and concluding that ideas are copies of things. Moreover as we have shown, our primary judgments of sense are beliefs founded on sensations without requiring ideas, and are beliefs not merely that something is determined, but that it is deter mined as existing; and, accordingly, our primary inference from these sensory judgments of existence are inferences that other things beyond sense are similarly determined as existing. First press your lips together and then press a pen between them: you will not be conscious of perceiving any ideas: you will be conscious first of perceiving one existing lip exerting pressure reciprocally with the other existing lip; then, on putting the pen between your lips, of perceiving each lip similarly exerting pressure, but not with the other; and consequently of inferring that each existing lip is exerting pressure reciprocally with another existing body, the pen.
Inference then, though it is accompanied by ideas, is not an ideal construction, nor a process from idea to idea, nor a process from idea to thing, but a process from direct to indirect beliefs in things, and originally in existing things. Logic cannot, it is true, decide what these things are, nor what the senses know about them, without appealing to metaphysics and psychology. But, as the science of inference, it can make sure that inference, on the one hand, starts from sensory judgments about sensible things and logically proceeds to inferential judgments about similar things beyond sense, and, on the other hand, cannot logically go beyond the similar. These are the limits within which logical inference works, because its nature essentially consists in proceeding from two judgments to another about similar things, existing or not.
6. Truth. Finally, though sensory judgment is always true of its sensible object, inferential judgments are not always true, but are true so far as they are logically inferred, however indirectly, from sense; and knowledge consists of sense, memory after sense and logical inference from sense, which, we must remember, is not merely the outer sense of our five senses, but also the inner sense of ourselves as conscious thinking persons. We come then at last to the old question. What is truth? Truth proper, as Aristotle said in the Metaphysics, is in the mind: it is not being, but ones signification. of being. Its requisites are that there are things to be known and powers of knowing things. It is an attribute of judgments and derivatively of propositions. That judgment is true which apprehends a thing as it is capable of being known to be; and that proposition is true which so asserts the thing to be. Or, to combine truth in thought and in speech, the true is what signifies a thing as it is capable of being known. Secondarily, the thing itself is ambiguously said to be true in the sense of being signified as it is. For example, as I am weary and am conscious of being weary, my judgment and proposition that I am weary are true because they signify what I am and know myself to be by direct consciousness; and my being weary is ambiguously said to be true because it is so signified.
But it will be said that Kant has proved that real truth, in the sense of the agreement of knowledge with the object, is unattainable, because we could compare knowledge with the object only by knowing both. Sigwart, indeed, adopting Kant's argument, concludes that we must be satisfied with consistency among the thoughts which presuppose an existent; this, too, is the reason why he thinks that induction is reduction, on the theory that we can show the necessary consequence of the given particular, but that truth of fact is unattainable. But Kant's criticism and Sigwart's corollary only derive plausibility from a false definition of truth. Truth is not the agreement of knowledge with an object beyond itself, and therefore ex hypothesi unknowable, but the agreement of our judgments with the objects of our knowledge. A judgment is true whenever it is a belief that a thing is determined as it is known to be by sense, or by memory after sense, or by inference from sense, however indirect the inference may be, and even when in the form of inference of non-existence it extends consequently from primary to secondary judgments. Thus the judgments this sensible pressure exists, that sensible pressure existed, other similar pressures exist, a conceivable centaur does not exist but is a figment, are all equally true, because they are in accordance with one or other of these kinds of knowledge. Consequently, as knowledge is attainable by sense, memory and inference, truth is also attainable, because, though we cannot test what we know by something else, we can test what we judge and assert by what we know. Not: that all inference is knowledge, but it is sometimes. The aim, of logic in general is to find the laws of all inference, which, so far as it obeys those laws, is always consistent, but is true or false according to its data as well as its consistency; and the aim of the special logic of knowledge is to find the laws of direct and indirect inferences from sense, because as sense produces sensory judgments which are always true of the sensible things actually perceived, inference from sense produces inferential judgments which, so far as they are consequent on sensory judgments, are always true of things similar to sensible things, by the very consistency of inference, or, as we say, by parity of reasoning.
We return then to the old view of Aristotle, that truth is believing in being; that sense is true of its immediate objects, and reasoning from sense true of its mediate objects; and that logic is the science of reasoning with a view to truth, or Logica est ars ratiocinandi, ut discernatur verum a falso. All we aspire to add is that, in order to attain to real truth, we must proceed gradually from sense, memory and experience through analogical particular inference, to inductive and deductive universal inference or reasoning. Logic is the science of all inference, beginning from sense and ending in reason.
In conclusion, the logic of the last quarter of the 19th century may be said to be animated by a spirit of inquiry, marred by a love of paradox and a corresponding hatred of tradition. But we have found, on the whole, that logical tradition rises superior to logical innovation. There are two old logics which still remain indispensable, Aristotle's Organon and Bacons Novum Organum. If, and only if, the study of deductive logic begins with Aristotle, and the study of inductive logic with Aristotle and Bacon, it will be profitable to add the works of the following recent German and English authors: Authorities.
J. Bergmann, Reine Logik (Berlin, 1879)
- - - - - - -, Die Grundprobleme der Logik (2nd ed., Berlin, 1895)
B. Bosanquet, Logic (Oxford, 1888)
- - - - - - -, The Essentials of Logic (London, 1895)
F. H. Bradley, The Principles of Logic (London, 1883)
F. Brentano, Psychologie vom empirischen Standpunkte (Vienna, 1874)
R. F. Clarke, Logic (London, 1889)
W. L. Davidson, The Logic of Definition (London, 1885);
E. Duhring, Logik und Wissenschaftstheorie (Leipzig, 1878)
B. Erdmann, Logik (Halle, 1892)
T. Fowler, Bacon's Novum Organum, edited, with introduction, notes, &c. (2nd ed., Oxford, 1889)
T. H. Green, Lectures on Logic, in Works, vol. iii. (London, 1886)
J. G. Hibben, Inductive Logic (Edinburgh and London, 1896)
F. Hillebrand, Die neuen Theorien der kategorischen Schlusse (Vienna, 1891)
L. T. Hobhouse, The Theory of Knowledge (London, 1896)
H. Hughes, The Theory of Inference (London, 1894)
E. Husserl, Logische Untersuchungen (Halle, 1891, 1901)
W. Jerusalem, Die Urtheilsfunction (Vienna and Leipzig, 1895)
W. Stanley Jevons, The Principles of Science (3rd ed., London, 1879)
- - - - - - -, Studies in Deductive Logic (London, 1880)
H. W. B. Joseph, Introduction to Logic (1906)
E. E. Constance Jones, Elements of Logic (Edinburgh, 1890)
J. N. Keynes, Studies and Exercises in Formal Logic (2nd ed., London, 1887)
F. A. Lange, Logische Studien (2nd ed., Leipzig, 1894)
T. Lipps, Grundzuge der Logik (Hamburg and Leipzig, 1893)
R. H. Lotze, Logik (2nd ad., Leipzig, 1881, English translation edited by B. Bosanquet, Oxford, 1884)
- - - - - -, Grundzuge der Logik (Diktate) (3rd ed., Leipzig, 1891, English translation by G. T. Ladd, Boston, 1887)
Werner Luthe, Beitrage zur Logik (Berlin, 1872, 1877)
Members of Johns Hopkins University, Studies in Logic (edited by C. S. Peirce, Boston, 1883)
J. B. Meyer, Ueberwegs System der Logik, funfte vermehrte Auflage (Bonn, 1882)
Max Muller, Science of Thought (London, 1887)
Carveth Read, On the Theory of Logic (London, 1878)
- - - - - - -, Logic, Deductive and Inductive (2nd ed;, London, 1901)
E. Schroder, Vorlesungen Uber die Algebra der Logik (Leipzig, 1890, 1891, 1895)
W. Schuppe, Erkenntnistheoretische Logik (Bonn, 1878)
- - - - - - -, Grundriss der Erkenninistheorie und Logik (Berlin, 1894)
R. Shute, A Discourse on Truth (London, 1877)
Alfred Sidgwick, Fallacies (London, 3883)
- - - - - - -, The Use of Words in Reasoning (London, 1901)
C. Sigwart, Logik (2nd ad., Freiburg-i.-Br. and Leipzig, 1893, English translation by Helen Dendy, London, 1895)
Uphues, Grundlehren der Logik (Breslau, 1883)
J. Veitch, Institutes of Logic (Edinburgh and London, 1885)
J. Venn, Symbolic Logic (2nd ed., London, 1894)
Alfred Sidgwick, The Principles of Empirical or Inductive Logic (London, 1889)
J. Volkelt, Erfahrung und Denken (Hamburg and Leipzig, 1886)
T. Welton, A Manual of Logic (London, 1891, 1896)
W. Windelband, Praeludien (Freiburg-i-Br., 1884)
W. Wundt, Logik (2nd ed., Stuttgart, 1893-1895)
Text-books are not comprised in this list. (T. CA.)
Here are some notes on some of the logicians or philosophers mentioned in the bibliography above. It is far from complete! (Many thanks to Irving Anellis for correcting and making many improvements upon material in an earlier version).
Thomas H. Case, the author of the article, was born in Liverpool on 14 July 1844 and died in 1925. He studied and taught at Cambridge University and later was Professor of Metaphysics and Morals at Oxford University from 1899 to 1910; then President of Christ’s College at Oxford University from 1910 to 1925. His specialisms included classical philosophy (he was also the author of the Encyclopaedia Britannica article on Aristotle (vol. 2, pp. 506-507) and the article "On the Development of Aristotle" (Mind n.s. 34 (1925), pp. 80-86)) and philosophy of mind and its relation to philosophy of science, his principal work being Physical Realism: Being an Analytical Philosophy from the Physical Objects of Science to the Physical Data of Sense (London/New York: Longmans, Green & Co., 1888), and he lectured on "The Scientific Method as a Mental Operation". His other works included Memoirs of a King’s College Chorister (Cambridge: Cambridge University Press, 1899) and A Brief History of the Proposal to Admit Women to Degrees at Cambridge in 1887-88 (Oxford: J. Parker & Co., 1896).
Ferdinand Canning Scott Schiller wrote the "Obituary of Thomas Case, 1844-1925" for The Pelican Record (of CorpusChristi College) 17 (December 1925), 52-53.
Case was a member of what Ivor Grattan-Guinness has called the school of "Oxbridge Logics" (see "Russell's Logicism versus the Oxbridge Logics 1880-1925: A Contribution to the Real History", Journal of the Archives of the Bertrand Russell Society (n.s.) 5, no. 2, 101-131) for an account of logic in Britain from the 1880s until the printing of the second edition of Whitehead and Russell’s Principia Mathematica, and of the general resistance to both algebraic logic and Frege-Russell logistic in the years when Aristotle remained dominant and the most influential logicians were still the neo-Hegelian British idealists Bradley, Bosanquet, and McTaggart.
Rudolf Hermann Lotze (1817-81) was a German philosopher and physiologist who tried to reconcile the Idealist tradition of Hegel with science. His work on logic was influential, and his textbook on logic was translated by Bosanquet in 1884. "His work is characteristic of the woolly and emotional nebulosities which in Germany followed the collapse of the idealist school". (R.G. Collingwood). Frege seems to have attended lectures by Lotze in his time at Gottingen.
Wilhelm Wundt (1832-1920) was the self-proclaimed founder of experimental psychology. He was self-taught as a philosopher, having originally decided to study medicine. His attention shifted to physiology, the field in which he lectured widely and published a number of articles during the years following his graduation. Psychology was just beginning to emerge as a distinct science, and much of Wundt's work anticipated the value of physiological methodology in dealing with psychological problems. According to Tappenden, Wundt's work is referred to by Frege ("On the foundations of geometry: First series") with apparent approval. However, his other references to "physiological psychology" are critical: what begins as realism (nerve fibres and ganglion cells) slides into an extreme form of idealism. See e.g. Logic, in Long, P. & White, R., Frege, Posthumous Writings 1969 128-49, p. 156.
L. T. Hobhouse (1864-1929) was a philosopher and Guardian journalist who was Professor of Sociology at the LSE. He advanced the idea in Mind in Evolution (1901) that society should be regarded as an organism, a product of evolution, with the individual as its basic unit, hence that society would improve over time as it evolved.
Bernard Bosanquet (1848-1923) was, with Bradley, one of the foremost contributors to the English Hegelian school of the late nineteenth century. He translated Lotze's work on Logic, and wrote a Logic of his own (1888).
Francis Herbert Bradley (1846-924) was, with Bosanquet, one of the foremost contributors to the English Hegelian school of the late nineteenth century. Appearance and Reality is probably his best known work. He argues that most things we are aware of are really appearances, whereas Reality, or the Absolute, is a single universal experience of which we are somehow components. This is a view that Frege disparages and condemns throughout his writing. However, some of Bradley's logical views, which are borrowed from the German school, are accepted by Frege. (As, for example, when he says in the Begriffschift that "The distinction between categorical, hypothetical and disjunctive judgments seems to me only to have grammatical significance").
Christoph Sigwart (1830-1904), following Herbart, advanced the view that the universal categorical judgment 'all M is P' is really the hypothetical judgment 'if anything is M it is P'. This view influenced not only German but also English logicians, such as Venn, Bradley and Bosanquet. This idea was essential to the development of modern logic. Thus a universal statement does not refer to a definite number of individuals, or to individuals at all, but rather to general ideas. The same view is found in Frege, who argues that "If I utter a sentence with the grammatical subject "all men", I do not wish to say something about some Central African chief wholly unknown to me". Universal statements connect concepts.
Sigwart is also quoted by Bosanquet in connection with the definite description "The centre of the material universe." "There cannot be two points, of which this content is true, but the meaning is still distinguishable from the particular instance, and is theoretically capable of having further particulars subsumed under it. Of course there may be two points in succsion - the centre may shift". (Bosanquet, Logic p. 47). This is similar to the example mentioned by Russell in "On Denoting".
Friedrich Adolf Trendelenburg (1802-1872). Frege refers once to Trendelenburg, and (apparently, according to Tappenden) adopts two distinctive words (‘lingua characterica’, ‘begriffsschrift’) from him.
Franz Hillebrand (1863-1926) was a pupil of Brentano.
Benno Erdmann (1851-1921) is for both Frege and Husserl the main advocate (& therefore chief culprit) of psychologism. Frege devotes ten sections (xiv-xxiv) of the Grundgesetze to a criticism of Erdmann's psychological logic.
John Venn (1834-1923) is, today, perhaps the best known of the logicians here. His Symbolic Logic, was first published in 1881. He also published a scathing review of Frege's Begriffschift in 1880. This was the only review which was published in English of a total of eight reviews that appeared in 1879 and 1880. This poor reception of his early work may explain why so little notice was taken of Frege, until Russell acknowledged his work in his appendix on Frege in the Principles of Mathematics (1903)
For historical accounts of the general reception of Frege’s work, see, e.g., Avrum Stroll, "On the First Flowering of Frege’s Reputation", Journal of the History of Philosophy 4 (1966), 72-81, Risto Vikko "The Reception of Frege’s Begriffsschrift", Historia Mathematica 25 (1998), 412-422.
For historical accounts to Frege by specific logicians, see, e.g. Benjamin S. Hawkins, Jr., "Peirce and Frege, A Question Unanswered", Modern Logic 3 (1993), 376-383; H. C. Kennedy, "Nine Letters from Giuseppe Peano to
Bertrand Russell", History and Philosophy of Logic 13 (1975), 205-220, and P.H. Nidditch, "Peano and the Recognition of Frege", Mind 72, 1963, 103-110.)
George Haywood Joyce was a neo-scholastic logician and author of Principles of Logic, available in part online here. It was reviewed in Mind (n.s.) 18 (1909), 290-291 by George Robert Thomson Ross; also in The Nation of 03/04/1909. Joyce was also the author of Principles of Natural Theology, London 1934; Christian Marriage: An Historical And Doctrinal Study (London & New York, 1933); and The Question of Miracles, Manresa Press, 1914.
Franz Brentano. Bibliographical material to follow, a discussion of his theory of existential import can be found here.
John Neville Keynes (1852-1949) is today best known as being the father of John Maynard Keynes. He was a lecturer in moral science at Cambridge from 1884 to 1911. His two main works are Studies and Exercises in Formal Logic (1884), , and a classic work on economic methodology, The Scope and Method of Political Economy (1891). His logic expressly avoided mathematical symbolism, and was one of the last works of logic to follow the Aristotelian model.
His extensive discussion of the problem of existential import, and of the ways of interpreting the traditional forms of opposition, remains a classic of the genre. (Part II, Chapter VIII, on "The Existential Import of Categorical Propositions" begins with the footnote "It may be advisable for students, on a first reading, to omit this chapter").
Arthur Prior had a great respect for Studies and Exercises in Formal Logic, and had planned a book on Keynes before a fire burned down the Prior family's house in early 1949.
Arthur Thomas Shearman wrote The Development of Symbolic Logic: A Critical-historical Study of the Logical Calculus (London: Williams and Norgate, 1906; reprinted: Dubuque, IA: Reprint Library, William C. Brown, 1988, and Bristol: Thoemmes Antiquarian Books, Ltd., 1990). This work also focuses on the traditionalists such as W. E. Johnson, and the algebraic logicians, such as Boole, De Morgan, and Jevons. In the final chapter he does mention Frege and Russell, only to say that their work was a continuation of the work of the traditional logicians already considered in the book! Shearman reviewed Russell's Principles of Mathematics in 1907.
Robert A. Adamson was a proponent of psychologistic logic who was interested especially in Bradley and Lotze. In his posthumously published A Short History of Logic (ed. Sorley, London 1911) he also took no notice of the work of Cantor, Frege, Peano, or Russell. Sorley, Adamson's editor, was a neo-Hegelian who was a follower of Lotze; Adamson's book, apart from the addition by Sorley of some book reviews and short articles written by Adamson, is essentially a reprint of Adamson's article "Logic" for the ninth edition, of 1882, for the Encyclopaedia Britannica [source: Irving Anellis].
Johnson was a logician and mathematical economist who worked at Cambridge. He published papers such as "The logical calculus" (1892) "Analysis of thinking" (1918), both of which appeared in Mind. Perhaps his most famous work on mathematical economics was "The pure theory of utility curves" which appeared in the Economic Journal of Science in 1913. (There is a long-standing debate about whether Johnson knew of Pareto’s work on this subject. This was set out in his Manuale di Economia Politica (1906), but Johnson does not cite Pareto). He is best known for his Logic (e-text in part here), published in 1922.
There is a bibliography of Johnson here which does not, however, mention Johnson's curious relationship with Wittgenstein. In February 1912, Russell arranged for Wittgenstein to be given formal tuition in logic by Johnson. This lasted only a few weeks, with Johnson terminating the arrangement in a letter to Wittgenstein. Wittgenstein later said (to Leavis) "I found in the first hour he had nothing to teach me". But Johnson said, sarcastically "At our first meeting he was teaching me". (Source, Rhees p.61, Russell, letter to Morrell 17.3.1912). Johnson considered Wittgenstein's return to Cambridge in 1929 "a disaster for Cambridge". He thought Wittgenstein to be "a man quite incapable of carrying on a discussion" (Rhees p.103).
Yet Wittgenstein, at least, seems to have retained some sort of affection for Johnson. He often mentioned Johnson in his letters to Keynes, and before World War I had arranged to give £200 a year to help Johnson continue his work in logic. The Tractatus was published in 1922, the same year as Johnson's book. Wittgenstein wonders in a letter to Ogden "what he [Johnson] will think of it". During his short friendship with Leavis, Wittgenstein often met at Johnson's 'at homes' – he seems to have admired Johnson more as a pianist than a logician. (These bits and pieces of information are to be found in Ray Monk's excellent bibliography of Wittgenstein, which he found in turn in Rush Rhees (ed.), Recollections of Wittgenstein Oxford 1984.
Tappenden, J., "Frege on Axioms, Indirect Proof, and Independence Arguments In Geometry", Notre Dame Journal of Formal Logic, Volume 41, Number 2, 2000 (tappen AT umich.edu)
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