. The ordinary categorical syllogism, that is, one which is formed of judgments in the sim ple form, x is y, although it is essentially grounded in the causal relation, nevertheless in a large measure conceals it. On the other hand, in the hypothetical syllogism, whose major premise is a hypothetical judgment, that is, of the form, if xis y, a is w, and whose minor premise affirms or denies one of these two clauses categorically, the causal relation is brought into the foreground, inasmuch as it states explicitly a consequence as dependent upon its ground. There is still another kind of syllogism known as the disjunctive syllogism in which the major premise is a disjunctive judgment that is of the form, x is either z or w, and the minor premise is an affirmation or denial of one of these alternatives while the conclusion is the corresponding denial or affir mation of the other. This affords a method of reasoning by elimination, and is most useful in deciding between possibilities. In this form of the syllogism the causal relation is not ex pressed on the surface at all, but is essen tially implied; for in order to state possibilities one must know all the underlying causal rela tions of the system in which these possibilities emerge. For instance, one cannot state that the contents of a stomach which have been chemically examined indicate the presence of arsenic or antimony unless there is known also the exact causal connection between these two poisons and the partially digested food; the disjunctive judgment standing as the major premise of the disjunctive syllogism always presupposes some definite knowledge of deter mining causal relations which exist in the sys tem under consideration.
The validity of the syllogism turns, as will be readily seen, upon the possibility of re ferring a special case to its appropriate univer sal. But there are often situations and cir cumstances when this method of reference is not at all possible and yet nevertheless a valid inference can be drawn. In other words, the syllogistic procedure by no means exhausts the possibility of deductive inference. There may be other relations which grow out of a system of interconnected parts and which are of such a nature as to warrant an inference from them. For instance, we may have the follow ing inference: The two angles of a triangle, A and B, equal 95 degrees. Therefore, the third angle must equal 85 degrees.
This is not a syllogism proper and yet is a perfectly valid course of reasoning. While there is no middle term there is an identical point of reference, namely, the given triangle; and there is also the universal upon which the inference depends, namely, the relations which underlie the very nature of the triangle itself and are rendered constant by it. Thus, all of the essentials of inference are found to be present in such a form of reasoning. Various kinds of inference may thus arise according to the different relations which may obtain in the system wherein they occur. To have a valid inference in any such case we must establish some identity of relation between the parts which we are comparing; otherwise we can not logically pass from one 'to the other. And identity of relationship can be established only in systems of such simplicity that no unknown elements which might enter to disturb the exist ing relations can be conceived. Our thought in other words must command the system com pletely; otherwise we are never justified in using our knowledge as the basis of reasoning.
In the inductive process as we have seen, the procedure is from particular instances to the universal which underlies them and which they illustrate; there is here, however, an evident break in the continuity of the logical process. The conclusion contains more than the prem ises; for in the universal reached by induction our knowledge goes beyond our actual experi ence. This is the so-called °inductive leap* or "inductive hazard.° It is not, however, a leap in the dark. Such it would be, were we com pelled to use the mere data of experience as the sole ground of our inferences. But it is pos sible to formulate as a postulate some univer sal truth which the mind is constrained to assume and which serves to bridge the gap between the particular and the universal. This postulate has been variously expressed by dif ferent authors, yet with substantially the same underlying significance in all. In the older logic this is put in the convenient formula of °the uniformity of nature," that is — beyond the sphere of experience phenomena are supposed to behave under like conditions in the same manner as in the sphere of immediate obser vation and experiment. In the modern logic, the phrase °uniformity of consciousness' takes the place of °uniformity of nature,' the latter being regarded as somewhat indefinite and as implying a view exclusively objective. By °um
formity of consciousness is meant, that our knowledge must be consistent throughout with itself, part to part, and parts to the whole, and that the world for us is the world as it is con structed and interpreted by our knowledge. Whenever a concrete instance is present in consciousness it is to be regarded as having its i appropriate place in a system of universal and necessary relations, so that a correct interpre tation of the concrete case must reveal the uni versal element which underlies it and gives it a place and meaning in our world of knowl edge. Nature after all is only another word for the world as we know it — a world of uni versal and necessary relations; otherwise it could not be a world of order and uniformity.
The one relation above all others which enables us to discover the universal significance of concrete instances is the relation of cause and effect. When in the phenomena of nature or the events of life, a simple causal relation can be discovered, even though it is illustrated only in a certain case, there is sufficient ground for the generalization of the connection thus discovered. The method of inductive investi gation, therefore, consists in various tests for the exhibition of true causal relations of such a simple nature as to furnish evidence that these relations are both definite and constant. A complex relation is too indefinite and vari able to warrant any generalization which is based upon it. But however complex the phe nomenon may be, it may be always subjected to some process of analysis which will reveal a more simple causal relation underlying it. Moreover, it is necessary by proper inductive tests to discriminate between a causal con nection and a mere coincidence. All of this is provided for in the so-called inductive methods — the method of agreement, of difference, of agreement and difference, of concomitant vari ation and of residues. These are essentially the methods of scientific procedure, the methods of research and experimentation. The func tion of hypothesis in inductive investigation must not be overlooked. An hypothesis is a supposition which is made concerning the prob able cause of a phenomenon either as prelim inary to an experiment which may prove or disprove the supposition or in the place of an experiment or systematic observation when such are impossible owing to the peculiar conditions of the phenomenon itself. In the first instance the function of hypothesis determines the line of experiment in a definite manner and does not leave the phenomenon in question to inde terminate and haphazard investigation. This may be illustrated by a quotation concerning Charles Darwin, taken from the 'Reminis cences" of his son, Francis Darwin: "He often said that no one could be a good ob server unless he was an active theorizer. It was as though he were charged with theoriz ing power ready to flow into any channel on the slightest disturbance, so that no fact how ever small could avoid releasing a stream of theory and then the fact became magnified into importance. In this way it naturally happened that many untenable theories occurred to him; but fortunately his richness of imagination was equalled by his power of judging and con demning the thoughts that occurred to him. He was just to his theories and did not con demn them unheard; and so it happened that he was willing to test what would seem to most people not at all worth testing." ('Life and Letters of Charles Darwin,' Vol. I, p. 126). But there is a second function of hypothesis— where an explanation is needed to account for phenomena which it is impossible to repro duce in the form of an experiment. We are not always able to perceive the relations be tween facts and yet we are constrained to think of them as related; but in order to syste matize. them we must supply the lacunce which appear in the phenomena as perceived. A sup position of this nature which is necessary in order to construct a body of facts into a sys tem is an hypothesis — as for instance the nebular hypothesis of Laplace. No course of reasoning, however, can be carried on to any extent or to any effect which does not combine the two processes of deduction and induction in a manner provided for by the complementary relation which they sustain one to the other. The combination of the deductive and inductive processes has been called by John Stuart Mill the deductive method simply. A more dis tinctive name, however, would be — the induc todeductive method. This combined method consists of three stages: 1. A preliminary process of induction whose results may be expressed tentatively at least in the form of a universal law or principle.