to consequent fact (ὁ τοῦ διότι συλλογισμός), and the other
regressing a posteriori from consequent fact to real ground
(ὁ τοῦ ὄτι συλλογισμός). 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:—
| 1. 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 (1) 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 (S), 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. |
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,
