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    "source_title": "Encyclopaedia Britannica (1911)",
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    "title": "INFERENCE",
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    "verified_text": "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 non-mortal professor. on the other hand, if on the plan of sigwart categorical universals were reducible to hypothetical, 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 p1 or p2 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 p1 or p2 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 p. | s is not m. | s is not m. :. neither s nor m | :. s may be p. | :. s may be p. is p. | | but (1) concludes with a mere repetition, (2) and (3) with a contingent \"may be,\" which, as aristotle says, also \"may not be,\" and therefore _nihil certo colligitur_. the same answer applies to schuppe's supposed syllogisms from two particular premises:-- (1) | (2) 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 be 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 connexion\" (_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 are fusible.\" the moment we stir a step further with wundt m the direction of a more general conclusion (_ein 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 connexion\" 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, but data for induction parading as syllogism. we must pass the same sentence on lotze's attempt to extend the second figure of the syllogism for inductive purposes, thus:-- s is m. q is m. r is m. :. every [sigma], which is common to s, q, r, is m. we could not have a more flagrant abuse of the rule _ne esto plus minusque in conclusione quam in praemissis_. as we see from lotze's own defence, the conclusion cannot be drawn without another premise or premises to the effect that \"s, q, r, are [sigma], and [sigma] 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. | x = y. | y = x. no s is m. | z = y. | y = z. :. no s is p. | :. x = z. | :. x = z. 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 (1) \"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:-- 1. _progression._ | 2. _regression._ | (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 ([greek: ho tou dioti syllogismos]), and the other regressing a posteriori from consequent fact to real ground ([greek: ho tou hoti syllogismos]). 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, 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, s-p, s-m, m-p, and becomes an inductive method in the form of jevons's inverse deduction, or hypothetical deduction, or analysis, m-p,",
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