Aristotle
17.]
[Footnote 25: Ibid. vi. p. 75, a. 8-37.
On the point last mentioned, M. Barthélemy St. Hilaire observes in
his note, p. 41: "Dans les questions de dialectique, la conclusion
est nécessaire en ce sens, qu'elle suit nécessairement des prémisses;
elle n'est pas du tout nécessaire en ce sens, que la chose qu'elle
exprime soit nécessaire. Ainsi il faut distinguer la nécessité de la
forme et la nécessité de la matière: ou comme disent les
scholastiques, _necessitas illationis et necessitas materiæ_. La
dialectique se contente de la première, mais la demonstration a
essentiellement besoin des deux."]
In every demonstration three things may be distinguished: (1) The
demonstrated conclusion, or Attribute essential to a certain genus;
(2) The Genus, of which the attributes _per se_ are the matter of
demonstration; (3) The Axioms, out of which, or through which, the
demonstration is obtained. These Axioms may be and are common to
several genera: but the demonstration cannot be transferred from one
genus to another; both the extremes as well as the middle term must
belong to the same genus. An arithmetical demonstration cannot be
transferred to magnitudes and their properties, except in so far as
magnitudes are numbers, which is partially true of some among them.
The demonstrations in arithmetic may indeed be transferred to
harmonics, because harmonics is subordinate to arithmetic; and, for
the like reason, demonstrations in geometry may be transferred to
mechanics and optics. But we cannot introduce into geometry any
property of lines, which does not belong to them _quâ_ lines; such,
for example, as that a straight line is the most beautiful of all
lines, or is the contrary of a circular line; for these predicates
belong to it, not _quâ_ line, but _quâ_ member of a different or more
extensive genus.[26] There can be no complete demonstration about
perishable things, or about any individual line, except in regard to
its attributes as member of the genus line. Where the conclusion is
not eternally true, but true at one time and not true at another,
this can only be because one of its premisses is not universal or
essential. Where both premisses are universal and essential, the
conclusion must be eternal or eternally true. As there is no
demonstration, so also there can be no definition, of perishable
attributes.[27]
[Footnote 26: Ibid. vii. p. 75, a. 38-b. 20. Mr. Poste, in his
translation, here cites (p. 50) a good illustrative passage from Dr.
Whewell's Philosophy of the Inductive Sciences, Book II. ii.:--"But,
in order that we may make any real advance in the discovery of truth,
our ideas must not only be clear; they must also be _appropriate_.
Each science has for its basis a different class of ideas; and the
steps which constitute the progress of one science can never be made
by employing the ideas of another kind of science. No genuine advance
could ever be obtained in Mechanics by applying to the subject the
ideas of space and time merely; no advance in Chemistry by the use of
mere mechanical conceptions; no discovery in Physiology by referring
facts to mere chemical and mechanical principles." &c.]
[Footnote 27: Aristot. Analyt. Post. I. viii. p. 75, b. 21-36.
Compare Metaphys. Z. p. 1040, a. 1: [Greek: dê=lon o(/ti ou)k a)\n
ei)/ê au)tô=n (tô=n phthartô=n) ou)/th' o(rismo\s ou)/t'
a)po/deixis]. Also Biese, Die Philosophie des Aristoteles, ch. iv. p.
249.]
For complete demonstration, it is not sufficient that the premisses
be true, immediate, and undemonstrable; they must, furthermore, be
essential and appropriate to the class in hand. Unless they be such,
you cannot be said to know the conclusion _absolutely_; you know it
only by accident. You can only know a conclusion when demonstrated
from its own appropriate premisses; and you know it best when it is
demonstrated from its highest premisses. It is sometimes difficult to
determine whether we really know or not; for we fancy that we know,
when we demonstrate from true and universal _principia_, without
being aware whether they are, or are not, the _principia_ appropriate
to the case.[28] But these _principia_ must always be assumed without
demonstration--the class whose essential constituent properties are
in question, the universal Axioms, and the Definition or meaning of
the attributes to be demonstrated. If these definitions and axioms
are not always formally enunciated, it is because we tacitly presume
them to be already known and admitted by the learner.[29] He may
indeed always refuse to grant them in express words, but they are
such that he cannot help granting them by internal assent in his
mind, to which every syllogism must address itself. When you assume a
premiss without demonstrating it, though it be really demonstrable,
this, if the learner is favourable and willing to grant it, is an
assumption or Hypothesis, valid relatively to him alone, but not
valid absolutely: if he is reluctant or adverse, it is a Postulate,
which you claim whether he is satisfied or not.[30] The Definition by
itself is not an hypothesis; for it neither affirms nor denies the
existence of anything. The pupil must indeed understand the terms of
it; but this alone is not an hypothesis, unless you call the fact
that the pupil comes to learn, an hypothesis.[31] The Hypothesis or
assumption is contained in the premisses, being that by which the
reason of the conclusion comes to be true. Some object that the
geometer makes a false hypothesis or assumption, when he declares a
given line drawn to be straight, or to be a foot long, though it is
neither one nor the other. But this objection has no pertinence,
since the geometer does not derive his conclusions from what is true
of the visible lines drawn before his eyes, but from what is true of
the lines conceived in his own mind, and signified or illustrated by
the visible diagrams.[32]
[Footnote 28: Ibid. ix. p. 75, b. 37-p. 76, a. 30.]
[Footnote 29: Ibid. x. p. 76, a. 31-b. 22.]
[Footnote 30: Aristot. Analyt. Post. I. x. p. 76, b. 29-34: [Greek:
e)a\n me\n dokou=nta lamba/nê| tô=| mantha/nonti, u(poti/thetai, kai\
e)/stin ou)/ch a(plô=s u(po/thesis, a)lla\ pro\s e)kei=non mo/non,
a)\n de\ ê)\ mêdemi/a=s e)nou/sês do/xês ê)\ kai\ e)nanti/as
e)nou/sês lamba/nê| to\ au)to/, ai)tei=tai. kai\ tou/tô| diaphe/rei
_u(po/thesis_ kai\ _ai)/têma_], &c. Themistius, Paraphras. p. 37,
Spengel.]
[Footnote 31: Ibid. p. 76, b. 36: [Greek: tou=to d' ou)ch
u(po/thesis, ei) mê\ kai\ _to\ a)kou/ein_ u(po/thesi/n tis ei)=nai
phê/sei]. For the meaning of [Greek: _to\ a)kou/ein_], compare
[Greek: o( a)kou/ôn], infra, Analyt. Post. I. xxiv. p. 85, b. 22.]
[Footnote 32: Ibid. p. 77, a. 1: [Greek: o( de\ geôme/três ou)de\n
sumperai/netai tô=| tê/nde ei)=nai tê\n grammê\n ê(\n au)to\s
e)/phthegktai, a)lla\ ta\ dia\ tou/tôn dêlou/mena.]
Themistius, Paraphr. p. 37: [Greek: ô(/sper ou)d' oi( geôme/trai
ke/chrêntai tai=s grammai=s u(pe\r ô(=n diale/gontai kai\
deiknu/ousin, a)ll' a(\s e)/chousin e)n tê=| psuchê=|, ô(=n ei)si\
su/mbola ai( grapho/menai.]
A similar doctrine is asserted, Analyt. Prior. I. xli. p. 49, b. 35,
and still more clearly in De Memoria et Reminiscentia, p. 450,
a. 2-12.]
The process of Demonstration neither requires, nor countenances, the
Platonic theory of Ideas--universal substances beyond and apart from
particulars. But it does require that we should admit universal
predications; that is, one and the same predicate truly applicable in
the same sense to many different particulars. Unless this be so,
there can be no universal major premiss, nor appropriate middle term,
nor valid demonstrative syllogism.[33]
[Footnote 33: Aristot. Analyt. Post. I. xi. p. 77, a. 5-9.]
The Maxim or Axiom of Contradiction, in its most general enunciation,
is never formally enunciated by any special science; but each of them
assumes the Maxim so far as applicable to its own purpose, whenever
the _Reductio ad Absurdum_ is introduced.[34] It is in this and the
other common principles or Axioms that all the sciences find their
point of contact and communion; and that Dialectic also comes into
communion with all of them, as also the science (First Philosophy)
that scrutinizes the validity or demonstrability of the Axioms.[35]
The dialectician is not confined to any one science, or to any
definite subject-matter. His liberty of interrogation is unlimited;
but his procedure is essentially interrogatory, and he is bound to
accept the answer of the respondent--whatever it be, affirmative or
negative--as premiss for any syllogism that he may construct. In this
way he can never be sure of demonstrating any thing; for the
affirmative and the negative will not be equally serviceable for that
purpose. There is indeed also, in discussions on the separate
sciences, a legitimate practice of scientific interrogation. Here the
questions proper to be put are limited in number, and the answers
proper to be made are determined beforehand by the truths of the
science--say Geometry; still, an answer thus correctly made will
serve to the interrogator as premiss for syllogistic
demonstration.[36] The respondent must submit to have such answer
tested by appeal to geometrical _principia_ and to other geometrical
propositions already proved as legitimate conclusions from the
_principia_; if he finds himself involved in contradictions, he is
confuted _quâ_ geometer, and must correct or modify his answer. But
he is not bound, _quâ_ geometer, to undergo scrutiny as to the
geometrical _principia_ themselves; this would carry the dialogue out
of the province of Geometry into that of First Philosophy and
Dialectic. Care, indeed, must be taken to keep both questions and
answers within the limits of the science. Now there can be no
security for this restriction, except in the scientific competence of
the auditors. Refrain, accordingly, from all geometrical discussions
among men ignorant of geometry and confine yourself to geometrical
auditors, who alone can distinguish what questions and answers are
really appropriate. And what is here said about geometry, is equally
true about the other special sciences.[37] Answers may be improper
either as foreign to the science under debate, or as appertaining to
the science, yet false as to the matter, or as equivocal in middle
term; though this last is less likely to occur in Geometry, since the
demonstrations are accompanied by diagrams, which help to render
conspicuous any such ambiguity.[38] To an inductive proposition,
bringing forward a single case as contributory to an ultimate
generalization, no general objection should be offered; the objection
should be reserved until the generalization itself is tendered.[39]
Sometimes the mistake is made of drawing an affirmative conclusion
from premisses in the Second figure; this is formally wrong, but the
conclusion may in some cases be true, if the major premiss happens to
be a reciprocating proposition, having its predicate co-extensive
with its subject. This, however, cannot be presumed; nor can a
conclusion be made to yield up its principles by necessary
reciprocation; for we have already observed that, though the truth of
the premisses certifies the truth of the conclusion, we cannot say
_vice versâ_ that the truth of the conclusion certifies the truth of
the premisses. Yet propositions are more frequently found to
reciprocate in scientific discussion than in Dialectic; because, in
the former, we take no account of accidental properties, but only of
definitions and what follows from them.[40]
[Footnote 34: Ibid. a. 10, seq.]
[Footnote 35: Ibid. a. 26-30: [Greek: kai\ ei)/ tis katho/lou
peirô=|to deiknu/nai ta\ koina/, oi(=on o(/ti a(/pan pha/nai ê)\
a)popha/nai, ê)\ o(/ti i)/sa a)po\ i)/sôn, ê)\ tô=n toiou/tôn