Aristotle
universal truth, and by thus knowing that _every_ triangle had its
three angles equal to two right angles; but you did not know it
absolutely, being ignorant that A B C was a triangle.[11]
[Footnote 11: Aristot. Analyt. Post. I. i. p. 71, a. 17-b. 8: [Greek:
e)/sti de\ gnôri/zein ta\ me\n pro/teron gnôri/zonta, tô=n de\ kai\
a)/ma lamba/nonta tê\n gnô=sin, oi(=on o(/sa tugcha/nei o)/nta u(po\
to\ katho/lou, ô(=n e)/chei tê\n gnô=sin. o(/ti me\n ga\r pa=n
tri/gônon e)/chei dusi\n o)rthai=s i)/sas, proê/|dei; o(/ti de\ to/de
to\ e)n tô=| ê(mikukli/ô| tri/gôno/n e)stin, a(/ma e)pago/menos
e)gnô/risen.--pri\n d' e)pachthê=nai ê)\ labei=n sullogismo/n,
tro/pon me/n tina i)/sôs phate/on e)pi/stasthai, tro/pon d' a)/llon
ou)/. o(\ ga\r mê\ ê)/|dei ei) e)/stin a(plô=s, tou=to pô=s ê)/|dei
o(/ti du/o o)rtha\s e)/chei a(plô=s? a)lla\ dê=lon ô(s _ô(di\ me\n
e)pi/statai, o(/ti katho/lou e)pi/statai, a(plô=s d' ou)k
e)pi/statai_.--ou)de\n (oi)=mai) kôlu/ei, o(\ mantha/nei, e)/stin ô(s
e)pi/stasthai, e)/sti d' ô(s a)gnoei=n; a)/topon ga\r ou)k ei)
oi)=de/ pôs o(\ mantha/nei, a)ll' ei) ô(di/, oi(=on ê(=| mantha/nei
kai\ ô(/s.] Compare also Anal. Post. I. xxiv. p. 86, a. 23, and
Metaph. A. ii. p. 982, a. 8; Anal. Prior. II. xxi. p. 67, a. 5-b.
10.)
Aristotle reports the solution given by others, but from which he
himself dissented, of the Platonic puzzle. The respondent was asked,
Do you know that every Dyad is even?--Yes. Some Dyad was then
produced, which the respondent did not know to be a Dyad; accordingly
he did not know it to be even. Now the critics alluded to by
Aristotle said that the respondent made a wrong answer; instead of
saying I know every Dyad is even, he ought to have said. Every Dyad
_which I know to be a Dyad_ is even. Aristotle pronounces that this
criticism is incorrect. The respondent knows the conclusion which had
previously been demonstrated to him; and that conclusion was, Every
triangle has its three angles equal to two right angles; it was not,
Every thing _which I know_ to be a triangle has its three angles
equal to two right angles. This last proposition had never been
demonstrated, nor even stated: [Greek: ou)demi/a ga\r pro/tasis
lamba/netai toiau/tê, o(/ti _o(\n su\ oi)=das_ a)rithmo/n, _ê)\ o(\
su\ oi)=das_ eu)thu/grammon, a)lla\ _kata\ panto/s_] (b. 3-5).
This discussion, in the commencement of the Analytica Posteriora
(combined with Analyt. Priora, II. xxi.), is interesting, because it
shows that even then the difficulties were felt, about the major
proposition of the Syllogism, which Mr. John Stuart Mill has so ably
cleared up, for the first time, in his System of Logic. See Book II.
ch. iii. of that work, especially as it stands in the sixth edition,
with the note there added, pp. 232-233. You affirm, in the major
proposition of the Syllogism, that every triangle has its three
angles equal to two right angles; does not this include the triangle
A, B, C, and is it not therefore a _petitio principii_? Or, if it be
not so, does it not assert more than you know? The Sophists (upon
whom both Plato and Aristotle are always severe, but who were
valuable contributors to the theory of Logic by fastening upon the
weak points) attacked it on this ground, and raised against it the
puzzle described by Aristotle (in this chapter), afterwards known as
the Sophism entitled [Greek: o( e)gkekalumme/nos] (see Themistius
Paraphras. I. i.; also 'Plato and the Other Companions of Sokrates,'
Vol. III. ch. xxxviii. p. 489). The critics whom Aristotle here cites
and disapproves, virtually admitted the pertinence of this puzzle by
modifying their assertion, and by cutting it down to "Everything
_which we know to be a triangle_ has its three angles equal to two
right angles." Aristotle finds fault with this modification, which,
however, is one way of abating the excess of absolute and peremptory
pretension contained in the major, and of intimating the want of a
minor to be added for interpreting and supplementing the major; while
Aristotle himself arrives at the same result by admitting that the
knowledge corresponding to the major proposition is not yet absolute,
but incomplete and qualified; and that it is only made absolute when
supplemented by a minor.
The very same point, substantially, is raised in the discussion
between Mr. John Stuart Mill and an opponent, in the note above
referred to. "A writer in the 'British Quarterly Review' endeavours
to show that there is no _petitio principii_ in the Syllogism, by
denying that the proposition All men are mortal, asserts or assumes
that Socrates is mortal. In support of this denial, he argues that we
may, and in fact do, admit the general proposition without having
particularly examined the case of Socrates, and even without knowing
whether the individual so named is a man or something else. But this
of course was never denied. That we can and do draw inferences
concerning cases specifically unknown to us, is the datum from which
all who discuss this subject must set out. The question is, in what
terms the evidence or ground on which we draw these conclusions may
best be designated--whether it is most correct to say that the
unknown case is proved by known cases, or that it is proved by a
general proposition including both sets of cases, the known and the
unknown? I contend for the former mode of expression. I hold it an
abuse of language to say, that the proof that Socrates is mortal, is
that all men are mortal. Turn it in what way we will, this seems to
me asserting that a thing is the proof of itself. Whoever pronounces
the words, All men are mortal, has affirmed that Socrates is mortal,
though he may never have heard of Socrates; for since Socrates,
whether known to be a man or not, really is a man, he is included in
the words, All men, and in every assertion of which they are the
subject. . . . . The reviewer acknowledges that the maxim (Dictum de
Omni et Nullo) as commonly expressed--'Whatever is true of a class is
true of everything included in the class,' is a mere identical
proposition, since the class _is_ nothing but the things included in
it. But he thinks this defect would be cured by wording the maxim
thus: 'Whatever is true of a class is true of everything which can be
shown to be a member of the class:' as if a thing could be shown to
be a member of the class without being one."
The qualified manner in which the maxim is here enunciated by the
reviewer (what _can be shown_ to be a member of the class)
corresponds with the qualification introduced by those critics whom
Aristotle impugns ([Greek: lu/ousi ga\r ou) pha/skontes ei)de/nai
pa=san dua/da a)rti/an ou)=san, a)ll' _ê(\n i)/sasin o(/ti dua/s_]);
and the reply of Mr. Mill would have suited for these critics as well
as for the reviewer. The puzzle started in the Platonic Menon is, at
bottom, founded on the same view as that of Mr. Mill, when he states
that the major proposition of the Syllogism includes beforehand the
conclusion. "The general principle, (says Mr. Mill, p. 205), instead
of being given as evidence of the particular case, cannot itself be
taken for true without exception, until every shadow of doubt which
could affect any case comprised in it is dispelled by evidence
_aliunde_; and then what remains for the syllogism to prove? From a
general principle we cannot infer any particulars but those which the
principle itself assumes as known."
To enunciate this in the language of the Platonic Menon, we learn
nothing by or through the evidence of the Syllogism, except a part of
what we have already professed ourselves to know by asserting the
major premiss.]
Aristotle proceeds to tell us what is meant by knowing a thing
_absolutely_ or completely ([Greek: a(plô=s]). It is when we believe
ourselves to know the cause or reason through which the matter known
exists, so that it cannot but be as it is. That is what
Demonstration, or Scientific Syllogism, teaches us;[12] a Syllogism
derived from premisses true, immediate, prior to, and more knowable
than the conclusion--causes of the conclusion, and specially
appropriate thereto. These premisses must be known beforehand without
being demonstrated (_i.e._ known not through a middle term); and must
be known not merely in the sense of understanding the signification
of the terms, but also in that of being able to affirm the truth of
the proposition. _Prior_ or _more knowable_ is understood here as
prior or more knowable _by nature_ (not _relatively to us_, according
to the antithesis formerly explained); first, most universal,
undemonstrable _principia_ are meant. Some of these are Axioms, which
the learner must "bring with him from home," or know before the
teacher can instruct him in any special science; some are Definitions
of the name and its essential meaning; others, again, are Hypotheses
or affirmations of the existence of the thing defined, which the
learner must accept upon the authority of the teacher.[13] As these
are the _principia_ of Demonstration, so it is necessary that the
learner should know them, not merely as well as the conclusions
demonstrated, but even better; and that among matters contradictory
to the _principia_ there should be none that he knows better or
trusts more.[14]
[Footnote 12: Aristot. Analyt. Post I. ii. p. 71, b. 9-17. Julius
Pacius says in a note, ad c. ii. p. 394: "Propositio demonstrativa
est prima, immediata, et indemonstrabilis. His tribus verbis
significatur una et eadem conditio; nam propositio prima est, quæ,
quod medio caret, demonstrari nequit."
So also Zabarella (In lib. I. Post. Anal. Comm., p. 340, Op. ed.
Venet. 1617): "Duæ illæ dictiones (_primis_ et _immediatis_) unam
tantum significant conditionem ordine secundam, non duas; idem namque
est, principia esse medio carentia, ac esse prima."]
[Footnote 13: Aristot. Analyt. Post. I. ii. p. 72, a. 1-24;
Themistius, Paraphr. I. ii. p. 10, ed. Spengel; Schol. p. 199, b. 44.
Themistius quotes the definition of an Axiom as given by
Theophrastus: [Greek: A)xi/ôma/ e)sti _do/xa_ tis], &c. This shows
the difficulty of adhering precisely to a scientific terminology.
Theophrastus explains an axiom to be a sort of [Greek: do/xa], thus
lapsing into the common loose use of the word. Yet still both he and
Aristotle declare [Greek: do/xa] to be of inferior intellectual worth
as compared with [Greek: e)pistê/mê] (Anal. Post. I. xxiii.), while
at the same time they declare the Axiom to be the very maximum of
scientific truth. Theophrastus gave, as examples of Axioms, the
**maxim of Contradiction, universally applicable, and, "If
equals be taken from equals the remainders will be equal," applicable
to homogeneous quantities. Even Aristotle himself sometimes falls
into the same vague employment of [Greek: do/xa], as including the
Axioms. See Metaphys. B. ii. p. 996, b. 28; [Greek: G]. iii. p. 1005,
b. 33.]
[Footnote 14: Aristot. Anal. Post. I. ii. p. 72, a. 25, b. 4. I
translate these words in conformity with Themistius, pp. 12-13, and
with Mr. Poste's translation, p. 43. Julius Pacius and M. Barthélemy
St. Hilaire render them somewhat differently. They also read [Greek:
a)meta/ptôtos], while Waitz and Firmin Didot read [Greek:
a)meta/peistos], which last seems preferable.]
In Aristotle's time two doctrines had been advanced, in opposition to
the preceding theory: (1) Some denied the necessity of any
indemonstrable _principia_, and affirmed the possibility of,
demonstrating backwards _ad infinitum_; (2) Others agreed in denying
the necessity of any indemonstrable _principia_, but contended that
demonstration in a circle is valid and legitimate--_e.g._ that A may
be demonstrated by means of B, and B by means of A. Against both
these doctrines Aristotle enters his protest. The first of them--the