Aristotle
démonstration ne sera en partie donnée que quand on aura une bonne
fois énuméré toutes les propositions indémontrables; et quand on aura
bien défini le caractère auquel on les reconnait. Nulle part on ne
trouve ni une semblable énumération, ni une semblable définition. On
reste à cet égard dans une position vague, et par cela même facile à
défendre."
It would seem, by these words, that M. Delboeuf stands in the most
direct opposition to Aristotle, who teaches us that the [Greek:
a)rchai\] or _principia_ from which demonstration starts cannot be
themselves demonstrated. But when we compare other passages of M.
Delboeuf's work, we find that, in rejecting all undemonstrable
propositions, what he really means is to reject all _self-evident
universal truths_, "C'est donc une véritable illusion d'admettre des
vérités évidentes par elles-mêmes. Il n'y a pas de proposition fausse
que nous ne soyons disposés d'admettre comme axiome, quand rien ne
nous a encore autorisés à la repousser" (p. ix.). This is quite true
in my opinion; but the immediate indemonstrable truths for which
Aristotle contends as [Greek: a)rchai\] of demonstration, are not
announced by him as _self-evident_, they are declared to be results
of sense and induction, to be raised from observation of particulars
multiplied, compared, and permanently formularized under the
intellectual _habitus_ called Noûs. By Demonstration Aristotle means
deduction in its most perfect form, beginning from these [Greek:
a)rchai\] which are inductively known but not demonstrable (_i. e._
not knowable deductively). And in this view the very able and
instructive treatise of M. Delboeuf mainly coincides, assigning even
greater preponderance to the inductive process, and approximating in
this respect to the important improvements in logical theory advanced
by Mr. John Stuart Mill.
Among the universal propositions which are not derived from
Induction, but which serve as [Greek: a)rchai\] for Deduction and
Demonstration, we may reckon the religious, ethical, æsthetical,
social, political, &c., beliefs received in each different community,
and impressed upon all newcomers born into it by the force of
precept, example, authority. Here the major premiss is felt by each
individual as carrying an authority of its own, stamped and enforced
by the sanction of society, and by the disgrace or other penalties in
store for those who disobey it. It is ready to be interpreted and
diversified by suitable minor premisses in all inferential
applications. But these [Greek: a)rchai\] for deduction, differing
widely at different times and places, though generated in the same
manner and enforced by the same sanction, would belong more properly
to the class which Aristotle terms [Greek: ta\ e)/ndoxa].]
We have thus recognized that there exist immediate (ultimate or
primary) propositions, wherein the conjunction between predicate and
subject is such that no intermediate term can be assigned between
them. When A is predicated both of B and C, this may perhaps be in
consequence of some common property possessed by B and C, and such
common property will form a middle term. For example, equality of
angles to two right angles belongs both to an isosceles and to a
scalene triangle, and it belongs to them by reason of their common
property--triangular figure; which last is thus the middle term. But
this need not be always the case.[66] It is possible that the two
propositions--A predicated of B, A predicated of C--may both of them
be immediate propositions; and that there may be no community of
nature between B and C. Whenever a middle term can be found,
demonstration is possible; but where no middle term can be found,
demonstration is impossible. The proposition, whether affirmative or
negative, is then an immediate or indivisible one. Such propositions,
and the terms of which they are composed, are the ultimate elements
or _principia_ of Demonstration. Predicate and subject are brought
constantly into closer and closer conjunction, until at last they
become one and indivisible.[67] Here we reach the unit or element of
the syllogizing process. In all scientific calculations there is
assumed an unit to start from, though in each branch of science it is
a different unit; _e.g._ in barology, the pound-weight; in harmonics,
the quarter-tone; in other branches of science, other units.[68]
Analytical research teaches us that the corresponding unit in
Syllogism is the affirmative or negative proposition which is
primary, immediate, indivisible. In Demonstration and Science it is
the Noûs or Intellect.[69]
[Footnote 66: Analyt. Post. I. xxiii. p. 84, b. 3-18. [Greek: tou=to
d' ou)k a)ei\ ou(/tôs e)/chei.]]
[Footnote 67: Ibid. b. 25-37. [Greek: a)ei\ to\ me/son puknou=tai,
e(/ôs a)diai/reta ge/nêtai kai\ e(/n. e)/sti d' e(/n, o(/tan a)/meson
ge/nêtai kai\ mi/a pro/tasis a(plô=s ê( a)/mesos.]]
[Footnote 68: Analyt. Post. I. xxiii. p. 84, b. 37: [Greek: kai\
ô(/sper e)n toi=s a)/llois ê( a)rchê\ a(plou=n, tou=to d' ou) tau)to\
pantachou=, a)ll' e)n barei= me\n mna=, e)n de\ me/lei di/esis,
a)/llo d' e)n a)/llô|, ou(/tôs e)n sullogismô=| to\ e(\n pro/tasis
a)/mesos, e)n d' a)podei/xei kai\ e)pistê/mê| o( nou=s.]]
[Footnote 69: Ibid. b. 35-p. 85, a. 1.]
Having thus, in the long preceding reasoning, sought to prove that
all demonstration must take its departure from primary undemonstrable
_principia_--from some premisses, affirmative and negative, which are
directly true in themselves, and not demonstrable through any middle
term or intervening propositions, Aristotle now passes to a different
enquiry. We have some demonstrations in which the conclusion is
Particular, others in which it is Universal: again, some Affirmative,
some Negative, Which of the two, in each of these alternatives, is
the best? We have also demonstrations Direct or Ostensive, and
demonstrations Indirect or by way of _Reductio ad Absurdum_. Which of
these two is the best? Both questions appear to have been subjected
to debate by contemporary philosophers.[70]
[Footnote 70: Ibid. xxiv. p. 85, a. 13-18. [Greek: a)mphisbêtei=tai
pote/ra belti/ôn; ô(s d' au(/tôs kai\ peri\ tê=s a)podeiknu/nai
legome/nês kai\ tê=s ei)s to\ a)du/naton a)gou/sês a)podei/xeôs.]]
Aristotle discusses these points dialectically (as indeed he points
out in the Topica that the comparison of two things generally, as to
better and worse, falls under the varieties of
**dialectical enquiry[71]), first stating and next refuting
the arguments on the weaker side. Some persons may think (he says)
that demonstration of the Particular is better than demonstration of
the Universal: first, because it conducts to fuller cognition of that
which the thing is in itself, and not merely that which it is
_quatenus_ member of a class; secondly, because demonstrations of the
Universal are apt to generate an illusory belief, that the Universal
is a distinct reality apart from and independent of all its
particulars (_i.e._, that figure in general has a real existence
apart from all particular figures, and number in general apart from
all particular numbers, &c.), while demonstrations of the Particular
do not lead to any such illusion.[72]
[Footnote 71: Aristot. Topic. III. i. p. 116, a. 1, seq.]
[Footnote 72: Analyt. Post. I. xxiv. p. 85, a. 20-b. 3. Themistius,
pp. 58-59, Spengel: [Greek: ou) ga\r o(mô/numon to\ katho/lou
e)sti/n, ou)de\ phônê\ mo/non, a)ll' u(po/stasis, ou) chôristê\ me\n
ô(/sper ou)de\ ta\ sumbebêko/ta, e)nargô=s d' ou)=n emphainome/nê
toi=s pra/gmasin.] The Scholastic doctrine of _Universalia in re_ is
here expressed very clearly by Themistius.]
To these arguments Aristotle replies:--1. It is not correct to say
that cognition of the Particular is more complete, or bears more upon
real existence, than cognition of the Universal. The reverse would be
nearer to the truth. To know that the isosceles, _quatenus_ triangle,
has its three angles equal to two right angles, is more complete
cognition than knowing simply that the isosceles has its three angles
equal to two right angles. 2. If the Universal be not an equivocal
term--if it represents one property and one definition common to many
particulars, it then has a real existence as much or more than any
one or any number of the particulars. For all these particulars are
perishable, but the class is imperishable. 3. He who believes that
the universal term has one meaning in all the particulars, need not
necessarily believe that it has any meaning _apart_ from all
particulars; he need not believe this about Quiddity, any more than
he believes it about Quality or Quantity. Or if he does believe so,
it is his own individual mistake, not imputable to the demonstration.
4. We have shown that a complete demonstration is one in which the
middle term is the cause or reason of the conclusion. Now the
Universal is most of the nature of Cause; for it represents the First
Essence or the _Per Se_, and is therefore its own cause, or has no
other cause behind it. The demonstration of the Universal has thus
more of the Cause or the _Why_, and is therefore better than the
demonstration of the Particular. 5. In the Final Cause or End of
action, there is always some ultimate end for the sake of which the
intermediate ends are pursued, and which, as it is better than they,
yields, when it is known, the only complete explanation of the
action. So it is also with the Formal Cause: there is one highest
form which contains the _Why_ of the subordinate forms, and the
knowledge of which is therefore better; as when, for example, the
exterior angles of a given isosceles triangle are seen to be equal to
four right angles, not because it is isosceles or triangle, but
because it is a rectilineal figure. 6. Particulars, as such, fall
into infinity of number, and are thus unknowable; the Universal tends
towards oneness and simplicity, and is thus essentially knowable,
more fully demonstrable than the infinity of particulars. The
demonstration thereof is therefore better. 7. It is also better, on
another ground; for he that knows the Universal does in a certain
sense know also the Particular;[73] but he that knows the Particular
cannot be said in any sense to know the Universal. 8. The
_principium_ or perfection of cognition is to be found in the
immediate proposition, true _per se_. When we demonstrate, and thus
employ a middle term, the nearer the middle term approaches to that
_principium_, the better the demonstration is. The demonstration of
the Universal is thus better and more accurate than that of the
Particular.[74]
[Footnote 73: Compare Analyt. Post. I. i. p. 71, a. 25; also
Metaphys. A. p. 981, a. 12.]
[Footnote 74: Analyt. Post. I. xxiv. p. 85, b. 4-p. 86, a. 21. Schol.
p. 233, b. 6: [Greek: o(moi/ôs de\ o)/ntôn gnôri/môn, ê( di'
e)latto/nôn me/sôn ai(retôte/ra; ma=llon ga\r e)ggute/rô tê=s tou=
nou= e)nergei/as.]]
Such are the several reasons enumerated by Aristotle in refutation of
the previous opinion stated in favour of the Particular. Evidently he
does not account them all of equal value: he intimates that some are
purely dialectical ([Greek: logika/]); and he insists most upon the
two following:--1. He that knows the Universal knows in a certain
sense the Particular; if he knows that every triangle has its three
angles equal to two right angles, he knows potentially that the
isosceles has its three angles equal to the same, though he may not
know as yet that the isosceles _is_ a triangle. But he that knows the
Particular does not in any way know the Universal, either actually or
potentially.[75] 2. The Universal is apprehended by Intellect or
Noûs, the highest of all cognitive powers; the Particular terminates
in sensation. Here, I presume, he means, that, in demonstration of