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350 BC
POSTERIOR ANALYTICS
by Aristotle
translated by G. R. G. Mure
Book I
1
ALL instruction given or received by way of argument proceeds from
pre-existent knowledge. This becomes evident upon a survey of all
the species of such instruction. The mathematical sciences and all
other speculative disciplines are acquired in this way, and so are the
two forms of dialectical reasoning, syllogistic and inductive; for
each of these latter make use of old knowledge to impart new, the
syllogism assuming an audience that accepts its premisses, induction
exhibiting the universal as implicit in the clearly known
particular. Again, the persuasion exerted by rhetorical arguments is
in principle the same, since they use either example, a kind of
induction, or enthymeme, a form of syllogism.
The pre-existent knowledge required is of two kinds. In some cases
admission of the fact must be assumed, in others comprehension of
the meaning of the term used, and sometimes both assumptions are
essential. Thus, we assume that every predicate can be either truly
affirmed or truly denied of any subject, and that 'triangle' means
so and so; as regards 'unit' we have to make the double assumption
of the meaning of the word and the existence of the thing. The
reason is that these several objects are not equally obvious to us.
Recognition of a truth may in some cases contain as factors both
previous knowledge and also knowledge acquired simultaneously with
that recognition-knowledge, this latter, of the particulars actually
falling under the universal and therein already virtually known. For
example, the student knew beforehand that the angles of every triangle
are equal to two right angles; but it was only at the actual moment at
which he was being led on to recognize this as true in the instance
before him that he came to know 'this figure inscribed in the
semicircle' to be a triangle. For some things (viz. the singulars
finally reached which are not predicable of anything else as
subject) are only learnt in this way, i.e. there is here no
recognition through a middle of a minor term as subject to a major.
Before he was led on to recognition or before he actually drew a
conclusion, we should perhaps say that in a manner he knew, in a
manner not.
If he did not in an unqualified sense of the term know the existence
of this triangle, how could he know without qualification that its
angles were equal to two right angles? No: clearly he knows not
without qualification but only in the sense that he knows universally.
If this distinction is not drawn, we are faced with the dilemma in the
Meno: either a man will learn nothing or what he already knows; for we
cannot accept the solution which some people offer. A man is asked,
'Do you, or do you not, know that every pair is even?' He says he does
know it. The questioner then produces a particular pair, of the
existence, and so a fortiori of the evenness, of which he was unaware.
The solution which some people offer is to assert that they do not
know that every pair is even, but only that everything which they know
to be a pair is even: yet what they know to be even is that of which
they have demonstrated evenness, i.e. what they made the subject of
their premiss, viz. not merely every triangle or number which they
know to be such, but any and every number or triangle without
reservation. For no premiss is ever couched in the form 'every
number which you know to be such', or 'every rectilinear figure
which you know to be such': the predicate is always construed as
applicable to any and every instance of the thing. On the other
hand, I imagine there is nothing to prevent a man in one sense knowing
what he is learning, in another not knowing it. The strange thing
would be, not if in some sense he knew what he was learning, but if he
were to know it in that precise sense and manner in which he was
learning it.
2
We suppose ourselves to possess unqualified scientific knowledge
of a thing, as opposed to knowing it in the accidental way in which
the sophist knows, when we think that we know the cause on which the
fact depends, as the cause of that fact and of no other, and, further,
that the fact could not be other than it is. Now that scientific
knowing is something of this sort is evident-witness both those who
falsely claim it and those who actually possess it, since the former
merely imagine themselves to be, while the latter are also actually,
in the condition described. Consequently the proper object of
unqualified scientific knowledge is something which cannot be other
than it is.
There may be another manner of knowing as well-that will be
discussed later. What I now assert is that at all events we do know by
demonstration. By demonstration I mean a syllogism productive of
scientific knowledge, a syllogism, that is, the grasp of which is eo
ipso such knowledge. Assuming then that my thesis as to the nature
of scientific knowing is correct, the premisses of demonstrated
knowledge must be true, primary, immediate, better known than and
prior to the conclusion, which is further related to them as effect to
cause. Unless these conditions are satisfied, the basic truths will
not be 'appropriate' to the conclusion. Syllogism there may indeed
be without these conditions, but such syllogism, not being
productive of scientific knowledge, will not be demonstration. The
premisses must be true: for that which is non-existent cannot be
known-we cannot know, e.g. that the diagonal of a square is
commensurate with its side. The premisses must be primary and
indemonstrable; otherwise they will require demonstration in order
to be known, since to have knowledge, if it be not accidental
knowledge, of things which are demonstrable, means precisely to have a
demonstration of them. The premisses must be the causes of the
conclusion, better known than it, and prior to it; its causes, since
we possess scientific knowledge of a thing only when we know its
cause; prior, in order to be causes; antecedently known, this
antecedent knowledge being not our mere understanding of the
meaning, but knowledge of the fact as well. Now 'prior' and 'better
known' are ambiguous terms, for there is a difference between what
is prior and better known in the order of being and what is prior
and better known to man. I mean that objects nearer to sense are prior
and better known to man; objects without qualification prior and
better known are those further from sense. Now the most universal
causes are furthest from sense and particular causes are nearest to
sense, and they are thus exactly opposed to one another. In saying
that the premisses of demonstrated knowledge must be primary, I mean
that they must be the 'appropriate' basic truths, for I identify
primary premiss and basic truth. A 'basic truth' in a demonstration is
an immediate proposition. An immediate proposition is one which has no
other proposition prior to it. A proposition is either part of an
enunciation, i.e. it predicates a single attribute of a single
subject. If a proposition is dialectical, it assumes either part
indifferently; if it is demonstrative, it lays down one part to the
definite exclusion of the other because that part is true. The term
'enunciation' denotes either part of a contradiction indifferently.
A contradiction is an opposition which of its own nature excludes a
middle. The part of a contradiction which conjoins a predicate with
a subject is an affirmation; the part disjoining them is a negation. I
call an immediate basic truth of syllogism a 'thesis' when, though
it is not susceptible of proof by the teacher, yet ignorance of it
does not constitute a total bar to progress on the part of the
pupil: one which the pupil must know if he is to learn anything
whatever is an axiom. I call it an axiom because there are such truths
and we give them the name of axioms par excellence. If a thesis
assumes one part or the other of an enunciation, i.e. asserts either
the existence or the non-existence of a subject, it is a hypothesis;
if it does not so assert, it is a definition. Definition is a 'thesis'
or a 'laying something down', since the arithmetician lays it down
that to be a unit is to be quantitatively indivisible; but it is not a
hypothesis, for to define what a unit is is not the same as to
affirm its existence.
Now since the required ground of our knowledge-i.e. of our
conviction-of a fact is the possession of such a syllogism as we
call demonstration, and the ground of the syllogism is the facts
constituting its premisses, we must not only know the primary
premisses-some if not all of them-beforehand, but know them better
than the conclusion: for the cause of an attribute's inherence in a
subject always itself inheres in the subject more firmly than that
attribute; e.g. the cause of our loving anything is dearer to us
than the object of our love. So since the primary premisses are the
cause of our knowledge-i.e. of our conviction-it follows that we
know them better-that is, are more convinced of them-than their
consequences, precisely because of our knowledge of the latter is
the effect of our knowledge of the premisses. Now a man cannot believe
in anything more than in the things he knows, unless he has either
actual knowledge of it or something better than actual knowledge.
But we are faced with this paradox if a student whose belief rests
on demonstration has not prior knowledge; a man must believe in
some, if not in all, of the basic truths more than in the
conclusion. Moreover, if a man sets out to acquire the scientific
knowledge that comes through demonstration, he must not only have a
better knowledge of the basic truths and a firmer conviction of them
than of the connexion which is being demonstrated: more than this,
nothing must be more certain or better known to him than these basic
truths in their character as contradicting the fundamental premisses
which lead to the opposed and erroneous conclusion. For indeed the
conviction of pure science must be unshakable.
3
Some hold that, owing to the necessity of knowing the primary
premisses, there is no scientific knowledge. Others think there is,
but that all truths are demonstrable. Neither doctrine is either
true or a necessary deduction from the premisses. The first school,
assuming that there is no way of knowing other than by
demonstration, maintain that an infinite regress is involved, on the
ground that if behind the prior stands no primary, we could not know
the posterior through the prior (wherein they are right, for one
cannot traverse an infinite series): if on the other hand-they say-the
series terminates and there are primary premisses, yet these are
unknowable because incapable of demonstration, which according to them
is the only form of knowledge. And since thus one cannot know the
primary premisses, knowledge of the conclusions which follow from them
is not pure scientific knowledge nor properly knowing at all, but
rests on the mere supposition that the premisses are true. The other
party agree with them as regards knowing, holding that it is only
possible by demonstration, but they see no difficulty in holding
that all truths are demonstrated, on the ground that demonstration may
be circular and reciprocal.
Our own doctrine is that not all knowledge is demonstrative: on
the contrary, knowledge of the immediate premisses is independent of
demonstration. (The necessity of this is obvious; for since we must
know the prior premisses from which the demonstration is drawn, and
since the regress must end in immediate truths, those truths must be
indemonstrable.) Such, then, is our doctrine, and in addition we
maintain that besides scientific knowledge there is its originative
source which enables us to recognize the definitions.
Now demonstration must be based on premisses prior to and better
known than the conclusion; and the same things cannot simultaneously
be both prior and posterior to one another: so circular
demonstration is clearly not possible in the unqualified sense of
'demonstration', but only possible if 'demonstration' be extended to
include that other method of argument which rests on a distinction
between truths prior to us and truths without qualification prior,
i.e. the method by which induction produces knowledge. But if we
accept this extension of its meaning, our definition of unqualified
knowledge will prove faulty; for there seem to be two kinds of it.
Perhaps, however, the second form of demonstration, that which
proceeds from truths better known to us, is not demonstration in the
unqualified sense of the term.
The advocates of circular demonstration are not only faced with
the difficulty we have just stated: in addition their theory reduces
to the mere statement that if a thing exists, then it does exist-an
easy way of proving anything. That this is so can be clearly shown
by taking three terms, for to constitute the circle it makes no
difference whether many terms or few or even only two are taken.
Thus by direct proof, if A is, B must be; if B is, C must be;
therefore if A is, C must be. Since then-by the circular proof-if A
is, B must be, and if B is, A must be, A may be substituted for C
above. Then 'if B is, A must be'='if B is, C must be', which above
gave the conclusion 'if A is, C must be': but C and A have been
identified. Consequently the upholders of circular demonstration are
in the position of saying that if A is, A must be-a simple way of
proving anything. Moreover, even such circular demonstration is
impossible except in the case of attributes that imply one another,
viz. 'peculiar' properties.
Now, it has been shown that the positing of one thing-be it one
term or one premiss-never involves a necessary consequent: two
premisses constitute the first and smallest foundation for drawing a
conclusion at all and therefore a fortiori for the demonstrative
syllogism of science. If, then, A is implied in B and C, and B and C
are reciprocally implied in one another and in A, it is possible, as
has been shown in my writings on the syllogism, to prove all the
assumptions on which the original conclusion rested, by circular
demonstration in the first figure. But it has also been shown that
in the other figures either no conclusion is possible, or at least
none which proves both the original premisses. Propositions the
terms of which are not convertible cannot be circularly demonstrated
at all, and since convertible terms occur rarely in actual
demonstrations, it is clearly frivolous and impossible to say that
demonstration is reciprocal and that therefore everything can be
demonstrated.
4
Since the object of pure scientific knowledge cannot be other than
it is, the truth obtained by demonstrative knowledge will be
necessary. And since demonstrative knowledge is only present when we
have a demonstration, it follows that demonstration is an inference
from necessary premisses. So we must consider what are the premisses
of demonstration-i.e. what is their character: and as a preliminary,
let us define what we mean by an attribute 'true in every instance
of its subject', an 'essential' attribute, and a 'commensurate and
universal' attribute. I call 'true in every instance' what is truly
predicable of all instances-not of one to the exclusion of
others-and at all times, not at this or that time only; e.g. if animal
is truly predicable of every instance of man, then if it be true to
say 'this is a man', 'this is an animal' is also true, and if the
one be true now the other is true now. A corresponding account holds
if point is in every instance predicable as contained in line. There
is evidence for this in the fact that the objection we raise against a
proposition put to us as true in every instance is either an
instance in which, or an occasion on which, it is not true.
Essential attributes are (1) such as belong to their subject as
elements in its essential nature (e.g. line thus belongs to
triangle, point to line; for the very being or 'substance' of triangle
and line is composed of these elements, which are contained in the
formulae defining triangle and line): (2) such that, while they belong
to certain subjects, the subjects to which they belong are contained
in the attribute's own defining formula. Thus straight and curved
belong to line, odd and even, prime and compound, square and oblong,
to number; and also the formula defining any one of these attributes
contains its subject-e.g. line or number as the case may be.
Extending this classification to all other attributes, I distinguish
those that answer the above description as belonging essentially to
their respective subjects; whereas attributes related in neither of
these two ways to their subjects I call accidents or 'coincidents';
e.g. musical or white is a 'coincident' of animal.
Further (a) that is essential which is not predicated of a subject
other than itself: e.g. 'the walking [thing]' walks and is white in
virtue of being something else besides; whereas substance, in the
sense of whatever signifies a 'this somewhat', is not what it is in
virtue of being something else besides. Things, then, not predicated
of a subject I call essential; things predicated of a subject I call
accidental or 'coincidental'.
In another sense again (b) a thing consequentially connected with
anything is essential; one not so connected is 'coincidental'. An
example of the latter is 'While he was walking it lightened': the
lightning was not due to his walking; it was, we should say, a
coincidence. If, on the other hand, there is a consequential
connexion, the predication is essential; e.g. if a beast dies when its
throat is being cut, then its death is also essentially connected with
the cutting, because the cutting was the cause of death, not death a
'coincident' of the cutting.
So far then as concerns the sphere of connexions scientifically
known in the unqualified sense of that term, all attributes which
(within that sphere) are essential either in the sense that their
subjects are contained in them, or in the sense that they are
contained in their subjects, are necessary as well as
consequentially connected with their subjects. For it is impossible
for them not to inhere in their subjects either simply or in the
qualified sense that one or other of a pair of opposites must inhere
in the subject; e.g. in line must be either straightness or curvature,
in number either oddness or evenness. For within a single identical
genus the contrary of a given attribute is either its privative or its
contradictory; e.g. within number what is not odd is even, inasmuch as
within this sphere even is a necessary consequent of not-odd. So,
since any given predicate must be either affirmed or denied of any
subject, essential attributes must inhere in their subjects of
necessity.
Thus, then, we have established the distinction between the
attribute which is 'true in every instance' and the 'essential'
attribute.
I term 'commensurately universal' an attribute which belongs to
every instance of its subject, and to every instance essentially and
as such; from which it clearly follows that all commensurate
universals inhere necessarily in their subjects. The essential
attribute, and the attribute that belongs to its subject as such,
are identical. E.g. point and straight belong to line essentially, for
they belong to line as such; and triangle as such has two right
angles, for it is essentially equal to two right angles.
An attribute belongs commensurately and universally to a subject
when it can be shown to belong to any random instance of that
subject and when the subject is the first thing to which it can be
shown to belong. Thus, e.g. (1) the equality of its angles to two
right angles is not a commensurately universal attribute of figure.
For though it is possible to show that a figure has its angles equal
to two right angles, this attribute cannot be demonstrated of any
figure selected at haphazard, nor in demonstrating does one take a
figure at random-a square is a figure but its angles are not equal
to two right angles. On the other hand, any isosceles triangle has its
angles equal to two right angles, yet isosceles triangle is not the
primary subject of this attribute but triangle is prior. So whatever
can be shown to have its angles equal to two right angles, or to
possess any other attribute, in any random instance of itself and
primarily-that is the first subject to which the predicate in question
belongs commensurately and universally, and the demonstration, in
the essential sense, of any predicate is the proof of it as
belonging to this first subject commensurately and universally:
while the proof of it as belonging to the other subjects to which it
attaches is demonstration only in a secondary and unessential sense.
Nor again (2) is equality to two right angles a commensurately
universal attribute of isosceles; it is of wider application.
5
We must not fail to observe that we often fall into error because
our conclusion is not in fact primary and commensurately universal
in the sense in which we think we prove it so. We make this mistake
(1) when the subject is an individual or individuals above which there
is no universal to be found: (2) when the subjects belong to different
species and there is a higher universal, but it has no name: (3)
when the subject which the demonstrator takes as a whole is really
only a part of a larger whole; for then the demonstration will be true
of the individual instances within the part and will hold in every
instance of it, yet the demonstration will not be true of this subject
primarily and commensurately and universally. When a demonstration
is true of a subject primarily and commensurately and universally,
that is to be taken to mean that it is true of a given subject
primarily and as such. Case (3) may be thus exemplified. If a proof
were given that perpendiculars to the same line are parallel, it might
be supposed that lines thus perpendicular were the proper subject of
the demonstration because being parallel is true of every instance
of them. But it is not so, for the parallelism depends not on these
angles being equal to one another because each is a right angle, but
simply on their being equal to one another. An example of (1) would be
as follows: if isosceles were the only triangle, it would be thought
to have its angles equal to two right angles qua isosceles. An
instance of (2) would be the law that proportionals alternate.
Alternation used to be demonstrated separately of numbers, lines,
solids, and durations, though it could have been proved of them all by
a single demonstration. Because there was no single name to denote
that in which numbers, lengths, durations, and solids are identical,
and because they differed specifically from one another, this property
was proved of each of them separately. To-day, however, the proof is
commensurately universal, for they do not possess this attribute qua
lines or qua numbers, but qua manifesting this generic character which
they are postulated as possessing universally. Hence, even if one
prove of each kind of triangle that its angles are equal to two
right angles, whether by means of the same or different proofs; still,
as long as one treats separately equilateral, scalene, and
isosceles, one does not yet know, except sophistically, that
triangle has its angles equal to two right angles, nor does one yet
know that triangle has this property commensurately and universally,
even if there is no other species of triangle but these. For one
does not know that triangle as such has this property, nor even that
'all' triangles have it-unless 'all' means 'each taken singly': if
'all' means 'as a whole class', then, though there be none in which
one does not recognize this property, one does not know it of 'all
triangles'.
When, then, does our knowledge fail of commensurate universality,
and when it is unqualified knowledge? If triangle be identical in
essence with equilateral, i.e. with each or all equilaterals, then
clearly we have unqualified knowledge: if on the other hand it be not,
and the attribute belongs to equilateral qua triangle; then our
knowledge fails of commensurate universality. 'But', it will be asked,
'does this attribute belong to the subject of which it has been
demonstrated qua triangle or qua isosceles? What is the point at which
the subject. to which it belongs is primary? (i.e. to what subject can
it be demonstrated as belonging commensurately and universally?)'
Clearly this point is the first term in which it is found to inhere as
the elimination of inferior differentiae proceeds. Thus the angles
of a brazen isosceles triangle are equal to two right angles: but
eliminate brazen and isosceles and the attribute remains. 'But'-you
may say-'eliminate figure or limit, and the attribute vanishes.' True,
but figure and limit are not the first differentiae whose
elimination destroys the attribute. 'Then what is the first?' If it is
triangle, it will be in virtue of triangle that the attribute
belongs to all the other subjects of which it is predicable, and
triangle is the subject to which it can be demonstrated as belonging
commensurately and universally.
6
Demonstrative knowledge must rest on necessary basic truths; for the
object of scientific knowledge cannot be other than it is. Now
attributes attaching essentially to their subjects attach
necessarily to them: for essential attributes are either elements in
the essential nature of their subjects, or contain their subjects as
elements in their own essential nature. (The pairs of opposites
which the latter class includes are necessary because one member or
the other necessarily inheres.) It follows from this that premisses of
the demonstrative syllogism must be connexions essential in the
sense explained: for all attributes must inhere essentially or else be
accidental, and accidental attributes are not necessary to their
subjects.
We must either state the case thus, or else premise that the
conclusion of demonstration is necessary and that a demonstrated
conclusion cannot be other than it is, and then infer that the
conclusion must be developed from necessary premisses. For though
you may reason from true premisses without demonstrating, yet if
your premisses are necessary you will assuredly demonstrate-in such
necessity you have at once a distinctive character of demonstration.
That demonstration proceeds from necessary premisses is also indicated
by the fact that the objection we raise against a professed
demonstration is that a premiss of it is not a necessary truth-whether
we think it altogether devoid of necessity, or at any rate so far as
our opponent's previous argument goes. This shows how naive it is to
suppose one's basic truths rightly chosen if one starts with a
proposition which is (1) popularly accepted and (2) true, such as
the sophists' assumption that to know is the same as to possess
knowledge. For (1) popular acceptance or rejection is no criterion
of a basic truth, which can only be the primary law of the genus
constituting the subject matter of the demonstration; and (2) not
all truth is 'appropriate'.
A further proof that the conclusion must be the development of
necessary premisses is as follows. Where demonstration is possible,
one who can give no account which includes the cause has no scientific
knowledge. If, then, we suppose a syllogism in which, though A
necessarily inheres in C, yet B, the middle term of the demonstration,
is not necessarily connected with A and C, then the man who argues
thus has no reasoned knowledge of the conclusion, since this
conclusion does not owe its necessity to the middle term; for though
the conclusion is necessary, the mediating link is a contingent
fact. Or again, if a man is without knowledge now, though he still
retains the steps of the argument, though there is no change in
himself or in the fact and no lapse of memory on his part; then
neither had he knowledge previously. But the mediating link, not being
necessary, may have perished in the interval; and if so, though
there be no change in him nor in the fact, and though he will still
retain the steps of the argument, yet he has not knowledge, and
therefore had not knowledge before. Even if the link has not
actually perished but is liable to perish, this situation is
possible and might occur. But such a condition cannot be knowledge.
When the conclusion is necessary, the middle through which it was
proved may yet quite easily be non-necessary. You can in fact infer
the necessary even from a non-necessary premiss, just as you can infer
the true from the not true. On the other hand, when the middle is
necessary the conclusion must be necessary; just as true premisses
always give a true conclusion. Thus, if A is necessarily predicated of
B and B of C, then A is necessarily predicated of C. But when the
conclusion is nonnecessary the middle cannot be necessary either.
Thus: let A be predicated non-necessarily of C but necessarily of B,
and let B be a necessary predicate of C; then A too will be a
necessary predicate of C, which by hypothesis it is not.
To sum up, then: demonstrative knowledge must be knowledge of a
necessary nexus, and therefore must clearly be obtained through a
necessary middle term; otherwise its possessor will know neither the
cause nor the fact that his conclusion is a necessary connexion.
Either he will mistake the non-necessary for the necessary and believe
the necessity of the conclusion without knowing it, or else he will
not even believe it-in which case he will be equally ignorant, whether
he actually infers the mere fact through middle terms or the
reasoned fact and from immediate premisses.
Of accidents that are not essential according to our definition of
essential there is no demonstrative knowledge; for since an
accident, in the sense in which I here speak of it, may also not
inhere, it is impossible to prove its inherence as a necessary
conclusion. A difficulty, however, might be raised as to why in
dialectic, if the conclusion is not a necessary connexion, such and
such determinate premisses should be proposed in order to deal with
such and such determinate problems. Would not the result be the same
if one asked any questions whatever and then merely stated one's
conclusion? The solution is that determinate questions have to be put,
not because the replies to them affirm facts which necessitate facts
affirmed by the conclusion, but because these answers are propositions
which if the answerer affirm, he must affirm the conclusion and affirm
it with truth if they are true.
Since it is just those attributes within every genus which are
essential and possessed by their respective subjects as such that
are necessary it is clear that both the conclusions and the
premisses of demonstrations which produce scientific knowledge are
essential. For accidents are not necessary: and, further, since
accidents are not necessary one does not necessarily have reasoned
knowledge of a conclusion drawn from them (this is so even if the
accidental premisses are invariable but not essential, as in proofs
through signs; for though the conclusion be actually essential, one
will not know it as essential nor know its reason); but to have
reasoned knowledge of a conclusion is to know it through its cause. We
may conclude that the middle must be consequentially connected with
the minor, and the major with the middle.
7
It follows that we cannot in demonstrating pass from one genus to
another. We cannot, for instance, prove geometrical truths by
arithmetic. For there are three elements in demonstration: (1) what is
proved, the conclusion-an attribute inhering essentially in a genus;
(2) the axioms, i.e. axioms which are premisses of demonstration;
(3) the subject-genus whose attributes, i.e. essential properties, are
revealed by the demonstration. The axioms which are premisses of
demonstration may be identical in two or more sciences: but in the
case of two different genera such as arithmetic and geometry you
cannot apply arithmetical demonstration to the properties of
magnitudes unless the magnitudes in question are numbers. How in
certain cases transference is possible I will explain later.
Arithmetical demonstration and the other sciences likewise
possess, each of them, their own genera; so that if the
demonstration is to pass from one sphere to another, the genus must be
either absolutely or to some extent the same. If this is not so,
transference is clearly impossible, because the extreme and the middle
terms must be drawn from the same genus: otherwise, as predicated,
they will not be essential and will thus be accidents. That is why
it cannot be proved by geometry that opposites fall under one science,
nor even that the product of two cubes is a cube. Nor can the
theorem of any one science be demonstrated by means of another
science, unless these theorems are related as subordinate to
superior (e.g. as optical theorems to geometry or harmonic theorems to
arithmetic). Geometry again cannot prove of lines any property which
they do not possess qua lines, i.e. in virtue of the fundamental
truths of their peculiar genus: it cannot show, for example, that
the straight line is the most beautiful of lines or the contrary of
the circle; for these qualities do not belong to lines in virtue of
their peculiar genus, but through some property which it shares with
other genera.
8
It is also clear that if the premisses from which the syllogism
proceeds are commensurately universal, the conclusion of such i.e.
in the unqualified sense-must also be eternal. Therefore no
attribute can be demonstrated nor known by strictly scientific
knowledge to inhere in perishable things. The proof can only be
accidental, because the attribute's connexion with its perishable
subject is not commensurately universal but temporary and special.
If such a demonstration is made, one premiss must be perishable and
not commensurately universal (perishable because only if it is
perishable will the conclusion be perishable; not commensurately
universal, because the predicate will be predicable of some
instances of the subject and not of others); so that the conclusion
can only be that a fact is true at the moment-not commensurately and
universally. The same is true of definitions, since a definition is
either a primary premiss or a conclusion of a demonstration, or else
only differs from a demonstration in the order of its terms.
Demonstration and science of merely frequent occurrences-e.g. of
eclipse as happening to the moon-are, as such, clearly eternal:
whereas so far as they are not eternal they are not fully
commensurate. Other subjects too have properties attaching to them
in the same way as eclipse attaches to the moon.
9
It is clear that if the conclusion is to show an attribute
inhering as such, nothing can be demonstrated except from its
'appropriate' basic truths. Consequently a proof even from true,
indemonstrable, and immediate premisses does not constitute knowledge.
Such proofs are like Bryson's method of squaring the circle; for
they operate by taking as their middle a common character-a character,
therefore, which the subject may share with another-and consequently
they apply equally to subjects different in kind. They therefore
afford knowledge of an attribute only as inhering accidentally, not as
belonging to its subject as such: otherwise they would not have been
applicable to another genus.
Our knowledge of any attribute's connexion with a subject is
accidental unless we know that connexion through the middle term in
virtue of which it inheres, and as an inference from basic premisses
essential and 'appropriate' to the subject-unless we know, e.g. the
property of possessing angles equal to two right angles as belonging
to that subject in which it inheres essentially, and as inferred
from basic premisses essential and 'appropriate' to that subject: so
that if that middle term also belongs essentially to the minor, the
middle must belong to the same kind as the major and minor terms.
The only exceptions to this rule are such cases as theorems in
harmonics which are demonstrable by arithmetic. Such theorems are
proved by the same middle terms as arithmetical properties, but with a
qualification-the fact falls under a separate science (for the subject
genus is separate), but the reasoned fact concerns the superior
science, to which the attributes essentially belong. Thus, even
these apparent exceptions show that no attribute is strictly
demonstrable except from its 'appropriate' basic truths, which,
however, in the case of these sciences have the requisite identity
of character.
It is no less evident that the peculiar basic truths of each
inhering attribute are indemonstrable; for basic truths from which
they might be deduced would be basic truths of all that is, and the
science to which they belonged would possess universal sovereignty.
This is so because he knows better whose knowledge is deduced from
higher causes, for his knowledge is from prior premisses when it
derives from causes themselves uncaused: hence, if he knows better
than others or best of all, his knowledge would be science in a higher
or the highest degree. But, as things are, demonstration is not
transferable to another genus, with such exceptions as we have
mentioned of the application of geometrical demonstrations to theorems
in mechanics or optics, or of arithmetical demonstrations to those
of harmonics.
It is hard to be sure whether one knows or not; for it is hard to be
sure whether one's knowledge is based on the basic truths
appropriate to each attribute-the differentia of true knowledge. We
think we have scientific knowledge if we have reasoned from true and
primary premisses. But that is not so: the conclusion must be
homogeneous with the basic facts of the science.
10
I call the basic truths of every genus those clements in it the
existence of which cannot be proved. As regards both these primary
truths and the attributes dependent on them the meaning of the name is
assumed. The fact of their existence as regards the primary truths
must be assumed; but it has to be proved of the remainder, the
attributes. Thus we assume the meaning alike of unity, straight, and
triangular; but while as regards unity and magnitude we assume also
the fact of their existence, in the case of the remainder proof is
required.
Of the basic truths used in the demonstrative sciences some are
peculiar to each science, and some are common, but common only in
the sense of analogous, being of use only in so far as they fall
within the genus constituting the province of the science in question.
Peculiar truths are, e.g. the definitions of line and straight;
common truths are such as 'take equals from equals and equals remain'.
Only so much of these common truths is required as falls within the
genus in question: for a truth of this kind will have the same force
even if not used generally but applied by the geometer only to
magnitudes, or by the arithmetician only to numbers. Also peculiar
to a science are the subjects the existence as well as the meaning
of which it assumes, and the essential attributes of which it
investigates, e.g. in arithmetic units, in geometry points and
lines. Both the existence and the meaning of the subjects are
assumed by these sciences; but of their essential attributes only
the meaning is assumed. For example arithmetic assumes the meaning
of odd and even, square and cube, geometry that of incommensurable, or
of deflection or verging of lines, whereas the existence of these
attributes is demonstrated by means of the axioms and from previous
conclusions as premisses. Astronomy too proceeds in the same way.
For indeed every demonstrative science has three elements: (1) that
which it posits, the subject genus whose essential attributes it
examines; (2) the so-called axioms, which are primary premisses of its
demonstration; (3) the attributes, the meaning of which it assumes.
Yet some sciences may very well pass over some of these elements; e.g.
we might not expressly posit the existence of the genus if its
existence were obvious (for instance, the existence of hot and cold is
more evident than that of number); or we might omit to assume
expressly the meaning of the attributes if it were well understood. In
the way the meaning of axioms, such as 'Take equals from equals and
equals remain', is well known and so not expressly assumed.
Nevertheless in the nature of the case the essential elements of
demonstration are three: the subject, the attributes, and the basic
premisses.
That which expresses necessary self-grounded fact, and which we must
necessarily believe, is distinct both from the hypotheses of a science
and from illegitimate postulate-I say 'must believe', because all
syllogism, and therefore a fortiori demonstration, is addressed not to
the spoken word, but to the discourse within the soul, and though we
can always raise objections to the spoken word, to the inward
discourse we cannot always object. That which is capable of proof
but assumed by the teacher without proof is, if the pupil believes and
accepts it, hypothesis, though only in a limited sense hypothesis-that
is, relatively to the pupil; if the pupil has no opinion or a contrary
opinion on the matter, the same assumption is an illegitimate
postulate. Therein lies the distinction between hypothesis and
illegitimate postulate: the latter is the contrary of the pupil's
opinion, demonstrable, but assumed and used without demonstration.
The definition-viz. those which are not expressed as statements that
anything is or is not-are not hypotheses: but it is in the premisses
of a science that its hypotheses are contained. Definitions require
only to be understood, and this is not hypothesis-unless it be
contended that the pupil's hearing is also an hypothesis required by
the teacher. Hypotheses, on the contrary, postulate facts on the being
of which depends the being of the fact inferred. Nor are the
geometer's hypotheses false, as some have held, urging that one must
not employ falsehood and that the geometer is uttering falsehood in
stating that the line which he draws is a foot long or straight,
when it is actually neither. The truth is that the geometer does not
draw any conclusion from the being of the particular line of which
he speaks, but from what his diagrams symbolize. A further distinction
is that all hypotheses and illegitimate postulates are either
universal or particular, whereas a definition is neither.
11
So demonstration does not necessarily imply the being of Forms nor a
One beside a Many, but it does necessarily imply the possibility of
truly predicating one of many; since without this possibility we
cannot save the universal, and if the universal goes, the middle
term goes witb. it, and so demonstration becomes impossible. We
conclude, then, that there must be a single identical term
unequivocally predicable of a number of individuals.
The law that it is impossible to affirm and deny simultaneously
the same predicate of the same subject is not expressly posited by any
demonstration except when the conclusion also has to be expressed in
that form; in which case the proof lays down as its major premiss that
the major is truly affirmed of the middle but falsely denied. It makes
no difference, however, if we add to the middle, or again to the minor
term, the corresponding negative. For grant a minor term of which it
is true to predicate man-even if it be also true to predicate
not-man of it--still grant simply that man is animal and not
not-animal, and the conclusion follows: for it will still be true to
say that Callias--even if it be also true to say that
not-Callias--is animal and not not-animal. The reason is that the
major term is predicable not only of the middle, but of something
other than the middle as well, being of wider application; so that the
conclusion is not affected even if the middle is extended to cover the
original middle term and also what is not the original middle term.
The law that every predicate can be either truly affirmed or truly
denied of every subject is posited by such demonstration as uses
reductio ad impossibile, and then not always universally, but so far
as it is requisite; within the limits, that is, of the genus-the
genus, I mean (as I have already explained), to which the man of
science applies his demonstrations. In virtue of the common elements
of demonstration-I mean the common axioms which are used as
premisses of demonstration, not the subjects nor the attributes
demonstrated as belonging to them-all the sciences have communion with
one another, and in communion with them all is dialectic and any
science which might attempt a universal proof of axioms such as the
law of excluded middle, the law that the subtraction of equals from
equals leaves equal remainders, or other axioms of the same kind.
Dialectic has no definite sphere of this kind, not being confined to a
single genus. Otherwise its method would not be interrogative; for the
interrogative method is barred to the demonstrator, who cannot use the
opposite facts to prove the same nexus. This was shown in my work on
the syllogism.
12
If a syllogistic question is equivalent to a proposition embodying
one of the two sides of a contradiction, and if each science has its
peculiar propositions from which its peculiar conclusion is developed,
then there is such a thing as a distinctively scientific question, and
it is the interrogative form of the premisses from which the
'appropriate' conclusion of each science is developed. Hence it is
clear that not every question will be relevant to geometry, nor to
medicine, nor to any other science: only those questions will be
geometrical which form premisses for the proof of the theorems of
geometry or of any other science, such as optics, which uses the
same basic truths as geometry. Of the other sciences the like is true.
Of these questions the geometer is bound to give his account, using
the basic truths of geometry in conjunction with his previous
conclusions; of the basic truths the geometer, as such, is not bound
to give any account. The like is true of the other sciences. There
is a limit, then, to the questions which we may put to each man of
science; nor is each man of science bound to answer all inquiries on
each several subject, but only such as fall within the defined field
of his own science. If, then, in controversy with a geometer qua
geometer the disputant confines himself to geometry and proves
anything from geometrical premisses, he is clearly to be applauded; if
he goes outside these he will be at fault, and obviously cannot even
refute the geometer except accidentally. One should therefore not
discuss geometry among those who are not geometers, for in such a
company an unsound argument will pass unnoticed. This is
correspondingly true in the other sciences.
Since there are 'geometrical' questions, does it follow that there
are also distinctively 'ungeometrical' questions? Further, in each
special science-geometry for instance-what kind of error is it that
may vitiate questions, and yet not exclude them from that science?
Again, is the erroneous conclusion one constructed from premisses
opposite to the true premisses, or is it formal fallacy though drawn
from geometrical premisses? Or, perhaps, the erroneous conclusion is
due to the drawing of premisses from another science; e.g. in a
geometrical controversy a musical question is distinctively
ungeometrical, whereas the notion that parallels meet is in one
sense geometrical, being ungeometrical in a different fashion: the
reason being that 'ungeometrical', like 'unrhythmical', is
equivocal, meaning in the one case not geometry at all, in the other
bad geometry? It is this error, i.e. error based on premisses of
this kind-'of' the science but false-that is the contrary of
science. In mathematics the formal fallacy is not so common, because
it is the middle term in which the ambiguity lies, since the major
is predicated of the whole of the middle and the middle of the whole
of the minor (the predicate of course never has the prefix 'all'); and
in mathematics one can, so to speak, see these middle terms with an
intellectual vision, while in dialectic the ambiguity may escape
detection. E.g. 'Is every circle a figure?' A diagram shows that
this is so, but the minor premiss 'Are epics circles?' is shown by the
diagram to be false.
If a proof has an inductive minor premiss, one should not bring an
'objection' against it. For since every premiss must be applicable
to a number of cases (otherwise it will not be true in every instance,
which, since the syllogism proceeds from universals, it must be), then
assuredly the same is true of an 'objection'; since premisses and
'objections' are so far the same that anything which can be validly
advanced as an 'objection' must be such that it could take the form of
a premiss, either demonstrative or dialectical. On the other hand,
arguments formally illogical do sometimes occur through taking as
middles mere attributes of the major and minor terms. An instance of
this is Caeneus' proof that fire increases in geometrical
proportion: 'Fire', he argues, 'increases rapidly, and so does
geometrical proportion'. There is no syllogism so, but there is a
syllogism if the most rapidly increasing proportion is geometrical and
the most rapidly increasing proportion is attributable to fire in
its motion. Sometimes, no doubt, it is impossible to reason from
premisses predicating mere attributes: but sometimes it is possible,
though the possibility is overlooked. If false premisses could never
give true conclusions 'resolution' would be easy, for premisses and
conclusion would in that case inevitably reciprocate. I might then
argue thus: let A be an existing fact; let the existence of A imply
such and such facts actually known to me to exist, which we may call
B. I can now, since they reciprocate, infer A from B.
Reciprocation of premisses and conclusion is more frequent in
mathematics, because mathematics takes definitions, but never an
accident, for its premisses-a second characteristic distinguishing
mathematical reasoning from dialectical disputations.
A science expands not by the interposition of fresh middle terms,
but by the apposition of fresh extreme terms. E.g. A is predicated
of B, B of C, C of D, and so indefinitely. Or the expansion may be
lateral: e.g. one major A, may be proved of two minors, C and E.
Thus let A represent number-a number or number taken
indeterminately; B determinate odd number; C any particular odd
number. We can then predicate A of C. Next let D represent determinate
even number, and E even number. Then A is predicable of E.
13
Knowledge of the fact differs from knowledge of the reasoned fact.
To begin with, they differ within the same science and in two ways:
(1) when the premisses of the syllogism are not immediate (for then
the proximate cause is not contained in them-a necessary condition
of knowledge of the reasoned fact): (2) when the premisses are
immediate, but instead of the cause the better known of the two
reciprocals is taken as the middle; for of two reciprocally predicable
terms the one which is not the cause may quite easily be the better
known and so become the middle term of the demonstration. Thus (2) (a)
you might prove as follows that the planets are near because they do
not twinkle: let C be the planets, B not twinkling, A proximity.
Then B is predicable of C; for the planets do not twinkle. But A is
also predicable of B, since that which does not twinkle is near--we
must take this truth as having been reached by induction or
sense-perception. Therefore A is a necessary predicate of C; so that
we have demonstrated that the planets are near. This syllogism,
then, proves not the reasoned fact but only the fact; since they are
not near because they do not twinkle, but, because they are near, do
not twinkle. The major and middle of the proof, however, may be
reversed, and then the demonstration will be of the reasoned fact.
Thus: let C be the planets, B proximity, A not twinkling. Then B is an
attribute of C, and A-not twinkling-of B. Consequently A is predicable
of C, and the syllogism proves the reasoned fact, since its middle
term is the proximate cause. Another example is the inference that the
moon is spherical from its manner of waxing. Thus: since that which so
waxes is spherical, and since the moon so waxes, clearly the moon is
spherical. Put in this form, the syllogism turns out to be proof of
the fact, but if the middle and major be reversed it is proof of the
reasoned fact; since the moon is not spherical because it waxes in a
certain manner, but waxes in such a manner because it is spherical.
(Let C be the moon, B spherical, and A waxing.) Again (b), in cases
where the cause and the effect are not reciprocal and the effect is
the better known, the fact is demonstrated but not the reasoned
fact. This also occurs (1) when the middle falls outside the major and
minor, for here too the strict cause is not given, and so the
demonstration is of the fact, not of the reasoned fact. For example,
the question 'Why does not a wall breathe?' might be answered,
'Because it is not an animal'; but that answer would not give the
strict cause, because if not being an animal causes the absence of
respiration, then being an animal should be the cause of
respiration, according to the rule that if the negation of causes
the non-inherence of y, the affirmation of x causes the inherence of
y; e.g. if the disproportion of the hot and cold elements is the cause
of ill health, their proportion is the cause of health; and
conversely, if the assertion of x causes the inherence of y, the
negation of x must cause y's non-inherence. But in the case given this
consequence does not result; for not every animal breathes. A
syllogism with this kind of cause takes place in the second figure.
Thus: let A be animal, B respiration, C wall. Then A is predicable
of all B (for all that breathes is animal), but of no C; and
consequently B is predicable of no C; that is, the wall does not
breathe. Such causes are like far-fetched explanations, which
precisely consist in making the cause too remote, as in Anacharsis'
account of why the Scythians have no flute-players; namely because
they have no vines.
Thus, then, do the syllogism of the fact and the syllogism of the
reasoned fact differ within one science and according to the
position of the middle terms. But there is another way too in which
the fact and the reasoned fact differ, and that is when they are
investigated respectively by different sciences. This occurs in the
case of problems related to one another as subordinate and superior,
as when optical problems are subordinated to geometry, mechanical
problems to stereometry, harmonic problems to arithmetic, the data
of observation to astronomy. (Some of these sciences bear almost the
same name; e.g. mathematical and nautical astronomy, mathematical
and acoustical harmonics.) Here it is the business of the empirical
observers to know the fact, of the mathematicians to know the reasoned
fact; for the latter are in possession of the demonstrations giving
the causes, and are often ignorant of the fact: just as we have
often a clear insight into a universal, but through lack of
observation are ignorant of some of its particular instances. These
connexions have a perceptible existence though they are manifestations
of forms. For the mathematical sciences concern forms: they do not
demonstrate properties of a substratum, since, even though the
geometrical subjects are predicable as properties of a perceptible
substratum, it is not as thus predicable that the mathematician
demonstrates properties of them. As optics is related to geometry,
so another science is related to optics, namely the theory of the
rainbow. Here knowledge of the fact is within the province of the
natural philosopher, knowledge of the reasoned fact within that of the
optician, either qua optician or qua mathematical optician. Many
sciences not standing in this mutual relation enter into it at points;
e.g. medicine and geometry: it is the physician's business to know
that circular wounds heal more slowly, the geometer's to know the
reason why.
14
Of all the figures the most scientific is the first. Thus, it is the
vehicle of the demonstrations of all the mathematical sciences, such
as arithmetic, geometry, and optics, and practically all of all
sciences that investigate causes: for the syllogism of the reasoned
fact is either exclusively or generally speaking and in most cases
in this figure-a second proof that this figure is the most scientific;
for grasp of a reasoned conclusion is the primary condition of
knowledge. Thirdly, the first is the only figure which enables us to
pursue knowledge of the essence of a thing. In the second figure no
affirmative conclusion is possible, and knowledge of a thing's essence
must be affirmative; while in the third figure the conclusion can be
affirmative, but cannot be universal, and essence must have a
universal character: e.g. man is not two-footed animal in any
qualified sense, but universally. Finally, the first figure has no
need of the others, while it is by means of the first that the other
two figures are developed, and have their intervals closepacked
until immediate premisses are reached.
Clearly, therefore, the first figure is the primary condition of
knowledge.
15
Just as an attribute A may (as we saw) be atomically connected
with a subject B, so its disconnexion may be atomic. I call 'atomic'
connexions or disconnexions which involve no intermediate term;
since in that case the connexion or disconnexion will not be
mediated by something other than the terms themselves. It follows that
if either A or B, or both A and B, have a genus, their disconnexion
cannot be primary. Thus: let C be the genus of A. Then, if C is not
the genus of B-for A may well have a genus which is not the genus of
B-there will be a syllogism proving A's disconnexion from B thus:
all A is C,
no B is C,
therefore no B is A.
Or if it is B which has a genus D, we have
all B is D,
no D is A,
therefore no B is A, by syllogism;
and the proof will be similar if both A and B have a genus. That the
genus of A need not be the genus of B and vice versa, is shown by
the existence of mutually exclusive coordinate series of
predication. If no term in the series ACD...is predicable of any
term in the series BEF...,and if G-a term in the former series-is
the genus of A, clearly G will not be the genus of B; since, if it
were, the series would not be mutually exclusive. So also if B has a
genus, it will not be the genus of A. If, on the other hand, neither A
nor B has a genus and A does not inhere in B, this disconnexion must
be atomic. If there be a middle term, one or other of them is bound to
have a genus, for the syllogism will be either in the first or the
second figure. If it is in the first, B will have a genus-for the
premiss containing it must be affirmative: if in the second, either
A or B indifferently, since syllogism is possible if either is
contained in a negative premiss, but not if both premisses are
negative.
Hence it is clear that one thing may be atomically disconnected from
another, and we have stated when and how this is possible.
16
Ignorance-defined not as the negation of knowledge but as a positive
state of mind-is error produced by inference.
(1) Let us first consider propositions asserting a predicate's
immediate connexion with or disconnexion from a subject. Here, it is
true, positive error may befall one in alternative ways; for it may
arise where one directly believes a connexion or disconnexion as
well as where one's belief is acquired by inference. The error,
however, that consists in a direct belief is without complication; but
the error resulting from inference-which here concerns us-takes many
forms. Thus, let A be atomically disconnected from all B: then the
conclusion inferred through a middle term C, that all B is A, will
be a case of error produced by syllogism. Now, two cases are possible.
Either (a) both premisses, or (b) one premiss only, may be false.
(a) If neither A is an attribute of any C nor C of any B, whereas
the contrary was posited in both cases, both premisses will be
false. (C may quite well be so related to A and B that C is neither
subordinate to A nor a universal attribute of B: for B, since A was
said to be primarily disconnected from B, cannot have a genus, and A
need not necessarily be a universal attribute of all things.
Consequently both premisses may be false.) On the other hand, (b)
one of the premisses may be true, though not either indifferently
but only the major A-C since, B having no genus, the premiss C-B
will always be false, while A-C may be true. This is the case if,
for example, A is related atomically to both C and B; because when the
same term is related atomically to more terms than one, neither of
those terms will belong to the other. It is, of course, equally the
case if A-C is not atomic.
Error of attribution, then, occurs through these causes and in
this form only-for we found that no syllogism of universal attribution
was possible in any figure but the first. On the other hand, an
error of non-attribution may occur either in the first or in the
second figure. Let us therefore first explain the various forms it
takes in the first figure and the character of the premisses in each
case.
(c) It may occur when both premisses are false; e.g. supposing A
atomically connected with both C and B, if it be then assumed that
no C is and all B is C, both premisses are false.
(d) It is also possible when one is false. This may be either
premiss indifferently. A-C may be true, C-B false-A-C true because A
is not an attribute of all things, C-B false because C, which never
has the attribute A, cannot be an attribute of B; for if C-B were
true, the premiss A-C would no longer be true, and besides if both
premisses were true, the conclusion would be true. Or again, C-B may
be true and A-C false; e.g. if both C and A contain B as genera, one
of them must be subordinate to the other, so that if the premiss takes
the form No C is A, it will be false. This makes it clear that whether
either or both premisses are false, the conclusion will equally be
false.
In the second figure the premisses cannot both be wholly false;
for if all B is A, no middle term can be with truth universally
affirmed of one extreme and universally denied of the other: but
premisses in which the middle is affirmed of one extreme and denied of
the other are the necessary condition if one is to get a valid
inference at all. Therefore if, taken in this way, they are wholly
false, their contraries conversely should be wholly true. But this
is impossible. On the other hand, there is nothing to prevent both
premisses being partially false; e.g. if actually some A is C and some
B is C, then if it is premised that all A is C and no B is C, both
premisses are false, yet partially, not wholly, false. The same is
true if the major is made negative instead of the minor. Or one
premiss may be wholly false, and it may be either of them. Thus,
supposing that actually an attribute of all A must also be an
attribute of all B, then if C is yet taken to be a universal attribute
of all but universally non-attributable to B, C-A will be true but C-B
false. Again, actually that which is an attribute of no B will not
be an attribute of all A either; for if it be an attribute of all A,
it will also be an attribute of all B, which is contrary to
supposition; but if C be nevertheless assumed to be a universal
attribute of A, but an attribute of no B, then the premiss C-B is true
but the major is false. The case is similar if the major is made the
negative premiss. For in fact what is an attribute of no A will not be
an attribute of any B either; and if it be yet assumed that C is
universally non-attributable to A, but a universal attribute of B, the
premiss C-A is true but the minor wholly false. Again, in fact it is
false to assume that that which is an attribute of all B is an
attribute of no A, for if it be an attribute of all B, it must be an
attribute of some A. If then C is nevertheless assumed to be an
attribute of all B but of no A, C-B will be true but C-A false.
It is thus clear that in the case of atomic propositions erroneous
inference will be possible not only when both premisses are false
but also when only one is false.
17
In the case of attributes not atomically connected with or
disconnected from their subjects, (a) (i) as long as the false
conclusion is inferred through the 'appropriate' middle, only the
major and not both premisses can be false. By 'appropriate middle' I
mean the middle term through which the contradictory-i.e. the
true-conclusion is inferrible. Thus, let A be attributable to B
through a middle term C: then, since to produce a conclusion the
premiss C-B must be taken affirmatively, it is clear that this premiss
must always be true, for its quality is not changed. But the major A-C
is false, for it is by a change in the quality of A-C that the
conclusion becomes its contradictory-i.e. true. Similarly (ii) if
the middle is taken from another series of predication; e.g. suppose D
to be not only contained within A as a part within its whole but
also predicable of all B. Then the premiss D-B must remain
unchanged, but the quality of A-D must be changed; so that D-B is
always true, A-D always false. Such error is practically identical
with that which is inferred through the 'appropriate' middle. On the
other hand, (b) if the conclusion is not inferred through the
'appropriate' middle-(i) when the middle is subordinate to A but is
predicable of no B, both premisses must be false, because if there
is to be a conclusion both must be posited as asserting the contrary
of what is actually the fact, and so posited both become false: e.g.
suppose that actually all D is A but no B is D; then if these
premisses are changed in quality, a conclusion will follow and both of
the new premisses will be false. When, however, (ii) the middle D is
not subordinate to A, A-D will be true, D-B false-A-D true because A
was not subordinate to D, D-B false because if it had been true, the
conclusion too would have been true; but it is ex hypothesi false.
When the erroneous inference is in the second figure, both premisses
cannot be entirely false; since if B is subordinate to A, there can be
no middle predicable of all of one extreme and of none of the other,
as was stated before. One premiss, however, may be false, and it may
be either of them. Thus, if C is actually an attribute of both A and
B, but is assumed to be an attribute of A only and not of B, C-A
will be true, C-B false: or again if C be assumed to be attributable
to B but to no A, C-B will be true, C-A false.
We have stated when and through what kinds of premisses error will
result in cases where the erroneous conclusion is negative. If the
conclusion is affirmative, (a) (i) it may be inferred through the
'appropriate' middle term. In this case both premisses cannot be false
since, as we said before, C-B must remain unchanged if there is to
be a conclusion, and consequently A-C, the quality of which is
changed, will always be false. This is equally true if (ii) the middle
is taken from another series of predication, as was stated to be the
case also with regard to negative error; for D-B must remain
unchanged, while the quality of A-D must be converted, and the type of
error is the same as before.
(b) The middle may be inappropriate. Then (i) if D is subordinate to
A, A-D will be true, but D-B false; since A may quite well be
predicable of several terms no one of which can be subordinated to
another. If, however, (ii) D is not subordinate to A, obviously A-D,
since it is affirmed, will always be false, while D-B may be either
true or false; for A may very well be an attribute of no D, whereas
all B is D, e.g. no science is animal, all music is science. Equally
well A may be an attribute of no D, and D of no B. It emerges, then,
that if the middle term is not subordinate to the major, not only both
premisses but either singly may be false.
Thus we have made it clear how many varieties of erroneous inference
are liable to happen and through what kinds of premisses they occur,
in the case both of immediate and of demonstrable truths.
18
It is also clear that the loss of any one of the senses entails
the loss of a corresponding portion of knowledge, and that, since we
learn either by induction or by demonstration, this knowledge cannot
be acquired. Thus demonstration develops from universals, induction
from particulars; but since it is possible to familiarize the pupil
with even the so-called mathematical abstractions only through
induction-i.e. only because each subject genus possesses, in virtue of
a determinate mathematical character, certain properties which can
be treated as separate even though they do not exist in isolation-it
is consequently impossible to come to grasp universals except
through induction. But induction is impossible for those who have
not sense-perception. For it is sense-perception alone which is
adequate for grasping the particulars: they cannot be objects of
scientific knowledge, because neither can universals give us knowledge
of them without induction, nor can we get it through induction without
sense-perception.
19
Every syllogism is effected by means of three terms. One kind of
syllogism serves to prove that A inheres in C by showing that A
inheres in B and B in C; the other is negative and one of its
premisses asserts one term of another, while the other denies one term
of another. It is clear, then, that these are the fundamentals and
so-called hypotheses of syllogism. Assume them as they have been
stated, and proof is bound to follow-proof that A inheres in C through
B, and again that A inheres in B through some other middle term, and
similarly that B inheres in C. If our reasoning aims at gaining
credence and so is merely dialectical, it is obvious that we have only
to see that our inference is based on premisses as credible as
possible: so that if a middle term between A and B is credible
though not real, one can reason through it and complete a
dialectical syllogism. If, however, one is aiming at truth, one must
be guided by the real connexions of subjects and attributes. Thus:
since there are attributes which are predicated of a subject
essentially or naturally and not coincidentally-not, that is, in the
sense in which we say 'That white (thing) is a man', which is not
the same mode of predication as when we say 'The man is white': the
man is white not because he is something else but because he is man,
but the white is man because 'being white' coincides with 'humanity'
within one substratum-therefore there are terms such as are
naturally subjects of predicates. Suppose, then, C such a term not
itself attributable to anything else as to a subject, but the
proximate subject of the attribute B--i.e. so that B-C is immediate;
suppose further E related immediately to F, and F to B. The first
question is, must this series terminate, or can it proceed to
infinity? The second question is as follows: Suppose nothing is
essentially predicated of A, but A is predicated primarily of H and of
no intermediate prior term, and suppose H similarly related to G and G
to B; then must this series also terminate, or can it too proceed to
infinity? There is this much difference between the questions: the
first is, is it possible to start from that which is not itself
attributable to anything else but is the subject of attributes, and
ascend to infinity? The second is the problem whether one can start
from that which is a predicate but not itself a subject of predicates,
and descend to infinity? A third question is, if the extreme terms are
fixed, can there be an infinity of middles? I mean this: suppose for
example that A inheres in C and B is intermediate between them, but
between B and A there are other middles, and between these again fresh
middles; can these proceed to infinity or can they not? This is the
equivalent of inquiring, do demonstrations proceed to infinity, i.e.
is everything demonstrable? Or do ultimate subject and primary
attribute limit one another?
I hold that the same questions arise with regard to negative
conclusions and premisses: viz. if A is attributable to no B, then
either this predication will be primary, or there will be an
intermediate term prior to B to which a is not attributable-G, let
us say, which is attributable to all B-and there may still be
another term H prior to G, which is attributable to all G. The same
questions arise, I say, because in these cases too either the series
of prior terms to which a is not attributable is infinite or it
terminates.
One cannot ask the same questions in the case of reciprocating
terms, since when subject and predicate are convertible there is
neither primary nor ultimate subject, seeing that all the
reciprocals qua subjects stand in the same relation to one another,
whether we say that the subject has an infinity of attributes or
that both subjects and attributes-and we raised the question in both
cases-are infinite in number. These questions then cannot be
asked-unless, indeed, the terms can reciprocate by two different
modes, by accidental predication in one relation and natural
predication in the other.
20
Now, it is clear that if the predications terminate in both the
upward and the downward direction (by 'upward' I mean the ascent to
the more universal, by 'downward' the descent to the more particular),
the middle terms cannot be infinite in number. For suppose that A is
predicated of F, and that the intermediates-call them BB'B"...-are
infinite, then clearly you might descend from and find one term
predicated of another ad infinitum, since you have an infinity of
terms between you and F; and equally, if you ascend from F, there
are infinite terms between you and A. It follows that if these
processes are impossible there cannot be an infinity of
intermediates between A and F. Nor is it of any effect to urge that
some terms of the series AB...F are contiguous so as to exclude
intermediates, while others cannot be taken into the argument at
all: whichever terms of the series B...I take, the number of
intermediates in the direction either of A or of F must be finite or
infinite: where the infinite series starts, whether from the first
term or from a later one, is of no moment, for the succeeding terms in
any case are infinite in number.
21
Further, if in affirmative demonstration the series terminates in
both directions, clearly it will terminate too in negative
demonstration. Let us assume that we cannot proceed to infinity either
by ascending from the ultimate term (by 'ultimate term' I mean a
term such as was, not itself attributable to a subject but itself
the subject of attributes), or by descending towards an ultimate
from the primary term (by 'primary term' I mean a term predicable of a
subject but not itself a subject). If this assumption is justified,
the series will also terminate in the case of negation. For a negative
conclusion can be proved in all three figures. In the first figure
it is proved thus: no B is A, all C is B. In packing the interval
B-C we must reach immediate propositions--as is always the case with
the minor premiss--since B-C is affirmative. As regards the other
premiss it is plain that if the major term is denied of a term D prior
to B, D will have to be predicable of all B, and if the major is
denied of yet another term prior to D, this term must be predicable of
all D. Consequently, since the ascending series is finite, the descent
will also terminate and there will be a subject of which A is
primarily non-predicable. In the second figure the syllogism is, all A
is B, no C is B,..no C is A. If proof of this is required, plainly
it may be shown either in the first figure as above, in the second
as here, or in the third. The first figure has been discussed, and
we will proceed to display the second, proof by which will be as
follows: all B is D, no C is D..., since it is required that B
should be a subject of which a predicate is affirmed. Next, since D is
to be proved not to belong to C, then D has a further predicate
which is denied of C. Therefore, since the succession of predicates
affirmed of an ever higher universal terminates, the succession of
predicates denied terminates too.
The third figure shows it as follows: all B is A, some B is not C.
Therefore some A is not C. This premiss, i.e. C-B, will be proved
either in the same figure or in one of the two figures discussed
above. In the first and second figures the series terminates. If we
use the third figure, we shall take as premisses, all E is B, some E
is not C, and this premiss again will be proved by a similar
prosyllogism. But since it is assumed that the series of descending
subjects also terminates, plainly the series of more universal
non-predicables will terminate also. Even supposing that the proof
is not confined to one method, but employs them all and is now in
the first figure, now in the second or third-even so the regress
will terminate, for the methods are finite in number, and if finite
things are combined in a finite number of ways, the result must be
finite.
Thus it is plain that the regress of middles terminates in the
case of negative demonstration, if it does so also in the case of
affirmative demonstration. That in fact the regress terminates in both
these cases may be made clear by the following dialectical
considerations.
22
In the case of predicates constituting the essential nature of a
thing, it clearly terminates, seeing that if definition is possible,
or in other words, if essential form is knowable, and an infinite
series cannot be traversed, predicates constituting a thing's
essential nature must be finite in number. But as regards predicates
generally we have the following prefatory remarks to make. (1) We
can affirm without falsehood 'the white (thing) is walking', and
that big (thing) is a log'; or again, 'the log is big', and 'the man
walks'. But the affirmation differs in the two cases. When I affirm
'the white is a log', I mean that something which happens to be
white is a log-not that white is the substratum in which log
inheres, for it was not qua white or qua a species of white that the
white (thing) came to be a log, and the white (thing) is
consequently not a log except incidentally. On the other hand, when
I affirm 'the log is white', I do not mean that something else,
which happens also to be a log, is white (as I should if I said 'the
musician is white,' which would mean 'the man who happens also to be a
musician is white'); on the contrary, log is here the substratum-the
substratum which actually came to be white, and did so qua wood or qua
a species of wood and qua nothing else.
If we must lay down a rule, let us entitle the latter kind of
statement predication, and the former not predication at all, or not
strict but accidental predication. 'White' and 'log' will thus serve
as types respectively of predicate and subject.
We shall assume, then, that the predicate is invariably predicated
strictly and not accidentally of the subject, for on such
predication demonstrations depend for their force. It follows from
this that when a single attribute is predicated of a single subject,
the predicate must affirm of the subject either some element
constituting its essential nature, or that it is in some way
qualified, quantified, essentially related, active, passive, placed,
or dated.
(2) Predicates which signify substance signify that the subject is
identical with the predicate or with a species of the predicate.
Predicates not signifying substance which are predicated of a
subject not identical with themselves or with a species of
themselves are accidental or coincidental; e.g. white is a
coincident of man, seeing that man is not identical with white or a
species of white, but rather with animal, since man is identical
with a species of animal. These predicates which do not signify
substance must be predicates of some other subject, and nothing can be
white which is not also other than white. The Forms we can dispense
with, for they are mere sound without sense; and even if there are
such things, they are not relevant to our discussion, since
demonstrations are concerned with predicates such as we have defined.
(3) If A is a quality of B, B cannot be a quality of A-a quality
of a quality. Therefore A and B cannot be predicated reciprocally of
one another in strict predication: they can be affirmed without
falsehood of one another, but not genuinely predicated of each
other. For one alternative is that they should be substantially
predicated of one another, i.e. B would become the genus or
differentia of A-the predicate now become subject. But it has been
shown that in these substantial predications neither the ascending
predicates nor the descending subjects form an infinite series; e.g.
neither the series, man is biped, biped is animal, &c., nor the series
predicating animal of man, man of Callias, Callias of a further.
subject as an element of its essential nature, is infinite. For all
such substance is definable, and an infinite series cannot be
traversed in thought: consequently neither the ascent nor the
descent is infinite, since a substance whose predicates were
infinite would not be definable. Hence they will not be predicated
each as the genus of the other; for this would equate a genus with one
of its own species. Nor (the other alternative) can a quale be
reciprocally predicated of a quale, nor any term belonging to an
adjectival category of another such term, except by accidental
predication; for all such predicates are coincidents and are
predicated of substances. On the other hand-in proof of the
impossibility of an infinite ascending series-every predication
displays the subject as somehow qualified or quantified or as
characterized under one of the other adjectival categories, or else is
an element in its substantial nature: these latter are limited in
number, and the number of the widest kinds under which predications
fall is also limited, for every predication must exhibit its subject
as somehow qualified, quantified, essentially related, acting or
suffering, or in some place or at some time.
I assume first that predication implies a single subject and a
single attribute, and secondly that predicates which are not
substantial are not predicated of one another. We assume this
because such predicates are all coincidents, and though some are
essential coincidents, others of a different type, yet we maintain
that all of them alike are predicated of some substratum and that a
coincident is never a substratum-since we do not class as a coincident
anything which does not owe its designation to its being something
other than itself, but always hold that any coincident is predicated
of some substratum other than itself, and that another group of
coincidents may have a different substratum. Subject to these
assumptions then, neither the ascending nor the descending series of
predication in which a single attribute is predicated of a single
subject is infinite. For the subjects of which coincidents are
predicated are as many as the constitutive elements of each individual
substance, and these we have seen are not infinite in number, while in
the ascending series are contained those constitutive elements with
their coincidents-both of which are finite. We conclude that there
is a given subject (D) of which some attribute (C) is primarily
predicable; that there must be an attribute (B) primarily predicable
of the first attribute, and that the series must end with a term (A)
not predicable of any term prior to the last subject of which it was
predicated (B), and of which no term prior to it is predicable.
The argument we have given is one of the so-called proofs; an
alternative proof follows. Predicates so related to their subjects
that there are other predicates prior to them predicable of those
subjects are demonstrable; but of demonstrable propositions one cannot
have something better than knowledge, nor can one know them without
demonstration. Secondly, if a consequent is only known through an
antecedent (viz. premisses prior to it) and we neither know this
antecedent nor have something better than knowledge of it, then we
shall not have scientific knowledge of the consequent. Therefore, if
it is possible through demonstration to know anything without
qualification and not merely as dependent on the acceptance of certain
premisses-i.e. hypothetically-the series of intermediate
predications must terminate. If it does not terminate, and beyond
any predicate taken as higher than another there remains another still
higher, then every predicate is demonstrable. Consequently, since
these demonstrable predicates are infinite in number and therefore
cannot be traversed, we shall not know them by demonstration. If,
therefore, we have not something better than knowledge of them, we
cannot through demonstration have unqualified but only hypothetical
science of anything.
As dialectical proofs of our contention these may carry
conviction, but an analytic process will show more briefly that
neither the ascent nor the descent of predication can be infinite in
the demonstrative sciences which are the object of our
investigation. Demonstration proves the inherence of essential
attributes in things. Now attributes may be essential for two reasons:
either because they are elements in the essential nature of their
subjects, or because their subjects are elements in their essential
nature. An example of the latter is odd as an attribute of
number-though it is number's attribute, yet number itself is an
element in the definition of odd; of the former, multiplicity or the
indivisible, which are elements in the definition of number. In
neither kind of attribution can the terms be infinite. They are not
infinite where each is related to the term below it as odd is to
number, for this would mean the inherence in odd of another
attribute of odd in whose nature odd was an essential element: but
then number will be an ultimate subject of the whole infinite chain of
attributes, and be an element in the definition of each of them.
Hence, since an infinity of attributes such as contain their subject
in their definition cannot inhere in a single thing, the ascending
series is equally finite. Note, moreover, that all such attributes
must so inhere in the ultimate subject-e.g. its attributes in number
and number in them-as to be commensurate with the subject and not of
wider extent. Attributes which are essential elements in the nature of
their subjects are equally finite: otherwise definition would be
impossible. Hence, if all the attributes predicated are essential
and these cannot be infinite, the ascending series will terminate, and
consequently the descending series too.
If this is so, it follows that the intermediates between any two
terms are also always limited in number. An immediately obvious
consequence of this is that demonstrations necessarily involve basic
truths, and that the contention of some-referred to at the outset-that
all truths are demonstrable is mistaken. For if there are basic
truths, (a) not all truths are demonstrable, and (b) an infinite
regress is impossible; since if either (a) or (b) were not a fact,
it would mean that no interval was immediate and indivisible, but that
all intervals were divisible. This is true because a conclusion is
demonstrated by the interposition, not the apposition, of a fresh
term. If such interposition could continue to infinity there might
be an infinite number of terms between any two terms; but this is
impossible if both the ascending and descending series of
predication terminate; and of this fact, which before was shown
dialectically, analytic proof has now been given.
23
It is an evident corollary of these conclusions that if the same
attribute A inheres in two terms C and D predicable either not at all,
or not of all instances, of one another, it does not always belong
to them in virtue of a common middle term. Isosceles and scalene
possess the attribute of having their angles equal to two right angles
in virtue of a common middle; for they possess it in so far as they
are both a certain kind of figure, and not in so far as they differ
from one another. But this is not always the case: for, were it so, if
we take B as the common middle in virtue of which A inheres in C and
D, clearly B would inhere in C and D through a second common middle,
and this in turn would inhere in C and D through a third, so that
between two terms an infinity of intermediates would fall-an
impossibility. Thus it need not always be in virtue of a common middle
term that a single attribute inheres in several subjects, since
there must be immediate intervals. Yet if the attribute to be proved
common to two subjects is to be one of their essential attributes, the
middle terms involved must be within one subject genus and be
derived from the same group of immediate premisses; for we have seen
that processes of proof cannot pass from one genus to another.
It is also clear that when A inheres in B, this can be
demonstrated if there is a middle term. Further, the 'elements' of
such a conclusion are the premisses containing the middle in question,
and they are identical in number with the middle terms, seeing that
the immediate propositions-or at least such immediate propositions
as are universal-are the 'elements'. If, on the other hand, there is
no middle term, demonstration ceases to be possible: we are on the way
to the basic truths. Similarly if A does not inhere in B, this can
be demonstrated if there is a middle term or a term prior to B in
which A does not inhere: otherwise there is no demonstration and a
basic truth is reached. There are, moreover, as many 'elements' of the
demonstrated conclusion as there are middle terms, since it is
propositions containing these middle terms that are the basic
premisses on which the demonstration rests; and as there are some
indemonstrable basic truths asserting that 'this is that' or that
'this inheres in that', so there are others denying that 'this is
that' or that 'this inheres in that'-in fact some basic truths will
affirm and some will deny being.
When we are to prove a conclusion, we must take a primary
essential predicate-suppose it C-of the subject B, and then suppose
A similarly predicable of C. If we proceed in this manner, no
proposition or attribute which falls beyond A is admitted in the
proof: the interval is constantly condensed until subject and
predicate become indivisible, i.e. one. We have our unit when the
premiss becomes immediate, since the immediate premiss alone is a
single premiss in the unqualified sense of 'single'. And as in other
spheres the basic element is simple but not identical in all-in a
system of weight it is the mina, in music the quarter-tone, and so
on--so in syllogism the unit is an immediate premiss, and in the
knowledge that demonstration gives it is an intuition. In
syllogisms, then, which prove the inherence of an attribute, nothing
falls outside the major term. In the case of negative syllogisms on
the other hand, (1) in the first figure nothing falls outside the
major term whose inherence is in question; e.g. to prove through a
middle C that A does not inhere in B the premisses required are, all B
is C, no C is A. Then if it has to be proved that no C is A, a
middle must be found between and C; and this procedure will never
vary.
(2) If we have to show that E is not D by means of the premisses,
all D is C; no E, or not all E, is C; then the middle will never
fall beyond E, and E is the subject of which D is to be denied in
the conclusion.
(3) In the third figure the middle will never fall beyond the limits
of the subject and the attribute denied of it.
24
Since demonstrations may be either commensurately universal or
particular, and either affirmative or negative; the question arises,
which form is the better? And the same question may be put in regard
to so-called 'direct' demonstration and reductio ad impossibile. Let
us first examine the commensurately universal and the particular
forms, and when we have cleared up this problem proceed to discuss
'direct' demonstration and reductio ad impossibile.
The following considerations might lead some minds to prefer
particular demonstration.
(1) The superior demonstration is the demonstration which gives us
greater knowledge (for this is the ideal of demonstration), and we
have greater knowledge of a particular individual when we know it in
itself than when we know it through something else; e.g. we know
Coriscus the musician better when we know that Coriscus is musical
than when we know only that man is musical, and a like argument
holds in all other cases. But commensurately universal
demonstration, instead of proving that the subject itself actually
is x, proves only that something else is x- e.g. in attempting to
prove that isosceles is x, it proves not that isosceles but only that
triangle is x- whereas particular demonstration proves that the
subject itself is x. The demonstration, then, that a subject, as such,
possesses an attribute is superior. If this is so, and if the
particular rather than the commensurately universal forms
demonstrates, particular demonstration is superior.
(2) The universal has not a separate being over against groups of
singulars. Demonstration nevertheless creates the opinion that its
function is conditioned by something like this-some separate entity
belonging to the real world; that, for instance, of triangle or of
figure or number, over against particular triangles, figures, and
numbers. But demonstration which touches the real and will not mislead
is superior to that which moves among unrealities and is delusory. Now
commensurately universal demonstration is of the latter kind: if we
engage in it we find ourselves reasoning after a fashion well
illustrated by the argument that the proportionate is what answers
to the definition of some entity which is neither line, number, solid,
nor plane, but a proportionate apart from all these. Since, then, such
a proof is characteristically commensurate and universal, and less
touches reality than does particular demonstration, and creates a
false opinion, it will follow that commensurate and universal is
inferior to particular demonstration.
We may retort thus. (1) The first argument applies no more to
commensurate and universal than to particular demonstration. If
equality to two right angles is attributable to its subject not qua
isosceles but qua triangle, he who knows that isosceles possesses that
attribute knows the subject as qua itself possessing the attribute, to
a less degree than he who knows that triangle has that attribute. To
sum up the whole matter: if a subject is proved to possess qua
triangle an attribute which it does not in fact possess qua
triangle, that is not demonstration: but if it does possess it qua
triangle the rule applies that the greater knowledge is his who
knows the subject as possessing its attribute qua that in virtue of
which it actually does possess it. Since, then, triangle is the
wider term, and there is one identical definition of triangle-i.e. the
term is not equivocal-and since equality to two right angles belongs
to all triangles, it is isosceles qua triangle and not triangle qua
isosceles which has its angles so related. It follows that he who
knows a connexion universally has greater knowledge of it as it in
fact is than he who knows the particular; and the inference is that
commensurate and universal is superior to particular demonstration.
(2) If there is a single identical definition i.e. if the
commensurate universal is unequivocal-then the universal will
possess being not less but more than some of the particulars, inasmuch
as it is universals which comprise the imperishable, particulars
that tend to perish.
(3) Because the universal has a single meaning, we are not therefore
compelled to suppose that in these examples it has being as a
substance apart from its particulars-any more than we need make a
similar supposition in the other cases of unequivocal universal
predication, viz. where the predicate signifies not substance but
quality, essential relatedness, or action. If such a supposition is
entertained, the blame rests not with the demonstration but with the
hearer.
(4) Demonstration is syllogism that proves the cause, i.e. the
reasoned fact, and it is rather the commensurate universal than the
particular which is causative (as may be shown thus: that which
possesses an attribute through its own essential nature is itself
the cause of the inherence, and the commensurate universal is primary;
hence the commensurate universal is the cause). Consequently
commensurately universal demonstration is superior as more
especially proving the cause, that is the reasoned fact.
(5) Our search for the reason ceases, and we think that we know,
when the coming to be or existence of the fact before us is not due to
the coming to be or existence of some other fact, for the last step of
a search thus conducted is eo ipso the end and limit of the problem.
Thus: 'Why did he come?' 'To get the money-wherewith to pay a
debt-that he might thereby do what was right.' When in this regress we
can no longer find an efficient or final cause, we regard the last
step of it as the end of the coming-or being or coming to be-and we
regard ourselves as then only having full knowledge of the reason
why he came.
If, then, all causes and reasons are alike in this respect, and if
this is the means to full knowledge in the case of final causes such
as we have exemplified, it follows that in the case of the other
causes also full knowledge is attained when an attribute no longer
inheres because of something else. Thus, when we learn that exterior
angles are equal to four right angles because they are the exterior
angles of an isosceles, there still remains the question 'Why has
isosceles this attribute?' and its answer 'Because it is a triangle,
and a triangle has it because a triangle is a rectilinear figure.'
If rectilinear figure possesses the property for no further reason, at
this point we have full knowledge-but at this point our knowledge
has become commensurately universal, and so we conclude that
commensurately universal demonstration is superior.
(6) The more demonstration becomes particular the more it sinks into
an indeterminate manifold, while universal demonstration tends to
the simple and determinate. But objects so far as they are an
indeterminate manifold are unintelligible, so far as they are
determinate, intelligible: they are therefore intelligible rather in
so far as they are universal than in so far as they are particular.
From this it follows that universals are more demonstrable: but
since relative and correlative increase concomitantly, of the more
demonstrable there will be fuller demonstration. Hence the
commensurate and universal form, being more truly demonstration, is
the superior.
(7) Demonstration which teaches two things is preferable to
demonstration which teaches only one. He who possesses
commensurately universal demonstration knows the particular as well,
but he who possesses particular demonstration does not know the
universal. So that this is an additional reason for preferring
commensurately universal demonstration. And there is yet this
further argument:
(8) Proof becomes more and more proof of the commensurate
universal as its middle term approaches nearer to the basic truth, and
nothing is so near as the immediate premiss which is itself the
basic truth. If, then, proof from the basic truth is more accurate
than proof not so derived, demonstration which depends more closely on
it is more accurate than demonstration which is less closely
dependent. But commensurately universal demonstration is characterized
by this closer dependence, and is therefore superior. Thus, if A had
to be proved to inhere in D, and the middles were B and C, B being the
higher term would render the demonstration which it mediated the
more universal.
Some of these arguments, however, are dialectical. The clearest
indication of the precedence of commensurately universal demonstration
is as follows: if of two propositions, a prior and a posterior, we
have a grasp of the prior, we have a kind of knowledge-a potential
grasp-of the posterior as well. For example, if one knows that the
angles of all triangles are equal to two right angles, one knows in
a sense-potentially-that the isosceles' angles also are equal to two
right angles, even if one does not know that the isosceles is a
triangle; but to grasp this posterior proposition is by no means to
know the commensurate universal either potentially or actually.
Moreover, commensurately universal demonstration is through and
through intelligible; particular demonstration issues in
sense-perception.
25
The preceding arguments constitute our defence of the superiority of
commensurately universal to particular demonstration. That affirmative
demonstration excels negative may be shown as follows.
(1) We may assume the superiority ceteris paribus of the
demonstration which derives from fewer postulates or hypotheses-in
short from fewer premisses; for, given that all these are equally well
known, where they are fewer knowledge will be more speedily
acquired, and that is a desideratum. The argument implied in our
contention that demonstration from fewer assumptions is superior may
be set out in universal form as follows. Assuming that in both cases
alike the middle terms are known, and that middles which are prior are
better known than such as are posterior, we may suppose two
demonstrations of the inherence of A in E, the one proving it
through the middles B, C and D, the other through F and G. Then A-D is
known to the same degree as A-E (in the second proof), but A-D is
better known than and prior to A-E (in the first proof); since A-E
is proved through A-D, and the ground is more certain than the
conclusion.
Hence demonstration by fewer premisses is ceteris paribus
superior. Now both affirmative and negative demonstration operate
through three terms and two premisses, but whereas the former
assumes only that something is, the latter assumes both that something
is and that something else is not, and thus operating through more
kinds of premiss is inferior.
(2) It has been proved that no conclusion follows if both
premisses are negative, but that one must be negative, the other
affirmative. So we are compelled to lay down the following
additional rule: as the demonstration expands, the affirmative
premisses must increase in number, but there cannot be more than one
negative premiss in each complete proof. Thus, suppose no B is A,
and all C is B. Then if both the premisses are to be again expanded, a
middle must be interposed. Let us interpose D between A and B, and E
between B and C. Then clearly E is affirmatively related to B and C,
while D is affirmatively related to B but negatively to A; for all B
is D, but there must be no D which is A. Thus there proves to be a
single negative premiss, A-D. In the further prosyllogisms too it is
the same, because in the terms of an affirmative syllogism the
middle is always related affirmatively to both extremes; in a negative
syllogism it must be negatively related only to one of them, and so
this negation comes to be a single negative premiss, the other
premisses being affirmative. If, then, that through which a truth is
proved is a better known and more certain truth, and if the negative
proposition is proved through the affirmative and not vice versa,
affirmative demonstration, being prior and better known and more
certain, will be superior.
(3) The basic truth of demonstrative syllogism is the universal
immediate premiss, and the universal premiss asserts in affirmative
demonstration and in negative denies: and the affirmative
proposition is prior to and better known than the negative (since
affirmation explains denial and is prior to denial, just as being is
prior to not-being). It follows that the basic premiss of
affirmative demonstration is superior to that of negative
demonstration, and the demonstration which uses superior basic
premisses is superior.
(4) Affirmative demonstration is more of the nature of a basic
form of proof, because it is a sine qua non of negative demonstration.
26
Since affirmative demonstration is superior to negative, it is
clearly superior also to reductio ad impossibile. We must first make
certain what is the difference between negative demonstration and
reductio ad impossibile. Let us suppose that no B is A, and that all C
is B: the conclusion necessarily follows that no C is A. If these
premisses are assumed, therefore, the negative demonstration that no C
is A is direct. Reductio ad impossibile, on the other hand, proceeds
as follows. Supposing we are to prove that does not inhere in B, we
have to assume that it does inhere, and further that B inheres in C,
with the resulting inference that A inheres in C. This we have to
suppose a known and admitted impossibility; and we then infer that A
cannot inhere in B. Thus if the inherence of B in C is not questioned,
A's inherence in B is impossible.
The order of the terms is the same in both proofs: they differ
according to which of the negative propositions is the better known,
the one denying A of B or the one denying A of C. When the falsity
of the conclusion is the better known, we use reductio ad
impossible; when the major premiss of the syllogism is the more
obvious, we use direct demonstration. All the same the proposition
denying A of B is, in the order of being, prior to that denying A of
C; for premisses are prior to the conclusion which follows from
them, and 'no C is A' is the conclusion, 'no B is A' one of its
premisses. For the destructive result of reductio ad impossibile is
not a proper conclusion, nor are its antecedents proper premisses.
On the contrary: the constituents of syllogism are premisses related
to one another as whole to part or part to whole, whereas the
premisses A-C and A-B are not thus related to one another. Now the
superior demonstration is that which proceeds from better known and
prior premisses, and while both these forms depend for credence on the
not-being of something, yet the source of the one is prior to that
of the other. Therefore negative demonstration will have an
unqualified superiority to reductio ad impossibile, and affirmative
demonstration, being superior to negative, will consequently be
superior also to reductio ad impossibile.
27
The science which is knowledge at once of the fact and of the
reasoned fact, not of the fact by itself without the reasoned fact, is
the more exact and the prior science.
A science such as arithmetic, which is not a science of properties
qua inhering in a substratum, is more exact than and prior to a
science like harmonics, which is a science of pr,operties inhering
in a substratum; and similarly a science like arithmetic, which is
constituted of fewer basic elements, is more exact than and prior to
geometry, which requires additional elements. What I mean by
'additional elements' is this: a unit is substance without position,
while a point is substance with position; the latter contains an
additional element.
28
A single science is one whose domain is a single genus, viz. all the
subjects constituted out of the primary entities of the genus-i.e. the
parts of this total subject-and their essential properties.
One science differs from another when their basic truths have
neither a common source nor are derived those of the one science
from those the other. This is verified when we reach the
indemonstrable premisses of a science, for they must be within one
genus with its conclusions: and this again is verified if the
conclusions proved by means of them fall within one genus-i.e. are
homogeneous.
29
One can have several demonstrations of the same connexion not only
by taking from the same series of predication middles which are
other than the immediately cohering term e.g. by taking C, D, and F
severally to prove A-B--but also by taking a middle from another
series. Thus let A be change, D alteration of a property, B feeling
pleasure, and G relaxation. We can then without falsehood predicate
D of B and A of D, for he who is pleased suffers alteration of a
property, and that which alters a property changes. Again, we can
predicate A of G without falsehood, and G of B; for to feel pleasure
is to relax, and to relax is to change. So the conclusion can be drawn
through middles which are different, i.e. not in the same series-yet
not so that neither of these middles is predicable of the other, for
they must both be attributable to some one subject.
A further point worth investigating is how many ways of proving
the same conclusion can be obtained by varying the figure,
30
There is no knowledge by demonstration of chance conjunctions; for
chance conjunctions exist neither by necessity nor as general
connexions but comprise what comes to be as something distinct from
these. Now demonstration is concerned only with one or other of
these two; for all reasoning proceeds from necessary or general
premisses, the conclusion being necessary if the premisses are
necessary and general if the premisses are general. Consequently, if
chance conjunctions are neither general nor necessary, they are not
demonstrable.
31
Scientific knowledge is not possible through the act of
perception. Even if perception as a faculty is of 'the such' and not
merely of a 'this somewhat', yet one must at any rate actually
perceive a 'this somewhat', and at a definite present place and
time: but that which is commensurately universal and true in all cases
one cannot perceive, since it is not 'this' and it is not 'now'; if it
were, it would not be commensurately universal-the term we apply to
what is always and everywhere. Seeing, therefore, that
demonstrations are commensurately universal and universals
imperceptible, we clearly cannot obtain scientific knowledge by the
act of perception: nay, it is obvious that even if it were possible to
perceive that a triangle has its angles equal to two right angles,
we should still be looking for a demonstration-we should not (as
some say) possess knowledge of it; for perception must be of a
particular, whereas scientific knowledge involves the recognition of
the commensurate universal. So if we were on the moon, and saw the
earth shutting out the sun's light, we should not know the cause of
the eclipse: we should perceive the present fact of the eclipse, but
not the reasoned fact at all, since the act of perception is not of
the commensurate universal. I do not, of course, deny that by watching
the frequent recurrence of this event we might, after tracking the
commensurate universal, possess a demonstration, for the
commensurate universal is elicited from the several groups of
singulars.
The commensurate universal is precious because it makes clear the
cause; so that in the case of facts like these which have a cause
other than themselves universal knowledge is more precious than
sense-perceptions and than intuition. (As regards primary truths there
is of course a different account to be given.) Hence it is clear
that knowledge of things demonstrable cannot be acquired by
perception, unless the term perception is applied to the possession of
scientific knowledge through demonstration. Nevertheless certain
points do arise with regard to connexions to be proved which are
referred for their explanation to a failure in sense-perception: there
are cases when an act of vision would terminate our inquiry, not
because in seeing we should be knowing, but because we should have
elicited the universal from seeing; if, for example, we saw the
pores in the glass and the light passing through, the reason of the
kindling would be clear to us because we should at the same time see
it in each instance and intuit that it must be so in all instances.
32
All syllogisms cannot have the same basic truths. This may be
shown first of all by the following dialectical considerations. (1)
Some syllogisms are true and some false: for though a true inference
is possible from false premisses, yet this occurs once only-I mean
if A for instance, is truly predicable of C, but B, the middle, is
false, both A-B and B-C being false; nevertheless, if middles are
taken to prove these premisses, they will be false because every
conclusion which is a falsehood has false premisses, while true
conclusions have true premisses, and false and true differ in kind.
Then again, (2) falsehoods are not all derived from a single identical
set of principles: there are falsehoods which are the contraries of
one another and cannot coexist, e.g. 'justice is injustice', and
'justice is cowardice'; 'man is horse', and 'man is ox'; 'the equal is
greater', and 'the equal is less.' From established principles we
may argue the case as follows, confining-ourselves therefore to true
conclusions. Not even all these are inferred from the same basic
truths; many of them in fact have basic truths which differ
generically and are not transferable; units, for instance, which are
without position, cannot take the place of points, which have
position. The transferred terms could only fit in as middle terms or
as major or minor terms, or else have some of the other terms
between them, others outside them.
Nor can any of the common axioms-such, I mean, as the law of
excluded middle-serve as premisses for the proof of all conclusions.
For the kinds of being are different, and some attributes attach to
quanta and some to qualia only; and proof is achieved by means of
the common axioms taken in conjunction with these several kinds and
their attributes.
Again, it is not true that the basic truths are much fewer than
the conclusions, for the basic truths are the premisses, and the
premisses are formed by the apposition of a fresh extreme term or
the interposition of a fresh middle. Moreover, the number of
conclusions is indefinite, though the number of middle terms is
finite; and lastly some of the basic truths are necessary, others
variable.
Looking at it in this way we see that, since the number of
conclusions is indefinite, the basic truths cannot be identical or
limited in number. If, on the other hand, identity is used in
another sense, and it is said, e.g. 'these and no other are the
fundamental truths of geometry, these the fundamentals of calculation,
these again of medicine'; would the statement mean anything except
that the sciences have basic truths? To call them identical because
they are self-identical is absurd, since everything can be
identified with everything in that sense of identity. Nor again can
the contention that all conclusions have the same basic truths mean
that from the mass of all possible premisses any conclusion may be
drawn. That would be exceedingly naive, for it is not the case in
the clearly evident mathematical sciences, nor is it possible in
analysis, since it is the immediate premisses which are the basic
truths, and a fresh conclusion is only formed by the addition of a new
immediate premiss: but if it be admitted that it is these primary
immediate premisses which are basic truths, each subject-genus will
provide one basic truth. If, however, it is not argued that from the
mass of all possible premisses any conclusion may be proved, nor yet
admitted that basic truths differ so as to be generically different
for each science, it remains to consider the possibility that, while
the basic truths of all knowledge are within one genus, special
premisses are required to prove special conclusions. But that this
cannot be the case has been shown by our proof that the basic truths
of things generically different themselves differ generically. For
fundamental truths are of two kinds, those which are premisses of
demonstration and the subject-genus; and though the former are common,
the latter-number, for instance, and magnitude-are peculiar.
33
Scientific knowledge and its object differ from opinion and the
object of opinion in that scientific knowledge is commensurately
universal and proceeds by necessary connexions, and that which is
necessary cannot be otherwise. So though there are things which are
true and real and yet can be otherwise, scientific knowledge clearly
does not concern them: if it did, things which can be otherwise
would be incapable of being otherwise. Nor are they any concern of
rational intuition-by rational intuition I mean an originative
source of scientific knowledge-nor of indemonstrable knowledge,
which is the grasping of the immediate premiss. Since then rational
intuition, science, and opinion, and what is revealed by these
terms, are the only things that can be 'true', it follows that it is
opinion that is concerned with that which may be true or false, and
can be otherwise: opinion in fact is the grasp of a premiss which is
immediate but not necessary. This view also fits the observed facts,
for opinion is unstable, and so is the kind of being we have described
as its object. Besides, when a man thinks a truth incapable of being
otherwise he always thinks that he knows it, never that he opines
it. He thinks that he opines when he thinks that a connexion, though
actually so, may quite easily be otherwise; for he believes that
such is the proper object of opinion, while the necessary is the
object of knowledge.
In what sense, then, can the same thing be the object of both
opinion and knowledge? And if any one chooses to maintain that all
that he knows he can also opine, why should not opinion be
knowledge? For he that knows and he that opines will follow the same
train of thought through the same middle terms until the immediate
premisses are reached; because it is possible to opine not only the
fact but also the reasoned fact, and the reason is the middle term; so
that, since the former knows, he that opines also has knowledge.
The truth perhaps is that if a man grasp truths that cannot be other
than they are, in the way in which he grasps the definitions through
which demonstrations take place, he will have not opinion but
knowledge: if on the other hand he apprehends these attributes as
inhering in their subjects, but not in virtue of the subjects'
substance and essential nature possesses opinion and not genuine
knowledge; and his opinion, if obtained through immediate premisses,
will be both of the fact and of the reasoned fact; if not so obtained,
of the fact alone. The object of opinion and knowledge is not quite
identical; it is only in a sense identical, just as the object of true
and false opinion is in a sense identical. The sense in which some
maintain that true and false opinion can have the same object leads
them to embrace many strange doctrines, particularly the doctrine that
what a man opines falsely he does not opine at all. There are really
many senses of 'identical', and in one sense the object of true and
false opinion can be the same, in another it cannot. Thus, to have a
true opinion that the diagonal is commensurate with the side would
be absurd: but because the diagonal with which they are both concerned
is the same, the two opinions have objects so far the same: on the
other hand, as regards their essential definable nature these
objects differ. The identity of the objects of knowledge and opinion
is similar. Knowledge is the apprehension of, e.g. the attribute
'animal' as incapable of being otherwise, opinion the apprehension
of 'animal' as capable of being otherwise-e.g. the apprehension that
animal is an element in the essential nature of man is knowledge;
the apprehension of animal as predicable of man but not as an
element in man's essential nature is opinion: man is the subject in
both judgements, but the mode of inherence differs.
This also shows that one cannot opine and know the same thing
simultaneously; for then one would apprehend the same thing as both
capable and incapable of being otherwise-an impossibility. Knowledge
and opinion of the same thing can co-exist in two different people
in the sense we have explained, but not simultaneously in the same
person. That would involve a man's simultaneously apprehending, e.g.
(1) that man is essentially animal-i.e. cannot be other than
animal-and (2) that man is not essentially animal, that is, we may
assume, may be other than animal.
Further consideration of modes of thinking and their distribution
under the heads of discursive thought, intuition, science, art,
practical wisdom, and metaphysical thinking, belongs rather partly
to natural science, partly to moral philosophy.
34
Quick wit is a faculty of hitting upon the middle term
instantaneously. It would be exemplified by a man who saw that the
moon has her bright side always turned towards the sun, and quickly
grasped the cause of this, namely that she borrows her light from him;
or observed somebody in conversation with a man of wealth and
divined that he was borrowing money, or that the friendship of these
people sprang from a common enmity. In all these instances he has seen
the major and minor terms and then grasped the causes, the middle
terms.
Let A represent 'bright side turned sunward', B 'lighted from the
sun', C the moon. Then B, 'lighted from the sun' is predicable of C,
the moon, and A, 'having her bright side towards the source of her
light', is predicable of B. So A is predicable of C through B.
Book II
1
THE kinds of question we ask are as many as the kinds of things
which we know. They are in fact four:-(1) whether the connexion of
an attribute with a thing is a fact, (2) what is the reason of the
connexion, (3) whether a thing exists, (4) What is the nature of the
thing. Thus, when our question concerns a complex of thing and
attribute and we ask whether the thing is thus or otherwise
qualified-whether, e.g. the sun suffers eclipse or not-then we are
asking as to the fact of a connexion. That our inquiry ceases with the
discovery that the sun does suffer eclipse is an indication of this;
and if we know from the start that the sun suffers eclipse, we do
not inquire whether it does so or not. On the other hand, when we know
the fact we ask the reason; as, for example, when we know that the sun
is being eclipsed and that an earthquake is in progress, it is the
reason of eclipse or earthquake into which we inquire.
Where a complex is concerned, then, those are the two questions we
ask; but for some objects of inquiry we have a different kind of
question to ask, such as whether there is or is not a centaur or a
God. (By 'is or is not' I mean 'is or is not, without further
qualification'; as opposed to 'is or is not [e.g.] white'.) On the
other hand, when we have ascertained the thing's existence, we inquire
as to its nature, asking, for instance, 'what, then, is God?' or 'what
is man?'.
2
These, then, are the four kinds of question we ask, and it is in the
answers to these questions that our knowledge consists.
Now when we ask whether a connexion is a fact, or whether a thing
without qualification is, we are really asking whether the connexion
or the thing has a 'middle'; and when we have ascertained either
that the connexion is a fact or that the thing is-i.e. ascertained
either the partial or the unqualified being of the thing-and are
proceeding to ask the reason of the connexion or the nature of the
thing, then we are asking what the 'middle' is.
(By distinguishing the fact of the connexion and the existence of
the thing as respectively the partial and the unqualified being of the
thing, I mean that if we ask 'does the moon suffer eclipse?', or 'does
the moon wax?', the question concerns a part of the thing's being; for
what we are asking in such questions is whether a thing is this or
that, i.e. has or has not this or that attribute: whereas, if we ask
whether the moon or night exists, the question concerns the
unqualified being of a thing.)
We conclude that in all our inquiries we are asking either whether
there is a 'middle' or what the 'middle' is: for the 'middle' here
is precisely the cause, and it is the cause that we seek in all our
inquiries. Thus, 'Does the moon suffer eclipse?' means 'Is there or is
there not a cause producing eclipse of the moon?', and when we have
learnt that t