POSTERIOR ANALYTICS

by Aristotle


BOOK I

Part 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. 


Part 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. 


Part 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. 


Part 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.


Part 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.


Part 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.


Part 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. 


Part 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.


Part 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. 


Part 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. 


Part 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. 


Part 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. 


Part 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. 


Part 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.


Part 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.


Part 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. 


Part 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.


Part 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. 


Part 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. 


Part 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.


Part 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.


Part 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. 


Part 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. 


Part 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. 


Part 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.


Part 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. 


Part 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. 


Part 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. 


Part 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, 


Part 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. 


Part 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. 


Part 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. 


Part 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. 


Part 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

Part 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?'.


Part 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 there is, our next question is, 'What, then, is this
cause? for the cause through which a thing is-not is this or that,
i.e. has this or that attribute, but without qualification is-and
the cause through which it is-not is without qualification, but is
this or that as having some essential attribute or some accident-are
both alike the middle'. By that which is without qualification I mean
the subject, e.g. moon or earth or sun or triangle; by that which
a subject is (in the partial sense) I mean a property, e.g. eclipse,
equality or inequality, interposition or non-interposition. For in
all these examples it is clear that the nature of the thing and the
reason of the fact are identical: the question 'What is eclipse?'
and its answer 'The privation of the moon's light by the interposition
of the earth' are identical with the question 'What is the reason
of eclipse?' or 'Why does the moon suffer eclipse?' and the reply
'Because of the failure of light through the earth's shutting it out'.
Again, for 'What is a concord? A commensurate numerical ratio of a
high and a low note', we may substitute 'What ratio makes a high and
a low note concordant? Their relation according to a commensurate
numerical ratio.' 'Are the high and the low note concordant?' is equivalent
to 'Is their ratio commensurate?'; and when we find that it is commensurate,
we ask 'What, then, is their ratio?'. 

Cases in which the 'middle' is sensible show that the object of our
inquiry is always the 'middle': we inquire, because we have not perceived
it, whether there is or is not a 'middle' causing, e.g. an eclipse.
On the other hand, if we were on the moon we should not be inquiring
either as to the fact or the reason, but both fact and reason would
be obvious simultaneously. For the act of perception would have enabled
us to know the universal too; since, the present fact of an eclipse
being evident, perception would then at the same time give us the
present fact of the earth's screening the sun's light, and from this
would arise the universal. 

Thus, as we maintain, to know a thing's nature is to know the reason
why it is; and this is equally true of things in so far as they are
said without qualification to he as opposed to being possessed of
some attribute, and in so far as they are said to be possessed of
some attribute such as equal to right angles, or greater or less.


Part 3

It is clear, then, that all questions are a search for a 'middle'.
Let us now state how essential nature is revealed and in what way
it can be reduced to demonstration; what definition is, and what things
are definable. And let us first discuss certain difficulties which
these questions raise, beginning what we have to say with a point
most intimately connected with our immediately preceding remarks,
namely the doubt that might be felt as to whether or not it is possible
to know the same thing in the same relation, both by definition and
by demonstration. It might, I mean, be urged that definition is held
to concern essential nature and is in every case universal and affirmative;
whereas, on the other hand, some conclusions are negative and some
are not universal; e.g. all in the second figure are negative, none
in the third are universal. And again, not even all affirmative conclusions
in the first figure are definable, e.g. 'every triangle has its angles
equal to two right angles'. An argument proving this difference between
demonstration and definition is that to have scientific knowledge
of the demonstrable is identical with possessing a demonstration of
it: hence if demonstration of such conclusions as these is possible,
there clearly cannot also be definition of them. If there could, one
might know such a conclusion also in virtue of its definition without
possessing the demonstration of it; for there is nothing to stop our
having the one without the other. 

Induction too will sufficiently convince us of this difference; for
never yet by defining anything-essential attribute or accident-did
we get knowledge of it. Again, if to define is to acquire knowledge
of a substance, at any rate such attributes are not substances.

It is evident, then, that not everything demonstrable can be defined.
What then? Can everything definable be demonstrated, or not? There
is one of our previous arguments which covers this too. Of a single
thing qua single there is a single scientific knowledge. Hence, since
to know the demonstrable scientifically is to possess the demonstration
of it, an impossible consequence will follow:-possession of its definition
without its demonstration will give knowledge of the demonstrable.

Moreover, the basic premisses of demonstrations are definitions, and
it has already been shown that these will be found indemonstrable;
either the basic premisses will be demonstrable and will depend on
prior premisses, and the regress will be endless; or the primary truths
will be indemonstrable definitions. 

But if the definable and the demonstrable are not wholly the same,
may they yet be partially the same? Or is that impossible, because
there can be no demonstration of the definable? There can be none,
because definition is of the essential nature or being of something,
and all demonstrations evidently posit and assume the essential nature-mathematical
demonstrations, for example, the nature of unity and the odd, and
all the other sciences likewise. Moreover, every demonstration proves
a predicate of a subject as attaching or as not attaching to it, but
in definition one thing is not predicated of another; we do not, e.g.
predicate animal of biped nor biped of animal, nor yet figure of plane-plane
not being figure nor figure plane. Again, to prove essential nature
is not the same as to prove the fact of a connexion. Now definition
reveals essential nature, demonstration reveals that a given attribute
attaches or does not attach to a given subject; but different things
require different demonstrations-unless the one demonstration is related
to the other as part to whole. I add this because if all triangles
have been proved to possess angles equal to two right angles, then
this attribute has been proved to attach to isosceles; for isosceles
is a part of which all triangles constitute the whole. But in the
case before us the fact and the essential nature are not so related
to one another, since the one is not a part of the other.

So it emerges that not all the definable is demonstrable nor all the
demonstrable definable; and we may draw the general conclusion that
there is no identical object of which it is possible to possess both
a definition and a demonstration. It follows obviously that definition
and demonstration are neither identical nor contained either within
the other: if they were, their objects would be related either as
identical or as whole and part. 


Part 4

So much, then, for the first stage of our problem. The next step is
to raise the question whether syllogism-i.e. demonstration-of the
definable nature is possible or, as our recent argument assumed, impossible.

We might argue it impossible on the following grounds:-(a) syllogism
proves an attribute of a subject through the middle term; on the other
hand (b) its definable nature is both 'peculiar' to a subject and
predicated of it as belonging to its essence. But in that case (1)
the subject, its definition, and the middle term connecting them must
be reciprocally predicable of one another; for if A is to C, obviously
A is 'peculiar' to B and B to C-in fact all three terms are 'peculiar'
to one another: and further (2) if A inheres in the essence of all
B and B is predicated universally of all C as belonging to C's essence,
A also must be predicated of C as belonging to its essence.

If one does not take this relation as thus duplicated-if, that is,
A is predicated as being of the essence of B, but B is not of the
essence of the subjects of which it is predicated-A will not necessarily
be predicated of C as belonging to its essence. So both premisses
will predicate essence, and consequently B also will be predicated
of C as its essence. Since, therefore, both premisses do predicate
essence-i.e. definable form-C's definable form will appear in the
middle term before the conclusion is drawn. 

We may generalize by supposing that it is possible to prove the essential
nature of man. Let C be man, A man's essential nature--two-footed
animal, or aught else it may be. Then, if we are to syllogize, A must
be predicated of all B. But this premiss will be mediated by a fresh
definition, which consequently will also be the essential nature of
man. Therefore the argument assumes what it has to prove, since B
too is the essential nature of man. It is, however, the case in which
there are only the two premisses-i.e. in which the premisses are primary
and immediate-which we ought to investigate, because it best illustrates
the point under discussion. 

Thus they who prove the essential nature of soul or man or anything
else through reciprocating terms beg the question. It would be begging
the question, for example, to contend that the soul is that which
causes its own life, and that what causes its own life is a self-moving
number; for one would have to postulate that the soul is a self-moving
number in the sense of being identical with it. For if A is predicable
as a mere consequent of B and B of C, A will not on that account be
the definable form of C: A will merely be what it was true to say
of C. Even if A is predicated of all B inasmuch as B is identical
with a species of A, still it will not follow: being an animal is
predicated of being a man-since it is true that in all instances to
be human is to be animal, just as it is also true that every man is
an animal-but not as identical with being man. 

We conclude, then, that unless one takes both the premisses as predicating
essence, one cannot infer that A is the definable form and essence
of C: but if one does so take them, in assuming B one will have assumed,
before drawing the conclusion, what the definable form of C is; so
that there has been no inference, for one has begged the question.


Part 5

Nor, as was said in my formal logic, is the method of division a process
of inference at all, since at no point does the characterization of
the subject follow necessarily from the premising of certain other
facts: division demonstrates as little as does induction. For in a
genuine demonstration the conclusion must not be put as a question
nor depend on a concession, but must follow necessarily from its premisses,
even if the respondent deny it. The definer asks 'Is man animal or
inanimate?' and then assumes-he has not inferred-that man is animal.
Next, when presented with an exhaustive division of animal into terrestrial
and aquatic, he assumes that man is terrestrial. Moreover, that man
is the complete formula, terrestrial-animal, does not follow necessarily
from the premisses: this too is an assumption, and equally an assumption
whether the division comprises many differentiae or few. (Indeed as
this method of division is used by those who proceed by it, even truths
that can be inferred actually fail to appear as such.) For why should
not the whole of this formula be true of man, and yet not exhibit
his essential nature or definable form? Again, what guarantee is there
against an unessential addition, or against the omission of the final
or of an intermediate determinant of the substantial being?

The champion of division might here urge that though these lapses
do occur, yet we can solve that difficulty if all the attributes we
assume are constituents of the definable form, and if, postulating
the genus, we produce by division the requisite uninterrupted sequence
of terms, and omit nothing; and that indeed we cannot fail to fulfil
these conditions if what is to be divided falls whole into the division
at each stage, and none of it is omitted; and that this-the dividendum-must
without further question be (ultimately) incapable of fresh specific
division. Nevertheless, we reply, division does not involve inference;
if it gives knowledge, it gives it in another way. Nor is there any
absurdity in this: induction, perhaps, is not demonstration any more
than is division, et it does make evident some truth. Yet to state
a definition reached by division is not to state a conclusion: as,
when conclusions are drawn without their appropriate middles, the
alleged necessity by which the inference follows from the premisses
is open to a question as to the reason for it, so definitions reached
by division invite the same question. 

Thus to the question 'What is the essential nature of man?' the divider
replies 'Animal, mortal, footed, biped, wingless'; and when at each
step he is asked 'Why?', he will say, and, as he thinks, proves by
division, that all animal is mortal or immortal: but such a formula
taken in its entirety is not definition; so that even if division
does demonstrate its formula, definition at any rate does not turn
out to be a conclusion of inference. 


Part 6

Can we nevertheless actually demonstrate what a thing essentially
and substantially is, but hypothetically, i.e. by premising (1) that
its definable form is constituted by the 'peculiar' attributes of
its essential nature; (2) that such and such are the only attributes
of its essential nature, and that the complete synthesis of them is
peculiar to the thing; and thus-since in this synthesis consists the
being of the thing-obtaining our conclusion? Or is the truth that,
since proof must be through the middle term, the definable form is
once more assumed in this minor premiss too? 

Further, just as in syllogizing we do not premise what syllogistic
inference is (since the premisses from which we conclude must be related
as whole and part), so the definable form must not fall within the
syllogism but remain outside the premisses posited. It is only against
a doubt as to its having been a syllogistic inference at all that
we have to defend our argument as conforming to the definition of
syllogism. It is only when some one doubts whether the conclusion
proved is the definable form that we have to defend it as conforming
to the definition of definable form which we assumed. Hence syllogistic
inference must be possible even without the express statement of what
syllogism is or what definable form is. 

The following type of hypothetical proof also begs the question. If
evil is definable as the divisible, and the definition of a thing's
contrary-if it has one the contrary of the thing's definition; then,
if good is the contrary of evil and the indivisible of the divisible,
we conclude that to be good is essentially to be indivisible. The
question is begged because definable form is assumed as a premiss,
and as a premiss which is to prove definable form. 'But not the same
definable form', you may object. That I admit, for in demonstrations
also we premise that 'this' is predicable of 'that'; but in this premiss
the term we assert of the minor is neither the major itself nor a
term identical in definition, or convertible, with the major.

Again, both proof by division and the syllogism just described are
open to the question why man should be animal-biped-terrestrial and
not merely animal and terrestrial, since what they premise does not
ensure that the predicates shall constitute a genuine unity and not
merely belong to a single subject as do musical and grammatical when
predicated of the same man. 


Part 7

How then by definition shall we prove substance or essential nature?
We cannot show it as a fresh fact necessarily following from the assumption
of premisses admitted to be facts-the method of demonstration: we
may not proceed as by induction to establish a universal on the evidence
of groups of particulars which offer no exception, because induction
proves not what the essential nature of a thing is but that it has
or has not some attribute. Therefore, since presumably one cannot
prove essential nature by an appeal to sense perception or by pointing
with the finger, what other method remains? 

To put it another way: how shall we by definition prove essential
nature? He who knows what human-or any other-nature is, must know
also that man exists; for no one knows the nature of what does not
exist-one can know the meaning of the phrase or name 'goat-stag' but
not what the essential nature of a goat-stag is. But further, if definition
can prove what is the essential nature of a thing, can it also prove
that it exists? And how will it prove them both by the same process,
since definition exhibits one single thing and demonstration another
single thing, and what human nature is and the fact that man exists
are not the same thing? Then too we hold that it is by demonstration
that the being of everything must be proved-unless indeed to be were
its essence; and, since being is not a genus, it is not the essence
of anything. Hence the being of anything as fact is matter for demonstration;
and this is the actual procedure of the sciences, for the geometer
assumes the meaning of the word triangle, but that it is possessed
of some attribute he proves. What is it, then, that we shall prove
in defining essential nature? Triangle? In that case a man will know
by definition what a thing's nature is without knowing whether it
exists. But that is impossible. 

Moreover it is clear, if we consider the methods of defining actually
in use, that definition does not prove that the thing defined exists:
since even if there does actually exist something which is equidistant
from a centre, yet why should the thing named in the definition exist?
Why, in other words, should this be the formula defining circle? One
might equally well call it the definition of mountain copper. For
definitions do not carry a further guarantee that the thing defined
can exist or that it is what they claim to define: one can always
ask why. 

Since, therefore, to define is to prove either a thing's essential
nature or the meaning of its name, we may conclude that definition,
if it in no sense proves essential nature, is a set of words signifying
precisely what a name signifies. But that were a strange consequence;
for (1) both what is not substance and what does not exist at all
would be definable, since even non-existents can be signified by a
name: (2) all sets of words or sentences would be definitions, since
any kind of sentence could be given a name; so that we should all
be talking in definitions, and even the Iliad would be a definition:
(3) no demonstration can prove that any particular name means any
particular thing: neither, therefore, do definitions, in addition
to revealing the meaning of a name, also reveal that the name has
this meaning. It appears then from these considerations that neither
definition and syllogism nor their objects are identical, and further
that definition neither demonstrates nor proves anything, and that
knowledge of essential nature is not to be obtained either by definition
or by demonstration. 


Part 8

We must now start afresh and consider which of these conclusions are
sound and which are not, and what is the nature of definition, and
whether essential nature is in any sense demonstrable and definable
or in none. 

Now to know its essential nature is, as we said, the same as to know
the cause of a thing's existence, and the proof of this depends on
the fact that a thing must have a cause. Moreover, this cause is either
identical with the essential nature of the thing or distinct from
it; and if its cause is distinct from it, the essential nature of
the thing is either demonstrable or indemonstrable. Consequently,
if the cause is distinct from the thing's essential nature and demonstration
is possible, the cause must be the middle term, and, the conclusion
proved being universal and affirmative, the proof is in the first
figure. So the method just examined of proving it through another
essential nature would be one way of proving essential nature, because
a conclusion containing essential nature must be inferred through
a middle which is an essential nature just as a 'peculiar' property
must be inferred through a middle which is a 'peculiar' property;
so that of the two definable natures of a single thing this method
will prove one and not the other. 

Now it was said before that this method could not amount to demonstration
of essential nature-it is actually a dialectical proof of it-so let
us begin again and explain by what method it can be demonstrated.
When we are aware of a fact we seek its reason, and though sometimes
the fact and the reason dawn on us simultaneously, yet we cannot apprehend
the reason a moment sooner than the fact; and clearly in just the
same way we cannot apprehend a thing's definable form without apprehending
that it exists, since while we are ignorant whether it exists we cannot
know its essential nature. Moreover we are aware whether a thing exists
or not sometimes through apprehending an element in its character,
and sometimes accidentally, as, for example, when we are aware of
thunder as a noise in the clouds, of eclipse as a privation of light,
or of man as some species of animal, or of the soul as a self-moving
thing. As often as we have accidental knowledge that the thing exists,
we must be in a wholly negative state as regards awareness of its
essential nature; for we have not got genuine knowledge even of its
existence, and to search for a thing's essential nature when we are
unaware that it exists is to search for nothing. On the other hand,
whenever we apprehend an element in the thing's character there is
less difficulty. Thus it follows that the degree of our knowledge
of a thing's essential nature is determined by the sense in which
we are aware that it exists. Let us then take the following as our
first instance of being aware of an element in the essential nature.
Let A be eclipse, C the moon, B the earth's acting as a screen. Now
to ask whether the moon is eclipsed or not is to ask whether or not
B has occurred. But that is precisely the same as asking whether A
has a defining condition; and if this condition actually exists, we
assert that A also actually exists. Or again we may ask which side
of a contradiction the defining condition necessitates: does it make
the angles of a triangle equal or not equal to two right angles? When
we have found the answer, if the premisses are immediate, we know
fact and reason together; if they are not immediate, we know the fact
without the reason, as in the following example: let C be the moon,
A eclipse, B the fact that the moon fails to produce shadows though
she is full and though no visible body intervenes between us and her.
Then if B, failure to produce shadows in spite of the absence of an
intervening body, is attributable A to C, and eclipse, is attributable
to B, it is clear that the moon is eclipsed, but the reason why is
not yet clear, and we know that eclipse exists, but we do not know
what its essential nature is. But when it is clear that A is attributable
to C and we proceed to ask the reason of this fact, we are inquiring
what is the nature of B: is it the earth's acting as a screen, or
the moon's rotation or her extinction? But B is the definition of
the other term, viz. in these examples, of the major term A; for eclipse
is constituted by the earth acting as a screen. Thus, (1) 'What is
thunder?' 'The quenching of fire in cloud', and (2) 'Why does it thunder?'
'Because fire is quenched in the cloud', are equivalent. Let C be
cloud, A thunder, B the quenching of fire. Then B is attributable
to C, cloud, since fire is quenched in it; and A, noise, is attributable
to B; and B is assuredly the definition of the major term A. If there
be a further mediating cause of B, it will be one of the remaining
partial definitions of A. 

We have stated then how essential nature is discovered and becomes
known, and we see that, while there is no syllogism-i.e. no demonstrative
syllogism-of essential nature, yet it is through syllogism, viz. demonstrative
syllogism, that essential nature is exhibited. So we conclude that
neither can the essential nature of anything which has a cause distinct
from itself be known without demonstration, nor can it be demonstrated;
and this is what we contended in our preliminary discussions.


Part 9

Now while some things have a cause distinct from themselves, others
have not. Hence it is evident that there are essential natures which
are immediate, that is are basic premisses; and of these not only
that they are but also what they are must be assumed or revealed in
some other way. This too is the actual procedure of the arithmetician,
who assumes both the nature and the existence of unit. On the other
hand, it is possible (in the manner explained) to exhibit through
demonstration the essential nature of things which have a 'middle',
i.e. a cause of their substantial being other than that being itself;
but we do not thereby demonstrate it. 


Part 10

Since definition is said to be the statement of a thing's nature,
obviously one kind of definition will be a statement of the meaning
of the name, or of an equivalent nominal formula. A definition in
this sense tells you, e.g. the meaning of the phrase 'triangular character'.
When we are aware that triangle exists, we inquire the reason why
it exists. But it is difficult thus to learn the definition of things
the existence of which we do not genuinely know-the cause of this
difficulty being, as we said before, that we only know accidentally
whether or not the thing exists. Moreover, a statement may be a unity
in either of two ways, by conjunction, like the Iliad, or because
it exhibits a single predicate as inhering not accidentally in a single
subject. 

That then is one way of defining definition. Another kind of definition
is a formula exhibiting the cause of a thing's existence. Thus the
former signifies without proving, but the latter will clearly be a
quasi-demonstration of essential nature, differing from demonstration
in the arrangement of its terms. For there is a difference between
stating why it thunders, and stating what is the essential nature
of thunder; since the first statement will be 'Because fire is quenched
in the clouds', while the statement of what the nature of thunder
is will be 'The noise of fire being quenched in the clouds'. Thus
the same statement takes a different form: in one form it is continuous
demonstration, in the other definition. Again, thunder can be defined
as noise in the clouds, which is the conclusion of the demonstration
embodying essential nature. On the other hand the definition of immediates
is an indemonstrable positing of essential nature. 

We conclude then that definition is (a) an indemonstrable statement
of essential nature, or (b) a syllogism of essential nature differing
from demonstration in grammatical form, or (c) the conclusion of a
demonstration giving essential nature. 

Our discussion has therefore made plain (1) in what sense and of what
things the essential nature is demonstrable, and in what sense and
of what things it is not; (2) what are the various meanings of the
term definition, and in what sense and of what things it proves the
essential nature, and in what sense and of what things it does not;
(3) what is the relation of definition to demonstration, and how far
the same thing is both definable and demonstrable and how far it is
not. 


Part 11

We think we have scientific knowledge when we know the cause, and
there are four causes: (1) the definable form, (2) an antecedent which
necessitates a consequent, (3) the efficient cause, (4) the final
cause. Hence each of these can be the middle term of a proof, for
(a) though the inference from antecedent to necessary consequent does
not hold if only one premiss is assumed-two is the minimum-still when
there are two it holds on condition that they have a single common
middle term. So it is from the assumption of this single middle term
that the conclusion follows necessarily. The following example will
also show this. Why is the angle in a semicircle a right angle?-or
from what assumption does it follow that it is a right angle? Thus,
let A be right angle, B the half of two right angles, C the angle
in a semicircle. Then B is the cause in virtue of which A, right angle,
is attributable to C, the angle in a semicircle, since B=A and the
other, viz. C,=B, for C is half of two right angles. Therefore it
is the assumption of B, the half of two right angles, from which it
follows that A is attributable to C, i.e. that the angle in a semicircle
is a right angle. Moreover, B is identical with (b) the defining form
of A, since it is what A's definition signifies. Moreover, the formal
cause has already been shown to be the middle. (c) 'Why did the Athenians
become involved in the Persian war?' means 'What cause originated
the waging of war against the Athenians?' and the answer is, 'Because
they raided Sardis with the Eretrians', since this originated the
war. Let A be war, B unprovoked raiding, C the Athenians. Then B,
unprovoked raiding, is true of C, the Athenians, and A is true of
B, since men make war on the unjust aggressor. So A, having war waged
upon them, is true of B, the initial aggressors, and B is true of
C, the Athenians, who were the aggressors. Hence here too the cause-in
this case the efficient cause-is the middle term. (d) This is no less
true where the cause is the final cause. E.g. why does one take a
walk after supper? For the sake of one's health. Why does a house
exist? For the preservation of one's goods. The end in view is in
the one case health, in the other preservation. To ask the reason
why one must walk after supper is precisely to ask to what end one
must do it. Let C be walking after supper, B the non-regurgitation
of food, A health. Then let walking after supper possess the property
of preventing food from rising to the orifice of the stomach, and
let this condition be healthy; since it seems that B, the non-regurgitation
of food, is attributable to C, taking a walk, and that A, health,
is attributable to B. What, then, is the cause through which A, the
final cause, inheres in C? It is B, the non-regurgitation of food;
but B is a kind of definition of A, for A will be explained by it.
Why is B the cause of A's belonging to C? Because to be in a condition
such as B is to be in health. The definitions must be transposed,
and then the detail will become clearer. Incidentally, here the order
of coming to be is the reverse of what it is in proof through the
efficient cause: in the efficient order the middle term must come
to be first, whereas in the teleological order the minor, C, must
first take place, and the end in view comes last in time.

The same thing may exist for an end and be necessitated as well. For
example, light shines through a lantern (1) because that which consists
of relatively small particles necessarily passes through pores larger
than those particles-assuming that light does issue by penetration-
and (2) for an end, namely to save us from stumbling. If then, a thing
can exist through two causes, can it come to be through two causes-as
for instance if thunder be a hiss and a roar necessarily produced
by the quenching of fire, and also designed, as the Pythagoreans say,
for a threat to terrify those that lie in Tartarus? Indeed, there
are very many such cases, mostly among the processes and products
of the natural world; for nature, in different senses of the term
'nature', produces now for an end, now by necessity. 

Necessity too is of two kinds. It may work in accordance with a thing's
natural tendency, or by constraint and in opposition to it; as, for
instance, by necessity a stone is borne both upwards and downwards,
but not by the same necessity. 

Of the products of man's intelligence some are never due to chance
or necessity but always to an end, as for example a house or a statue;
others, such as health or safety, may result from chance as well.

It is mostly in cases where the issue is indeterminate (though only
where the production does not originate in chance, and the end is
consequently good), that a result is due to an end, and this is true
alike in nature or in art. By chance, on the other hand, nothing comes
to be for an end. 


Part 12

The effect may be still coming to be, or its occurrence may
be past or future, yet the cause will be the same as when it is actually
existent-for it is the middle which is the cause-except that if the
effect actually exists the cause is actually existent, if it is coming
to be so is the cause, if its occurrence is past the cause is past,
if future the cause is future. For example, the moon was eclipsed
because the earth intervened, is becoming eclipsed because the earth
is in process of intervening, will be eclipsed because the earth will
intervene, is eclipsed because the earth intervenes. 

To take a second example: assuming that the definition of ice is solidified
water, let C be water, A solidified, B the middle, which is the cause,
namely total failure of heat. Then B is attributed to C, and A, solidification,
to B: ice when B is occurring, has formed when B has occurred, and
will form when B shall occur. 

This sort of cause, then, and its effect come to be simultaneously
when they are in process of becoming, and exist simultaneously when
they actually exist; and the same holds good when they are past and
when they are future. But what of cases where they are not simultaneous?
Can causes and effects different from one another form, as they seem
to us to form, a continuous succession, a past effect resulting from
a past cause different from itself, a future effect from a future
cause different from it, and an effect which is coming-to-be from
a cause different from and prior to it? Now on this theory it is from
the posterior event that we reason (and this though these later events
actually have their source of origin in previous events--a fact which
shows that also when the effect is coming-to-be we still reason from
the posterior event), and from the event we cannot reason (we cannot
argue that because an event A has occurred, therefore an event B has
occurred subsequently to A but still in the past-and the same holds
good if the occurrence is future)-cannot reason because, be the time
interval definite or indefinite, it will never be possible to infer
that because it is true to say that A occurred, therefore it is true
to say that B, the subsequent event, occurred; for in the interval
between the events, though A has already occurred, the latter statement
will be false. And the same argument applies also to future events;
i.e. one cannot infer from an event which occurred in the past that
a future event will occur. The reason of this is that the middle must
be homogeneous, past when the extremes are past, future when they
are future, coming to be when they are coming-to-be, actually existent
when they are actually existent; and there cannot be a middle term
homogeneous with extremes respectively past and future. And it is
a further difficulty in this theory that the time interval can be
neither indefinite nor definite, since during it the inference will
be false. We have also to inquire what it is that holds events together
so that the coming-to-be now occurring in actual things follows upon
a past event. It is evident, we may suggest, that a past event and
a present process cannot be 'contiguous', for not even two past events
can be 'contiguous'. For past events are limits and atomic; so just
as points are not 'contiguous' neither are past events, since both
are indivisible. For the same reason a past event and a present process
cannot be 'contiguous', for the process is divisible, the event indivisible.
Thus the relation of present process to past event is analogous to
that of line to point, since a process contains an infinity of past
events. These questions, however, must receive a more explicit treatment
in our general theory of change. 

The following must suffice as an account of the manner in which the
middle would be identical with the cause on the supposition that coming-to-be
is a series of consecutive events: for in the terms of such a series
too the middle and major terms must form an immediate premiss; e.g.
we argue that, since C has occurred, therefore A occurred: and C's
occurrence was posterior, A's prior; but C is the source of the inference
because it is nearer to the present moment, and the starting-point
of time is the present. We next argue that, since D has occurred,
therefore C occurred. Then we conclude that, since D has occurred,
therefore A must have occurred; and the cause is C, for since D has
occurred C must have occurred, and since C has occurred A must previously
have occurred. 

If we get our middle term in this way, will the series terminate in
an immediate premiss, or since, as we said, no two events are 'contiguous',
will a fresh middle term always intervene because there is an infinity
of middles? No: though no two events are 'contiguous', yet we must
start from a premiss consisting of a middle and the present event
as major. The like is true of future events too, since if it is true
to say that D will exist, it must be a prior truth to say that A will
exist, and the cause of this conclusion is C; for if D will exist,
C will exist prior to D, and if C will exist, A will exist prior to
it. And here too the same infinite divisibility might be urged, since
future events are not 'contiguous'. But here too an immediate basic
premiss must be assumed. And in the world of fact this is so: if a
house has been built, then blocks must have been quarried and shaped.
The reason is that a house having been built necessitates a foundation
having been laid, and if a foundation has been laid blocks must have
been shaped beforehand. Again, if a house will be built, blocks will
similarly be shaped beforehand; and proof is through the middle in
the same way, for the foundation will exist before the house.

Now we observe in Nature a certain kind of circular process of coming-to-be;
and this is possible only if the middle and extreme terms are reciprocal,
since conversion is conditioned by reciprocity in the terms of the
proof. This-the convertibility of conclusions and premisses-has been
proved in our early chapters, and the circular process is an instance
of this. In actual fact it is exemplified thus: when the earth had
been moistened an exhalation was bound to rise, and when an exhalation
had risen cloud was bound to form, and from the formation of cloud
rain necessarily resulted and by the fall of rain the earth was necessarily
moistened: but this was the starting-point, so that a circle is completed;
for posit any one of the terms and another follows from it, and from
that another, and from that again the first. 

Some occurrences are universal (for they are, or come-to-be what they
are, always and in ever case); others again are not always what they
are but only as a general rule: for instance, not every man can grow
a beard, but it is the general rule. In the case of such connexions
the middle term too must be a general rule. For if A is predicated
universally of B and B of C, A too must be predicated always and in
every instance of C, since to hold in every instance and always is
of the nature of the universal. But we have assumed a connexion which
is a general rule; consequently the middle term B must also be a general
rule. So connexions which embody a general rule-i.e. which exist or
come to be as a general rule-will also derive from immediate basic
premisses. 


Part 13

We have already explained how essential nature is set out in the terms
of a demonstration, and the sense in which it is or is not demonstrable
or definable; so let us now discuss the method to be adopted in tracing
the elements predicated as constituting the definable form.

Now of the attributes which inhere always in each several thing there
are some which are wider in extent than it but not wider than its
genus (by attributes of wider extent mean all such as are universal
attributes of each several subject, but in their application are not
confined to that subject). while an attribute may inhere in every
triad, yet also in a subject not a triad-as being inheres in triad
but also in subjects not numbers at all-odd on the other hand is an
attribute inhering in every triad and of wider application (inhering
as it does also in pentad), but which does not extend beyond the genus
of triad; for pentad is a number, but nothing outside number is odd.
It is such attributes which we have to select, up to the exact point
at which they are severally of wider extent than the subject but collectively
coextensive with it; for this synthesis must be the substance of the
thing. For example every triad possesses the attributes number, odd,
and prime in both senses, i.e. not only as possessing no divisors,
but also as not being a sum of numbers. This, then, is precisely what
triad is, viz. a number, odd, and prime in the former and also the
latter sense of the term: for these attributes taken severally apply,
the first two to all odd numbers, the last to the dyad also as well
as to the triad, but, taken collectively, to no other subject. Now
since we have shown above' that attributes predicated as belonging
to the essential nature are necessary and that universals are necessary,
and since the attributes which we select as inhering in triad, or
in any other subject whose attributes we select in this way, are predicated
as belonging to its essential nature, triad will thus possess these
attributes necessarily. Further, that the synthesis of them constitutes
the substance of triad is shown by the following argument. If it is
not identical with the being of triad, it must be related to triad
as a genus named or nameless. It will then be of wider extent than
triad-assuming that wider potential extent is the character of a genus.
If on the other hand this synthesis is applicable to no subject other
than the individual triads, it will be identical with the being of
triad, because we make the further assumption that the substance of
each subject is the predication of elements in its essential nature
down to the last differentia characterizing the individuals. It follows
that any other synthesis thus exhibited will likewise be identical
with the being of the subject. 

The author of a hand-book on a subject that is a generic whole should
divide the genus into its first infimae species-number e.g. into triad
and dyad-and then endeavour to seize their definitions by the method
we have described-the definition, for example, of straight line or
circle or right angle. After that, having established what the category
is to which the subaltern genus belongs-quantity or quality, for instance-he
should examine the properties 'peculiar' to the species, working through
the proximate common differentiae. He should proceed thus because
the attributes of the genera compounded of the infimae species will
be clearly given by the definitions of the species; since the basic
element of them all is the definition, i.e. the simple infirma species,
and the attributes inhere essentially in the simple infimae species,
in the genera only in virtue of these. 

Divisions according to differentiae are a useful accessory to this
method. What force they have as proofs we did, indeed, explain above,
but that merely towards collecting the essential nature they may be
of use we will proceed to show. They might, indeed, seem to be of
no use at all, but rather to assume everything at the start and to
be no better than an initial assumption made without division. But,
in fact, the order in which the attributes are predicated does make
a difference--it matters whether we say animal-tame-biped, or biped-animal-tame.
For if every definable thing consists of two elements and 'animal-tame'
forms a unity, and again out of this and the further differentia man
(or whatever else is the unity under construction) is constituted,
then the elements we assume have necessarily been reached by division.
Again, division is the only possible method of avoiding the omission
of any element of the essential nature. Thus, if the primary genus
is assumed and we then take one of the lower divisions, the dividendum
will not fall whole into this division: e.g. it is not all animal
which is either whole-winged or split-winged but all winged animal,
for it is winged animal to which this differentiation belongs. The
primary differentiation of animal is that within which all animal
falls. The like is true of every other genus, whether outside animal
or a subaltern genus of animal; e.g. the primary differentiation of
bird is that within which falls every bird, of fish that within which
falls every fish. So, if we proceed in this way, we can be sure that
nothing has been omitted: by any other method one is bound to omit
something without knowing it. 

To define and divide one need not know the whole of existence. Yet
some hold it impossible to know the differentiae distinguishing each
thing from every single other thing without knowing every single other
thing; and one cannot, they say, know each thing without knowing its
differentiae, since everything is identical with that from which it
does not differ, and other than that from which it differs. Now first
of all this is a fallacy: not every differentia precludes identity,
since many differentiae inhere in things specifically identical, though
not in the substance of these nor essentially. Secondly, when one
has taken one's differing pair of opposites and assumed that the two
sides exhaust the genus, and that the subject one seeks to define
is present in one or other of them, and one has further verified its
presence in one of them; then it does not matter whether or not one
knows all the other subjects of which the differentiae are also predicated.
For it is obvious that when by this process one reaches subjects incapable
of further differentiation one will possess the formula defining the
substance. Moreover, to postulate that the division exhausts the genus
is not illegitimate if the opposites exclude a middle; since if it
is the differentia of that genus, anything contained in the genus
must lie on one of the two sides. 

In establishing a definition by division one should keep three objects
in view: (1) the admission only of elements in the definable form,
(2) the arrangement of these in the right order, (3) the omission
of no such elements. The first is feasible because one can establish
genus and differentia through the topic of the genus, just as one
can conclude the inherence of an accident through the topic of the
accident. The right order will be achieved if the right term is assumed
as primary, and this will be ensured if the term selected is predicable
of all the others but not all they of it; since there must be one
such term. Having assumed this we at once proceed in the same way
with the lower terms; for our second term will be the first of the
remainder, our third the first of those which follow the second in
a 'contiguous' series, since when the higher term is excluded, that
term of the remainder which is 'contiguous' to it will be primary,
and so on. Our procedure makes it clear that no elements in the definable
form have been omitted: we have taken the differentia that comes first
in the order of division, pointing out that animal, e.g. is divisible
exhaustively into A and B, and that the subject accepts one of the
two as its predicate. Next we have taken the differentia of the whole
thus reached, and shown that the whole we finally reach is not further
divisible-i.e. that as soon as we have taken the last differentia
to form the concrete totality, this totality admits of no division
into species. For it is clear that there is no superfluous addition,
since all these terms we have selected are elements in the definable
form; and nothing lacking, since any omission would have to be a genus
or a differentia. Now the primary term is a genus, and this term taken
in conjunction with its differentiae is a genus: moreover the differentiae
are all included, because there is now no further differentia; if
there were, the final concrete would admit of division into species,
which, we said, is not the case. 

To resume our account of the right method of investigation: We must
start by observing a set of similar-i.e. specifically identical-individuals,
and consider what element they have in common. We must then apply
the same process to another set of individuals which belong to one
species and are generically but not specifically identical with the
former set. When we have established what the common element is in
all members of this second species, and likewise in members of further
species, we should again consider whether the results established
possess any identity, and persevere until we reach a single formula,
since this will be the definition of the thing. But if we reach not
one formula but two or more, evidently the definiendum cannot be one
thing but must be more than one. I may illustrate my meaning as follows.
If we were inquiring what the essential nature of pride is, we should
examine instances of proud men we know of to see what, as such, they
have in common; e.g. if Alcibiades was proud, or Achilles and Ajax
were proud, we should find on inquiring what they all had in common,
that it was intolerance of insult; it was this which drove Alcibiades
to war, Achilles wrath, and Ajax to suicide. We should next examine
other cases, Lysander, for example, or Socrates, and then if these
have in common indifference alike to good and ill fortune, I take
these two results and inquire what common element have equanimity
amid the vicissitudes of life and impatience of dishonour. If they
have none, there will be two genera of pride. Besides, every definition
is always universal and commensurate: the physician does not prescribe
what is healthy for a single eye, but for all eyes or for a determinate
species of eye. It is also easier by this method to define the single
species than the universal, and that is why our procedure should be
from the several species to the universal genera-this for the further
reason too that equivocation is less readily detected in genera than
in infimae species. Indeed, perspicuity is essential in definitions,
just as inferential movement is the minimum required in demonstrations;
and we shall attain perspicuity if we can collect separately the definition
of each species through the group of singulars which we have established
e.g. the definition of similarity not unqualified but restricted to
colours and to figures; the definition of acuteness, but only of sound-and
so proceed to the common universal with a careful avoidance of equivocation.
We may add that if dialectical disputation must not employ metaphors,
clearly metaphors and metaphorical expressions are precluded in definition:
otherwise dialectic would involve metaphors. 


Part 14

In order to formulate the connexions we wish to prove we have to select
our analyses and divisions. The method of selection consists in laying
down the common genus of all our subjects of investigation-if e.g.
they are animals, we lay down what the properties are which inhere
in every animal. These established, we next lay down the properties
essentially connected with the first of the remaining classes-e.g.
if this first subgenus is bird, the essential properties of every
bird-and so on, always characterizing the proximate subgenus. This
will clearly at once enable us to say in virtue of what character
the subgenera-man, e.g. or horse-possess their properties. Let A be
animal, B the properties of every animal, C D E various species of
animal. Then it is clear in virtue of what character B inheres in
D-namely A-and that it inheres in C and E for the same reason: and
throughout the remaining subgenera always the same rule applies.

We are now taking our examples from the traditional class-names, but
we must not confine ourselves to considering these. We must collect
any other common character which we observe, and then consider with
what species it is connected and what.properties belong to it. For
example, as the common properties of horned animals we collect the
possession of a third stomach and only one row of teeth. Then since
it is clear in virtue of what character they possess these attributes-namely
their horned character-the next question is, to what species does
the possession of horns attach? 

Yet a further method of selection is by analogy: for we cannot find
a single identical name to give to a squid's pounce, a fish's spine,
and an animal's bone, although these too possess common properties
as if there were a single osseous nature. 


Part 15

Some connexions that require proof are identical in that they possess
an identical 'middle' e.g. a whole group might be proved through 'reciprocal
replacement'-and of these one class are identical in genus, namely
all those whose difference consists in their concerning different
subjects or in their mode of manifestation. This latter class may
be exemplified by the questions as to the causes respectively of echo,
of reflection, and of the rainbow: the connexions to be proved which
these questions embody are identical generically, because all three
are forms of repercussion; but specifically they are different.

Other connexions that require proof only differ in that the 'middle'
of the one is subordinate to the 'middle' of the other. For example:
Why does the Nile rise towards the end of the month? Because towards
its close the month is more stormy. Why is the month more stormy towards
its close? Because the moon is waning. Here the one cause is subordinate
to the other. 


Part 16

The question might be raised with regard to cause and effect whether
when the effect is present the cause also is present; whether, for
instance, if a plant sheds its leaves or the moon is eclipsed, there
is present also the cause of the eclipse or of the fall of the leaves-the
possession of broad leaves, let us say, in the latter case, in the
former the earth's interposition. For, one might argue, if this cause
is not present, these phenomena will have some other cause: if it
is present, its effect will be at once implied by it-the eclipse by
the earth's interposition, the fall of the leaves by the possession
of broad leaves; but if so, they will be logically coincident and
each capable of proof through the other. Let me illustrate: Let A
be deciduous character, B the possession of broad leaves, C vine.
Now if A inheres in B (for every broad-leaved plant is deciduous),
and B in C (every vine possessing broad leaves); then A inheres in
C (every vine is deciduous), and the middle term B is the cause. But
we can also demonstrate that the vine has broad leaves because it
is deciduous. Thus, let D be broad-leaved, E deciduous, F vine. Then
E inheres in F (since every vine is deciduous), and D in E (for every
deciduous plant has broad leaves): therefore every vine has broad
leaves, and the cause is its deciduous character. If, however, they
cannot each be the cause of the other (for cause is prior to effect,
and the earth's interposition is the cause of the moon's eclipse and
not the eclipse of the interposition)-if, then, demonstration through
the cause is of the reasoned fact and demonstration not through the
cause is of the bare fact, one who knows it through the eclipse knows
the fact of the earth's interposition but not the reasoned fact. Moreover,
that the eclipse is not the cause of the interposition, but the interposition
of the eclipse, is obvious because the interposition is an element
in the definition of eclipse, which shows that the eclipse is known
through the interposition and not vice versa. 

On the other hand, can a single effect have more than one cause? One
might argue as follows: if the same attribute is predicable of more
than one thing as its primary subject, let B be a primary subject
in which A inheres, and C another primary subject of A, and D and
E primary subjects of B and C respectively. A will then inhere in
D and E, and B will be the cause of A's inherence in D, C of A's inherence
in E. The presence of the cause thus necessitates that of the effect,
but the presence of the effect necessitates the presence not of all
that may cause it but only of a cause which yet need not be the whole
cause. We may, however, suggest that if the connexion to be proved
is always universal and commensurate, not only will the cause be a
whole but also the effect will be universal and commensurate. For
instance, deciduous character will belong exclusively to a subject
which is a whole, and, if this whole has species, universally and
commensurately to those species-i.e. either to all species of plant
or to a single species. So in these universal and commensurate connexions
the 'middle' and its effect must reciprocate, i.e. be convertible.
Supposing, for example, that the reason why trees are deciduous is
the coagulation of sap, then if a tree is deciduous, coagulation must
be present, and if coagulation is present-not in any subject but in
a tree-then that tree must be deciduous. 


Part 17

Can the cause of an identical effect be not identical in every instance
of the effect but different? Or is that impossible? Perhaps it is
impossible if the effect is demonstrated as essential and not as inhering
in virtue of a symptom or an accident-because the middle is then the
definition of the major term-though possible if the demonstration
is not essential. Now it is possible to consider the effect and its
subject as an accidental conjunction, though such conjunctions would
not be regarded as connexions demanding scientific proof. But if they
are accepted as such, the middle will correspond to the extremes,
and be equivocal if they are equivocal, generically one if they are
generically one. Take the question why proportionals alternate. The
cause when they are lines, and when they are numbers, is both different
and identical; different in so far as lines are lines and not numbers,
identical as involving a given determinate increment. In all proportionals
this is so. Again, the cause of likeness between colour and colour
is other than that between figure and figure; for likeness here is
equivocal, meaning perhaps in the latter case equality of the ratios
of the sides and equality of the angles, in the case of colours identity
of the act of perceiving them, or something else of the sort. Again,
connexions requiring proof which are identical by analogy middles
also analogous. 

The truth is that cause, effect, and subject are reciprocally predicable
in the following way. If the species are taken severally, the effect
is wider than the subject (e.g. the possession of external angles
equal to four right angles is an attribute wider than triangle or
are), but it is coextensive with the species taken collectively (in
this instance with all figures whose external angles are equal to
four right angles). And the middle likewise reciprocates, for the
middle is a definition of the major; which is incidentally the reason
why all the sciences are built up through definition. 

We may illustrate as follows. Deciduous is a universal attribute of
vine, and is at the same time of wider extent than vine; and of fig,
and is of wider extent than fig: but it is not wider than but coextensive
with the totality of the species. Then if you take the middle which
is proximate, it is a definition of deciduous. I say that, because
you will first reach a middle next the subject, and a premiss asserting
it of the whole subject, and after that a middle-the coagulation of
sap or something of the sort-proving the connexion of the first middle
with the major: but it is the coagulation of sap at the junction of
leaf-stalk and stem which defines deciduous. 

If an explanation in formal terms of the inter-relation of cause and
effect is demanded, we shall offer the following. Let A be an attribute
of all B, and B of every species of D, but so that both A and B are
wider than their respective subjects. Then B will be a universal attribute
of each species of D (since I call such an attribute universal even
if it is not commensurate, and I call an attribute primary universal
if it is commensurate, not with each species severally but with their
totality), and it extends beyond each of them taken separately.

Thus, B is the cause of A's inherence in the species of D: consequently
A must be of wider extent than B; otherwise why should B be the cause
of A's inherence in D any more than A the cause of B's inherence in
D? Now if A is an attribute of all the species of E, all the species
of E will be united by possessing some common cause other than B:
otherwise how shall we be able to say that A is predicable of all
of which E is predicable, while E is not predicable of all of which
A can be predicated? I mean how can there fail to be some special
cause of A's inherence in E, as there was of A's inherence in all
the species of D? Then are the species of E, too, united by possessing
some common cause? This cause we must look for. Let us call it C.

We conclude, then, that the same effect may have more than one cause,
but not in subjects specifically identical. For instance, the cause
of longevity in quadrupeds is lack of bile, in birds a dry constitution-or
certainly something different. 


Part 18

If immediate premisses are not reached at once, and there is not merely
one middle but several middles, i.e. several causes; is the cause
of the property's inherence in the several species the middle which
is proximate to the primary universal, or the middle which is proximate
to the species? Clearly the cause is that nearest to each species
severally in which it is manifested, for that is the cause of the
subject's falling under the universal. To illustrate formally: C is
the cause of B's inherence in D; hence C is the cause of A's inherence
in D, B of A's inherence in C, while the cause of A's inherence in
B is B itself. 


Part 19

As regards syllogism and demonstration, the definition of, and the
conditions required to produce each of them, are now clear, and with
that also the definition of, and the conditions required to produce,
demonstrative knowledge, since it is the same as demonstration. As
to the basic premisses, how they become known and what is the developed
state of knowledge of them is made clear by raising some preliminary
problems. 

We have already said that scientific knowledge through demonstration
is impossible unless a man knows the primary immediate premisses.
But there are questions which might be raised in respect of the apprehension
of these immediate premisses: one might not only ask whether it is
of the same kind as the apprehension of the conclusions, but also
whether there is or is not scientific knowledge of both; or scientific
knowledge of the latter, and of the former a different kind of knowledge;
and, further, whether the developed states of knowledge are not innate
but come to be in us, or are innate but at first unnoticed. Now it
is strange if we possess them from birth; for it means that we possess
apprehensions more accurate than demonstration and fail to notice
them. If on the other hand we acquire them and do not previously possess
them, how could we apprehend and learn without a basis of pre-existent
knowledge? For that is impossible, as we used to find in the case
of demonstration. So it emerges that neither can we possess them from
birth, nor can they come to be in us if we are without knowledge of
them to the extent of having no such developed state at all. Therefore
we must possess a capacity of some sort, but not such as to rank higher
in accuracy than these developed states. And this at least is an obvious
characteristic of all animals, for they possess a congenital discriminative
capacity which is called sense-perception. But though sense-perception
is innate in all animals, in some the sense-impression comes to persist,
in others it does not. So animals in which this persistence does not
come to be have either no knowledge at all outside the act of perceiving,
or no knowledge of objects of which no impression persists; animals
in which it does come into being have perception and can continue
to retain the sense-impression in the soul: and when such persistence
is frequently repeated a further distinction at once arises between
those which out of the persistence of such sense-impressions develop
a power of systematizing them and those which do not. So out of sense-perception
comes to be what we call memory, and out of frequently repeated memories
of the same thing develops experience; for a number of memories constitute
a single experience. From experience again-i.e. from the universal
now stabilized in its entirety within the soul, the one beside the
many which is a single identity within them all-originate the skill
of the craftsman and the knowledge of the man of science, skill in
the sphere of coming to be and science in the sphere of being.

We conclude that these states of knowledge are neither innate in a
determinate form, nor developed from other higher states of knowledge,
but from sense-perception. It is like a rout in battle stopped by
first one man making a stand and then another, until the original
formation has been restored. The soul is so constituted as to be capable
of this process. 

Let us now restate the account given already, though with insufficient
clearness. When one of a number of logically indiscriminable particulars
has made a stand, the earliest universal is present in the soul: for
though the act of sense-perception is of the particular, its content
is universal-is man, for example, not the man Callias. A fresh stand
is made among these rudimentary universals, and the process does not
cease until the indivisible concepts, the true universals, are established:
e.g. such and such a species of animal is a step towards the genus
animal, which by the same process is a step towards a further generalization.

Thus it is clear that we must get to know the primary premisses by
induction; for the method by which even sense-perception implants
the universal is inductive. Now of the thinking states by which we
grasp truth, some are unfailingly true, others admit of error-opinion,
for instance, and calculation, whereas scientific knowing and intuition
are always true: further, no other kind of thought except intuition
is more accurate than scientific knowledge, whereas primary premisses
are more knowable than demonstrations, and all scientific knowledge
is discursive. From these considerations it follows that there will
be no scientific knowledge of the primary premisses, and since except
intuition nothing can be truer than scientific knowledge, it will
be intuition that apprehends the primary premisses-a result which
also follows from the fact that demonstration cannot be the originative
source of demonstration, nor, consequently, scientific knowledge of
scientific knowledge.If, therefore, it is the only other kind of true
thinking except scientific knowing, intuition will be the originative
source of scientific knowledge. And the originative source of science
grasps the original basic premiss, while science as a whole is similarly
related as originative source to the whole body of fact.

END


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