Like his teacher Plato, Aristotle's impact on philosophy can scarcely be overstated. From the Classical era until the Renaissance, he was the most relied-upon authority in logic, ethics, aesthetics, rhetoric, and political theory. He also provided the fundamentals of scientific thinking, both in his discussions of logic and mathematics and in his works of physics as well as plant and animal biology.
The selection below, from Posterior Analytics, contains Aristotle's explanation of scientific knowledge. Science in this context must be understood to have a broader meaning then our modern usage, applying to all forms of systematic knowledge (including logic and mathematics) not just the empirical fields we call science today. Even with this larger application, however, Aristotle establishes several principles that remain fundamental to the practice and philosophy of science today. In particular, he values the observation and explanation of the natural world. Science is more than the mere reportage of observed facts; scientific knowledge organizes facts into a series of arguments in order to explain what is not known by what is known.
Taken from Aristotle’s Posterior Analytics, trans. E.S. Bouchier, B.A. Oxford: Blackwell, 1901. The Online Library of Liberty. Web.
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Previous knowledge is required for all scientific studies or methods of instruction. Examples from Mathematics, Dialectic and Rhetoric. Previous knowledge as variously expressed in theses concerning either the existence of a thing or the meaning of the word denoting it. Learning consists in the conversion of universal into particular knowledge.
All communications of knowledge from teacher to pupil by way of reasoning pre-suppose some pre-existing knowledge. The truth of this statement may be seen from a complete enumeration of instances:—it is thus that the mathematical sciences are attained and every art also. The same is the case with dialectical arguments whether proceeding by means of the syllogism or of induction, for the former kind makes such assumptions as people who understand the meaning admit, the latter uses the recognized clearness of the particular as an indication of the universal, so that both convey their information by means of things already known. So too orators produce conviction in a like manner, using either Example, which is equivalent to induction, or Enthymeme, which corresponds to syllogism.
Pre-existing knowledge of two kinds is required: one must either assume beforehand that something exists, or one must understand what the word means, while sometimes both sorts of knowledge are required. As an example of the first case we may take the necessity for previously knowing the proposition ‘everything must be either affirmed or denied.’ Of the second case an instance would be the knowledge of the meaning conveyed by the word ‘triangle’; of the combination of both kinds, the knowledge both of what ‘Unit’ means, and of the fact that ‘Unit’ exists. The distinction is necessary, since the grounds of certainty differ in the two cases.
Some facts become known as a result of previously acquired knowledge, while others are learned at the moment of perceiving the object. This latter happens in the case of all things comprised under a universal, with which one is already acquainted. It is known to the pupil, before perceiving any particular triangle, that the interior angles of every triangle are equal to two right angles; but it is only at the moment of sense-perception that he learns that this figure inscribed in the semi-circle is a triangle.
In some cases knowledge is only acquired in this latter way, and the particular is not learned by means of a middle term: that is to say, in the cases where we touch the concrete particular, that is in the case of things which are not predicable of any subject. We ought to admit that, even before arriving at particulars, and so obtaining a syllogism, we do, from one point of view perhaps, possess knowledge, although from another we do not. For how, it may be asked, when he did not know whether the thing existed at all or not, could he have known absolutely that it contains two right angles? The answer is that he knows it from a particular point of view, in that he knows the universal, but he does not know it absolutely. On any other view we shall have the dilemma of the Meno—a man will either learn nothing at all or only what he knows before. This difficulty must not be solved as some try to do. The question is asked, ‘Do you or do you not know every dyad to be even?’ On receiving an affirmative reply they bring forward some dyad of the existence of which the other was ignorant, and so could not have known it to be even. The solution suggested is to say that one does not know every dyad to be even, but only that which one knows to be a dyad. On the other hand one knows that of which one possesses or has received a demonstration, and no demonstration concerns merely (e.g.) every triangle, or number, one may happen to know, but every possible triangle or number. No demonstrative proposition is taken as referring to ‘any number you may know of,’ or ‘any straight line you may know of,’ but to the entire subject. Nothing, however, I should suppose, precludes our knowing already what we learn from one point of view and not knowing it from another. The absurdity would consist not in having some sort of knowledge of what one learns, but in having knowledge of it in a certain respect—I mean in the very same respect and manner in which one learns it.
Scientific knowledge of a thing consists in knowing its cause demonstratively. The principles required for Demonstration. Meaning of ‘Thesis,’ ‘Hypothesis,’ ‘Axiom,’ ‘Definition.’
We suppose ourselves to know anything absolutely and not accidentally after the manner of the sophists, when we consider ourselves to know that the ground from which the thing arises is the ground of it, and that the fact cannot be otherwise. Science must clearly consist in this, for those who suppose themselves to have scientific knowledge of anything without really having it imagine that they are in the position described above, while those who do possess such knowledge are actually in that position in relation to the object.
Hence it follows that everything which admits of absolute knowledge is necessary. We will discuss later the question as to whether there is any other manner of knowing a thing, but at any rate we hold that that ‘knowledge comes through demonstration.’ By ‘demonstration’ I mean a scientific syllogism, and by ‘scientific’ a syllogism the mere possession of which makes us know.
If then the definition of knowledge be such as we have stated, the premises of demonstrative knowledge must needs be true, primary, immediate, better known than, anterior to, and the cause of, the conclusion, for under these conditions the principles will also be appropriate to the conclusion. One may, indeed, have a syllogism without these conditions, but not demonstration, for it will not produce scientific knowledge. The premises must be true, because it is impossible to know that which is not, e.g. that the diagonal of a square is commensurate with the side. The conclusion must proceed from primary premises that are indemonstrable premises, for one cannot know things of which one can give no demonstration, since to know demonstrable things in any real sense is just to have a demonstration of them. The premises must be Causal, Better known and Anterior; Causal, because we only know a thing when we have learned its cause, Anterior because anteriority is implied by causation, previously known not only in our second sense, viz. that their meaning is understood, but that one knows that they exist.
Now the expressions ‘anterior’ and ‘better known’ have each a double meaning; things which are naturally anterior are not the same as things anterior to us, nor yet are things naturally better known better known to us. I mean by things anterior, or better known, ‘to us,’ such as are nearer our sense-perception, while things which are absolutely anterior or better known are such as are more removed from it. Those things are the furthest removed from it which are most Universal, nearest to it stands the Particular, and these two are diametrically opposed.
The phrase ‘the conclusion must result from primary principles’ means that it must come from elements appropriate to itself, (for I attach the same meaning to primary principle [πρω̂τον] and to element [ἀρχή]). Now the element of demonstration is an immediate proposition; ‘immediate’ meaning a proposition with no other proposition anterior to it. A premise is either of the two parts of a predication, wherein one predicate is asserted of one subject. A dialectical premise is one which offers an alternative between the two parts of the predication, a demonstrative premise is one which lays down definitely that one of them is true.
Predication is either part of a Contradiction. Contradiction is an opposition of propositions which excludes any intermediate proposition. That part of a Contradiction which affirms one thing of another is Affirmation, that which denies one thing of another is Negation.
I apply the name Thesis to an immediate syllogistic principle which cannot be proved, and the previous possession of which is a necessary condition for learning something, but not all. That which is an indispensable antecedent to the acquisition of any knowledge I call an Axiom; for there are some principles of this kind, and ‘axiom’ is the name generally applied to them.
A Thesis which embodies one or other part of a predication (that is that the subject does, or does not, exist) is a Hypothesis; one which makes no such assertion a Definition. Definition is really a kind of Thesis; e.g. the arithmetician ‘lays it down’ that Unity is indivisibility in respect to quantity, but this is not a Hypothesis, for the nature of unity and the fact of its existence are not one and the same question.
Since then belief and knowledge with regard to any subject result from the possession of a demonstrative syllogism, and since a syllogism is demonstrative when the principles from which it is drawn are true, we must not merely have a previous knowledge of some or all of these primary principles, but have a higher knowledge of them than of the conclusion.
The Cause always possesses the quality which it impresses on a subject in a higher degree than that subject; thus, that for which we love anything is dear in a higher degree than the actual object of our love. Hence if our knowledge and belief is due to its primary principles, we have a higher knowledge of these latter and believe more firmly in them, because the thing itself is a consequence of them. Now it is not possible to believe less in what one knows than in what one neither knows nor has attained to by some higher faculty than knowledge. But this will happen unless he whose belief is produced by demonstration has a previous knowledge of the primary principles, for it is more needful to believe in these principles, either all or some, than in the conclusion to which they lead.
Now in order to attain to that knowledge which comes by demonstration one must not only be better acquainted with and believe more firmly in the elementary principles than in the conclusion, but nothing must be better known nor more firmly believed in than the opposites of those principles from which a false conclusion contrary to the science itself can be educed; that is to say if he who possesses absolute knowledge is to be quite immovable in his opinions.
Certain objections met. (1) That first principles are hypothetical; (2) That their consequences establish one another by a circular proof.
Now some persons, because of the necessity of knowledge of the primary principles, infer that knowledge does not exist, while others suppose that it does exist and that everything whatever is capable of demonstration. Neither of these views is either true or necessary. Those who assume that knowledge is not possible at all, think that it would involve an infinite regress, since one cannot know the later terms of a series by means of the earlier when such a series has no primary terms. In this they are right, for it is impossible to complete the infinite. But if there be a limit to the regress, and primaries do exist, they say that these must be unknowable, supposing that they admit of no demonstration, which is the only way of knowing they allow to exist. But if it be impossible to learn these primary principles, one cannot know their results either absolutely or in any proper sense, but only hypothetically, viz. on the assumption that such principles do exist.
The other party agrees with them in holding that knowledge can only be attained by demonstration, but considers that there is nothing to prevent a demonstration of everything being given, maintaining that demonstration may proceed in a circle, all things being proved reciprocally.
We, on the other hand, hold that not every form of knowledge is demonstrative, but that the knowledge of ultimate principles is indemonstrable. The necessity of this fact is obvious, for if one must needs know the antecedent principles and those on which the demonstration rests, and if in this process we at last reach ultimates, these ultimates must necessarily be indemonstrable. Our view then is not only that knowledge exists, but that there is something prior to science by means of which we acquire knowledge of these ultimates. On the other hand it is clear that absolute demonstration cannot proceed in a circle if it be admitted that the demonstration must be drawn from anterior and better known principles than itself; for it is impossible for the same things to be both anterior and posterior in relation to the same objects, except from a different point of view, e.g. some things may be anterior relatively to us and others absolutely anterior, a distinction which inductive proof illustrates. If this be so the definition of absolute knowledge might be considered defective, since it really has a double sense; or that second kind of demonstration drawn from principles better known in relation to us is ambiguous.
Those who hold that demonstration proceeds in a circle not only meet with the difficulty already mentioned, but really say that ‘this is if this is,’—an easy method of proving anything whatsoever. This appears plainly when three terms are assumed (for it is immaterial whether one says that the proof passes through many or few terms before returning to the starting point, as also whether it be through a few or two only). For when:
|If A is, B must be|
|and||If B is, C must be|
|Then||If A is, C will be|
|And when||If A is, B must be|
|and||If B is, A must be|
(for that is how the circular proof proceeds). Let A be placed in the position C held before. Then to say that ‘If B is, A must be,’ is equivalent to saying that C must be, and this proves that ‘If A is, C must be’; and C is here identical with A.
Thus those who hold that the demonstration proceeds in a circle simply declare that if A is, A must be—an easy method of proving anything.
Nor is even this proof possible except in the case of reciprocals such as Properties. It has been already shewn (Prior An. II. 5) that it is never necessary that a conclusion should follow when only one thing is assumed (by ‘one thing’ I mean one term or one proposition); such can only happen when there are at least two antecedent propositions capable of producing a syllogism.
If then A be a consequence of B and of C, and these latter consequences of each other, and also of A, it is possible to prove reciprocally all the questions that can be raised, in the first figure, as has been shewn in the treatise on the Syllogism (Prior An. II. 5). But it has also been shewn that in the other figures no circular demonstration can be effected, or none concerning the premises in question.
Circular demonstration is never admissible in the case of terms not reciprocal. Hence, as few such terms occur in demonstrations, it is clearly useless and untrue to maintain that demonstration consists in proving each term of a series by means of the others, and that consequently everything is demonstrable.
Demonstration deals with necessary truths. The definition of ‘distributively true,’ ‘essential,’ ‘universal.’
Now since the object of absolute knowledge can never undergo change, the objects of demonstrative knowledge must be necessary. Knowledge becomes demonstrative when we possess a demonstration of it, and hence demonstration is a conclusion drawn from necessary premises. We must now then state from what premises and conclusions demonstrations may be drawn; and first let us define what we mean by ‘Distributively true,’ ‘Essential’ and ‘Universal.’
By ‘Distributively true,’ I mean a quality which is not merely present in some instances and absent in others, or present at some times and absent at others; e.g. if the quality ‘Animal’ be distributively predicable of man, if it be true to say ‘this is a man,’ it must also be true to say ‘this is an animal’; and if he be the one now, then he must be the other now; so too if ‘Point’ be true of every line. An empirical proof of this is the fact that when the question is raised whether one thing is true of another distributively, our objections take the form of asserting that it is not true of some particular instance or at some particular time.
I. ‘Essential’ qualities are all those which enter into the essence of a thing, (as ‘line’ does into that of ‘triangle,’ and ‘point’ into that of ‘line’; for ‘line’ and ‘point’ belong to the essence of ‘triangle’ and line respectively), and are mentioned in their definition.
II. Essential qualities are, further, attributes of subjects in the definition of which the subject is mentioned, thus ‘Straight’ or ‘Curved’ are essential attributes of ‘Line’; ‘Odd’ or ‘Even’ of ‘Number’; as also ‘Prime’ and ‘Compound,’ ‘Equilateral’ (as 3) and ‘Scalene’ (as 6); in all these cases the things form part of the definition of the real nature of the attributes mentioned, these things being in the first instance ‘Line,’ in the second ‘Number.’ So too in other instances I call attributes which inhere in either of these ways ‘essential,’ while attributes which do not belong to the subject in either of these ways I call ‘accidental’; e.g. ‘Musical’ or ‘White’ as applied to ‘Animal.’
III. Thirdly, essential is that which is not predicated of anything other than itself as attribute of subject; thus if I say, ‘the walking thing,’ some other independent thing is ‘walking’ or is ‘white.’ On the other hand substances and everything which denotes a particular object are not what they are in virtue of being anything else but what they are. Things then which are not predicable of any subject I call ‘essential,’ those which are so predicable ‘accidental’ [in the sense of dependent].
IV. In a fourth sense the attribute which exists in a subject as a result of itself is essential, while that which is not self-caused is accidental. E.g. Suppose lightning to appear while a person is walking. This is accidental, for the lightning is not caused by his walking, but, as we say, ‘it was a coincidence.’ If, however, the attribute be self-caused it is essential: e.g. if someone is wounded and dies, his death is an essential consequence of the wound, since it has been caused by it:—the wound and death are not an accidental coincidence. In the case then of the objects of absolute knowledge, that which is called ‘essential’ in the sense of inhering in the attributes or of having the latter inhering in it is self-caused and necessary; for it must inhere either unconditionally or as one of a pair of contraries, e.g. as either straight or curved inhere in line, odd or even in number. Contrariety consists in either the privation or the contradiction of a quality in the case of homogeneous subjects: e.g. in the case of numbers ‘even’ is that which is not ‘odd,’ in so far as one of these qualities is necessarily present in a subject. Hence, if one of these qualities must be either affirmed or denied, essential attributes are necessary. This then may suffice for the definition of Distributive and Essential.
By ‘Universal’ I mean that which is true of every case of the subject and of the subject essentially and as such. It is clear then that all universal attributes inhere in things necessarily. Now ‘essentially’ and ‘as such’ are identical expressions: e.g. Point and Straight are essential attributes of line, in that they are attributes of it as such. Or again the possession of two right angles is an attribute of triangle as such, for the angles of a triangle are essentially equal to two right angles. The condition of universality is satisfied only when it is proved to be predicable of any member that may be taken at random of the class in question, but of no higher class; e.g. the possession of two right angles is not a universal attribute of figure, for though one may demonstrate of a particular figure that it has two right angles, it cannot be done of any and every figure, nor does the demonstrator make use of any and every figure, for a square is a figure, but its angles are not equal to two right angles. Any and every isosceles triangle has its angles equal to two right angles, but it is not a primary, ‘triangle’ standing yet higher. Thus any primary taken at random which is shewn to have its angles equal to two right angles, or to possess any other quality, is the primary subject of the universal predicate, and it is to that demonstration primarily and essentially applies; to everything else it applies only in a sense. Nor is this quality of having its angles equal to two right angles a universal attribute of isosceles triangle, but is of a wider application.
Demonstration must disregard all accidental circumstances, and aim at the discovery of the essential and universal.
We must not fail to notice that mistakes frequently arise from the primary universal not being really demonstrated in the way in which it is thought to be demonstrated. We fall into this mistake firstly when no universal can be found above the particular or particulars: secondly, when such a universal is found applicable to specifically different subjects, but yet has no name; thirdly, when the universal to be demonstrated stands to the true universal in the relation of part to whole.
In this last case the demonstration is indeed applicable to all the particular parts, but will not contain a primary universal. I consider the demonstration to be primary and essential when it is a demonstration of a primary universal. If then it were to be proved that perpendiculars to the same line are parallel, it might be thought that this was the primary subject of the demonstration because it is true in the case of all right angles so formed. This, however, is not the whole truth. The lines are parallel not because each of the angles at their base is a right angle, and consequently equal to the other, but because such angles are in all cases equal to two right angles.
So, too, if there were no other kind of triangle than the isosceles it might be supposed that the quality of possessing angles equal to two right angles was true of the subject as isosceles. Again, the law that proportionals, whether numbers, lines, solids, or periods of time, may be permuted, would be a case, as it used to be proved, viz., of each case separately, though it may really be proved of all together by means of a single demonstration; but since no single designation included magnitudes, times and solids, and since these differ specifically, they were treated of separately. The law is now, however, proved universally. It does not apply to numbers or lines as such, but only because it belongs to the universal conception as such in which all are supposed to be. Hence even if it be proved of equilateral, scalene and isosceles triangles separately, whether by means of the same or by different proofs that every one has its angles equal to two right angles, one will not know except accidentally, that triangle possesses this quality nor will one know it of the universal triangle, even though there is no other sort of triangle than those mentioned. One does not in fact know it of triangle as such, nor yet of every individual triangle, except distributively, nor does one know it of every triangle ideally, even if there is no triangle of which one does not know it.
When, we may ask, is our knowledge not universal and when is it absolute? It is clear that our knowledge of the law would be universal if triangularity and equilateral triangularity were identical in conception. If, however, the two concepts be not identical but diverse, and if the quality in question belong to triangle as such, then a knowledge of the law as relating merely to a particular form of triangle is not universal. Now does this quality belong to triangle as such, or to isosceles triangle as such? Further, what is its essential primary subject? Also, when does the demonstration of this establish anything universal? Clearly when, after the elimination of accidental qualities, the quality to be demonstrated is found to belong to the subject and to no higher subject. For example, the quality of having its angles equal to two right angles will be found to belong to bronze isosceles triangle, but will still be present when the qualities ‘bronze’ and ‘isosceles’ are eliminated; so too, it may be said they will cease to be present when Form or Limit are eliminated. But they are not the first conditions of such disappearance. What then will first produce this result? If it is triangle, the quality of having two right angles belongs to the particular kinds of triangles as a result of its belonging essentially to triangle, and the demonstration in regard to triangle is a universal demonstration.
For necessary conclusions necessary premises are required.
If then demonstrative knowledge be derived from necessary principles (and that which one knows is never contingent), and if the essential attributes of a subject be necessary (and essential attributes either inhere in the definition of the subject, or, in cases where one of a pair of opposites must necessarily be true, have the subjects inhering in their definition), then it is clear that the demonstrative syllogism must proceed from necessary premises Every attribute is predicable either in the way mentioned or accidentally, but accidental attributes are not necessary. We should then either express ourselves as above or lay it down as an elementary principle that demonstration is something necessary, and that if a thing has been demonstrated it can never be other than it is; and consequently that the demonstrative syllogism must proceed from necessary premises. It is indeed possible to syllogize from true premises without demonstrating anything, but not so if the premises be also necessary, for this very necessity is the characteristic of demonstration.
An empirical confirmation of the view that demonstration results from necessary premises is that when we bring forward objections against persons who imagine themselves to be producing a demonstration, we bring our objections in the form ‘There is no necessity.’ Whether we hold that the things in question are really contingent or only considered to be so for the sake of a particular argument. It is clear from this that it is folly to suppose oneself to have made a good choice of scientific principles so long as the premise be generally accepted and also true, after the manner of the sophists who assume that ‘Knowing is identical with possessing knowledge.’ It is not in fact that which is generally accepted or rejected which constitutes a principle, but the primary properties of the genus with which the demonstration deals; nor is everything which is true also appropriate to the conclusion to be demonstrated.
It is also clear from the following considerations that the syllogism can proceed from necessary premises only. If one who, in a case where demonstration is possible, is not acquainted with the cause, can have no real knowledge of the demonstration, then one who knows that A is necessarily predicable of C, whilst B, the middle term by means of which the demonstration has been effected, is not necessary, must be ignorant of the cause of the thing, for in this case the conclusion is not rendered necessary by the middle term; in fact the middle, since it is not necessary, may not exist at all, but the conclusion is necessary.
Moreover if one who now knows (accidentally) the cause of a necessary conclusion remains unchanged while the thing itself remains unchanged, and if, though he has not forgotten it, yet he has no real knowledge of it, then he can never have had any real knowledge of it before. When the middle term is not anything necessary, it may pass away. In such a case, if the man remain unchanged while the thing remains unchanged, he may hold fast the cause of the thing, but he has no real knowledge of the thing itself, nor has he ever had such knowledge. But if the thing denoted by the middle term has not passed away, but yet is capable of doing so, that which results from it is only the possible, not the necessary; and when one’s inference is derived only from the possible one cannot be said to have knowledge in the true sense of the word. When the conclusion is necessary there is nothing to prevent the middle term, by means of which the conclusion was proved from being necessary, for it is possible to infer the necessary from the not necessary, just as one may infer the true from the untrue.
But when the middle term is necessary the conclusion also is necessary, just as true premises always produce a true conclusion. Thus, suppose A to be necessarily predicable of B, and B of C; A then must be necessarily predicable of C. But when the conclusion is not necessary, it is impossible that the middle should be necessary.
Suppose that, Some C is A, and again that All B is A, and that All C is B. But then All C will be A, which is contrary to our original hypothesis.
Since then that which one knows demonstratively must be necessary, it is clear that one ought to obtain the demonstration by means of a necessary middle term. Otherwise one will neither know the cause of the thing demonstrated nor the necessity of its being what it is, but one will either think one knows it without doing so (that is if one supposes to be necessary that which is not necessary), or one will think one knows it in a different way if one knows the fact of the conclusion with the help of middle terms, and when one knows its cause without the help of middle terms. Now there is no demonstrative science of accidents (attributes) which are not essential according to our definition of ‘essential.’ It is not in this connection possible to prove that the conclusion is necessarily true, for the accidental may not be true; (it is of accidents of this kind that I am speaking).
A difficulty might perhaps be raised as to why accidental premises are asked for for the purposes of a conclusion, if the conclusion drawn from them be not necessary; for it might be maintained that it would make no difference if any sort of premise were brought forward and then the conclusion were subjoined. Premises should however be laid down not because the conclusion is necessarily true because of them, but because the person who admits the premises must necessarily admit the conclusion, and his admission will be correct if the premises are true.
Now since only the essential attributes of any genus and those belonging to it as such are necessary, it is clear that scientific demonstrations both deal with and are drawn from essential attributes. As accidental attributes are not necessary one does not require to know the cause of the conclusion, not even if this be an eternal attribute without being essential, as in the case of syllogisms based on universal concomitance. In this latter connection the essential will be known, but not the fact that it is essential, nor yet why it is so. (By ‘knowledge of why it is essential’ I mean ‘knowing its cause.’) In order then to possess knowledge of this sort the middle term must result from the nature of the minor, and the major from the nature of the middle.
Premises must be homogeneous with the conclusion. No transference of premises from one genus to another is valid unless the one is subaltern to the other.
It is not possible to arrive at a demonstration by using for one’s proof a different genus from that of the subject in question; e.g. one cannot demonstrate a geometrical problem by means of arithmetic. There are three elements in demonstrations:—(1) the conclusion which is demonstrated, i.e., an essential attribute of some genus; (2) axioms or self-evident principles from which the proof proceeds; (3) the genus in question whose properties, i.e. essential attributes, are set forth by the demonstrations. Now the axioms which form the grounds of the demonstration may be identical for different genera; but in cases where the genera differ, as do arithmetic and geometry, it is not possible, e.g. to adapt an arithmetical demonstration to attributes of spatial magnitudes, unless such magnitudes happen to be numbers. That such transference is possible in certain connections I will explain later (cf. Chap. IX.).
Arithmetical demonstration is restricted to the genus with which it is properly concerned, and so with other sciences. Hence if a demonstration is to be transferred from one science to another the subjects must be the same either absolutely or in some respect. Otherwise such a transference is clearly impossible, for the extremes and the middle terms must necessarily belong to the same genus, for if not they would not be essentially but only accidentally predicable of the subject.
Hence one cannot shew by means of geometry that opposites are dealt with by a single science nor yet that two cubes when multiplied together produce another cube. Nor can one prove what belongs to one science by means of another except when one is subordinate to the other, as optics are to geometry and harmonics to arithmetic.
Neither is geometry concerned with the question of an attribute of line which does not inhere in it as such, and does not result from the special principles of geometry, as for instance the question whether the straight line is the most beautiful kind of line, or whether the straight line is the opposite of a circumference, for these qualities of beauty and opposition do not belong to line as a result of its particular genus, but because it has some qualities in common with other subjects.
The conclusion of a demonstration must be of everlasting application. Perishable things are, strictly speaking, indemonstrable. This applies also to definitions, which are a partial demonstration.
It is clear that if the premises from which the syllogism proceeds are universal, the conclusion of such a demonstration and of demonstration in general must be eternal. There is then no knowledge properly speaking of perishable things, but only accidentally, because the knowledge of perishable things is not universal but under restrictions of time and manner. When this is the case, the minor premise at least must be other than universal and must be perishable:—perishable because then the conclusion will contain a similar element, other than universal because then the predication will apply to some and not others of the subjects in question; so that no universal conclusion can be drawn but only one referring to this or that definite time. The same holds good with regard to definitions, seeing that definition is either the starting point of a demonstration, or itself a demonstration which differs from definition only in the way in which it is expressed or, lastly, in form a conclusion of a demonstration.
Demonstrations and sciences concerning things which occur only frequently (e.g. lunar eclipses) are clearly of everlasting application in so far as they are demonstrations, while in so far as they are not of everlasting application they are particular. As in the case of eclipses so is it with other subjects of the kind.
All demonstration is derived from special principles, themselves indemonstrable, the knowledge of which, in each genus, is the supreme knowledge on which the whole deduction depends.
Since it is clear that nothing can be demonstrated except from its own elementary principles, that is to say when the thing demonstrated is an essential attribute of the subject, it does not suffice for the possession of knowledge that a thing shall have been demonstrated from true, indemonstrable and ultimate premises. Otherwise demonstrations would be admissible resembling that of Bryson demonstrating the squaring of the circle. Now such arguments demonstrate by means of a common principle which will apply to another science as well, so that the same arguments are of service in other sciences distinct in kind. Thus we have no essential but only an accidental knowledge of the thing, for otherwise the demonstration would not also be applicable to another kind of subjects.
We have more than an accidental knowledge of anything when we see it in the light of its essential nature, after starting from the elementary principles of the things as such. Thus we know the law that a triangle has two right angles when we know of what figure this is an essential attribute and know it after starting from the principles peculiar to Triangle. Hence if the attribute is essentially an attribute of the subject, the middle term of the demonstration must necessarily be included in the same genus, or, if not, one of the genera must be subordinate to the other, as when proportions in harmonics are proved by means of arithmetical premises. Such relations are proved in the same way as in arithmetic, but there is a difference between the two cases, for the question of the Fact falls under the one science (since the subjects of the two sciences differ generically) but the Cause is established by the superior science, to which the properties in question are essential. It is plain even from the case of the subordinate sciences that no absolute demonstration of a thing can be attained save by starting from its own elementary principles. In this case, however, the elementary principles of the sciences in question are not mutually exclusive.
If this be admitted it is also clear that it is impossible to demonstrate the special elementary principles of each science, for the principles of such a demonstration would be the elementary principles of everything, and the science formed by them would be the universal master science; seeing that one who learns a thing through the recognition of higher causes has a better knowledge of it, and the principles through which he learns the thing are anterior when they are causes not themselves produced by any higher cause. If then his knowledge be of this higher kind it must have attained to the highest possible degree, and if this subjective knowledge of his constitutes a science, that science must be higher than any other, and in fact the highest science.
The demonstration of one thing is not applicable to another genus except in the case already mentioned, as illustrated by the application of geometrical demonstrations to mechanical or optical, or of arithmetical demonstrations to harmonic theorems.
Now it is hard to decide if we really know a thing or not, for it is hard to decide whether our knowledge is derived from the elementary principles of the subject or not, and it is in this that knowledge consists. We imagine that, if we possess a syllogism drawn from true and primary premises, we really possess knowledge. This, however, is not the case, for the conclusions should belong to the same genus as the primary principles.
Such indemonstrable principles may be either peculiar to each science or common to several sciences, though common only by analogy. All demonstration involves three things:—the object demonstrated, common axioms or principles, and the special modifications or properties of the subject genus. The distinction between Hypothesis and Petition.
I mean by the elementary principles in each genus those whose existence it is not possible to prove. Now the meaning of the primary principles and that of their consequences are assumed; the existence of the elementary principles must also be assumed, that of everything else proved. For instance the meaning of Unit, or Straight, or Triangle must be assumed, that Unit and Magnitude exist must also be assumed, everything else must be proved.
Of the principles employed in demonstrative science some are peculiar to each science, others are common to all, i.e. common in the sense of analogous, since their use is confined to each genus as comprehended by a particular science. Principles peculiar to one science are such as the proposition ‘Line, or Straight, is of such and such a nature’; common principles are such as, ‘If one take equals from equals the remainders are equal.’ Each of these principles is taken as applicable to all cases belonging to the particular genus; for its results will be the same whether it be treated universally or only particularly, e.g. in geometry to spatial magnitudes or in arithmetic to numbers.
Those principles too are peculiar whose existence is assumed not demonstrated, namely those whose essential attributes are investigated by the science; as arithmetic investigates units, geometry points and lines, for these sciences assume that the thing in question exists, and that it is identical with some particular object. They likewise assume the meaning of the essential attributes of the thing, as arithmetic assumes the meaning of Odd, Even, Square or Cube, and geometry that of Incommensurable, and Inclined or at an Angle, while the existence of these qualities is shewn by means of the common principles and the conclusions already demonstrated. The same thing applies to astronomy.
In short in every demonstrative science there are three elements: (1) the things whose existence it assumes, namely the subject or genus, the essential attributes of which are investigated by the science; (2) what are called ‘Common Axioms’ which the demonstration uses as its primary principles; and (3) Properties, the meaning of which is assumed.
However nothing prevents some sciences from overlooking one or other of these elements; e.g. a science may not expressly assume the existence of the subject genus if this be self-evident (for the existence of Number is more obvious than that of Cold or Heat), or it may not assume the meaning of the properties if it is obvious, just as in the case of their common principles the sciences do not assume the meaning of ‘taking equals from equals,’ because this is known. None the less however there are naturally these three elements in a science:—the subject of proof, the things proved and the grounds of proof.
That which must needs exist and must necessarily be supposed to exist is neither Hypothesis nor Petition but Axiom. Demonstration is not concerned with the outward expression of an idea but with its inner significance, for that is the case with syllogism in general, and one may always raise objections to the external expression but not always to the inner significance.
Everything which, being capable of proof, is assumed without being proved, if admitted by the learner is a Hypothesis, which hypothesis is not an absolute hypothesis but only one with reference to the person who accepts it.
If however something be assumed with regard to which the learner has no opinion or a contrary one it is a Petition. This then is the difference between hypothesis and petition; petition being that which is somewhat opposed to the learner's opinion, or, in a wider sense, whatever, though capable of demonstration, is assumed and employed without any proof.
Definitions are not hypotheses, since it is not asserted that their subjects do or do not exist. Hypotheses are formulated as propositions, Definitions require only to be understood, and no Hypothesis consists in that alone, unless it be maintained that mere Hearing is a Hypothesis. Hypotheses are the premises from the existence of which the conclusion is inferred.
The hypotheses of the geometrician are not, as some assert, false, saying that, though one ought not to make use of false propositions, yet the geometrician calls a line a foot long which is not a foot long, or declares that he has drawn a straight line, though the line is really not straight. The geometrician in reality draws no conclusion from the fact of the particular line that he draws actually possessing the quality which he names, but from the existence of the things which that line represents.
Moreover all postulates and hypotheses are universal or particular; definitions are neither.
[The possibility of Demonstration presupposes the validity of universal predicates, but does not require Platonic ideas]. The ‘Common Axioms’ are expressly formulated in exceptional cases. They connect the sciences with one another, and with Dialectic and Metaphysics, thus giving unity to all forms of true Thought.
[It does not follow, if demonstration is to exist, that there must be Ideas or a Unity outside the many individual things, but it does follow that some unity must be truly predicable of the many. If no such unity existed we should have no universal; and without a universal there could be no middle term and consequently no demonstration. Since demonstration does exist there must be some self-identical unity, a real and no mere nominal unity, predicable of many individual things.] No demonstration lays down that it is impossible both to affirm and to deny a quality of a thing at the same time, unless it is necessary to present the conclusion in a corresponding form by the help of that axiom. In that event the conclusion is proved by our assuming that the major is predicable of the middle term, and that to deny the major of the middle is untrue. It makes no difference if the thing denoted by the middle be assumed to exist or to be non-existent, and the same applies to the thing denoted by the minor. If it be granted that Man is such and such; i.e. if, though Not-man be also such and such, it be simply granted that man is animal and not not-animal; then Callias [being man] will be animal and not not-animal, even though not-Callias be also man. The reason of this is that the major is not only predicated of the middle but of something else outside it, because it has a wider application, so that it makes no difference to the conclusion whether the middle be an affirmative or a negative expression.
Demonstration by means of reduction to absurdity assumes the truth of the law ‘everything may be either affirmed or denied of a subject,’ and this not always in a universal sense but simply to the extent required, namely so as to be applicable to the particular genus in question. I mean by ‘applying to the genus,’ that genus with which one’s demonstration is concerned, as has been remarked above. (Chap. X.).
All sciences overlap as far as their common principles are concerned. (By these I mean the principles used by them as the grounds of demonstration, not the subjects of the demonstration nor yet the thing demonstrated). Now dialectic is common to all the sciences, and if one were to try and give a universal proof of the common principles of science, such as ‘Everything can be either affirmed or denied,’ or ‘if equals be taken from equals,’ or some maxim of that kind [the resulting science would similarly be common to all sciences]. But dialectic does not deal with any definite objects of this sort nor with any single genus. Otherwise it would not have used the interrogative form, for this cannot be employed for purposes of demonstration; since the same thing cannot be proved from opposite propositions. This has been proved in the treatise on the syllogism. (Prior An. II. 15).
Corresponding to the special principles of a science are special questions which must not be transferred from one genus to another, so that no discussion of a science with persons ignorant of it can lead to valid results. Two kinds of opposites to a science exist:—questions or demonstrations entirely outside its range and those which involve a breach of some of its laws.
If a syllogistic question be the same as one of the members of an alternative, and if there be premises in each science from which the syllogism belonging specially to each science may be deduced, there must be some scientific question from which the special syllogism corresponding to each science is derived.
It is plain then that not every question can be a geometrical or a medical question, and similarly with all other special sciences, but only those questions can be geometrical proceeding from which some of the matters connected with geometry are proved, or something proved on the same principles as geometry; e.g. optical theorems. The same is the case with other sciences. Now with regard to these questions, in the case of geometry they must be explained in accordance with the principles and conclusions of geometry, but no account need be given of the principles themselves by the geometrician as such, and this applies to other sciences also.
One should not then ask every possible question of a person acquainted with a particular science, nor need he answer every question asked of him, but only a question concerning the definite subject of the particular science. If one enter into a discussion with a geometrician as such, it is clear that the proof he gives will be a sound one if drawn according to these principles, otherwise unsound. It is also clear that in such circumstances one cannot confute a geometrician except accidentally, so that we must not discuss geometry before persons ignorant of that science, for any unsound arguments put forward will remain unnoticed. The same is the case with other sciences.
Since then there are geometrical questions, it may be asked whether there are also ungeometrical, and what kind of ignorance in connection with each science causes certain questions to bear the same relation to that science as ungeometrical bear to geometrical questions. Further is a syllogism resting on ignorance a syllogism formed from premises which contradict the science it belongs to, or rather a fallacy which nevertheless does belong to the science in question, e.g. geometry? Or, again, is a question belonging to another pursuit, such as a musical question, ungeometrical as regards geometry? Again, is the supposition that parallel lines can meet in one sense geometrical and from another point of view ungeometrical? ‘Ungeometrical’ is in fact an ambiguous expression, as is ‘unrhythmical.’ One thing may be ungeometrical or unrhythmical from not possessing the quality in question at all, another from having it defectively. So too the form of ignorance resulting from bad or defective principles is contrary to Science. In mathematical sciences the fallacy is more easily perceived than in other sciences, because in them the middle term is always expressed twice, something being predicated distributively of the middle term, and the latter in turn predicated distributively of another subject. The predicate is not however used distributively. In mathematics one may, as it were, see by an immediate act of thought the relations of the middle term, while in words they remain unnoticed. E.g. as regards the question, ‘Is every circle a figure?’ If one describes a circle on paper it clearly is so. If the conclusion be drawn ‘then the epic cycle is a figure,’ this is clearly untrue.
No objection should be raised to a science on the ground that its premises are inductive, for just as nothing can be a premise which does not apply to several instances (otherwise it would not be universally predicable, and Syllogism is drawn from universals), so an objection must have a universal application. Premises and the objections to them correspond to one another, and any objection one urges against a premise should be capable of serving either as a demonstrative or as a dialectical premise.
The laws of the syllogism are violated when the common attribute of both major and minor terms is treated as their predicate. An instance is the syllogism of Caeneus that ‘fire increases in geometrical proportion’; ‘for,’ as he says, ‘fire increases rapidly and so does geometrical proportion.’ No syllogism can, however, be formed thus. The truth is: if the proportion which increases most quickly in respect to quantity be the geometrical, and if fire be that which increases most quickly in respect to motion....
Thus it is sometimes impossible to draw a conclusion from two premises of this kind, at other times it is possible, though the possibility may not be observed. If it were impossible to draw any true conclusion from false premises, it would be easy to bring the syllogism to a conclusion, for it would necessarily be convertible. For instance let A exist by hypothesis, and when A exists let something else (B for instance) exist also, which one knows in this instance does exist. By conversion then it may be shewn from B that A exists. Conversion is more frequent in pure mathematics because these admit of no accidental qualities (and in this differ from dialectical arguments) but only of definitions.
Mathematical science is advanced not by the use of a number of middle terms, but by the subsumption of one term under another (as A under B, B under C, C under D, and so to infinity). The process may also take two directions, A being predicable both of C and E. Suppose A represents any number definite or indefinite.
B any odd number of definite magnitude.
C any odd number whatsoever.
(Then A will be seen to be predicable of C).
Again:— Let D be an even definite number.
E any even number whatsoever.
Then A is predicable of E.
The objects of knowledge are four in number:—a thing’s existence, its cause, the question whether it is, and its nature.
The subjects of enquiry are equal in number to the objects of scientific knowledge. We enquire about four things, the fact of the phenomenon, its cause, whether it exists and what its nature is. Now when we ask whether a thing is this or that, taking two alternatives, e.g. asking whether the sun is eclipsed or not, we ask about the fact. A proof of this is that when we find that it is eclipsed we abandon this line of enquiry. Also if we know from the first that it is eclipsed we do not ask whether it is eclipsed or not. Next, after learning the fact of the phenomenon we seek for the cause of it. For example, when we know that the sun is eclipsed or that the earth does move, we go on to seek for the cause of the eclipse or of the movement.
These questions concerning the fact and cause stand towards each other in the relation here stated, but in some questions the enquiry proceeds differently: namely whether a thing exists at all or not; e.g. as to whether or not a centaur or a god is. By ‘whether it is or not’ I mean is absolutely, not whether a thing is, e.g., white or not white. When we know that the thing does exist we enquire about its nature, asking, for instance, ‘What then is a god, or what is a man?’
The first and third of these questions and also the second and fourth may be identified. Hence all scientific enquiry consists in investigating whether there is a middle term, and what the middle term is, for the middle is identical with the cause.
These or such as these are the subjects about which we enquire and which we know when we have found what we sought. Now when we ask about the fact, or enquire whether the thing has absolute existence, we enquire whether it has a middle term, but when we have learned the fact and solved the question as to its absolute or partial existence, then we ask what the middle term is. My phrase ‘partial existence’ would be illustrated by the questions ‘Does the moon wax?’ or ‘Is the moon eclipsed?’ In questions of this sort we do really ask whether a thing exists or not. ‘Absolute existence’ might be illustrated by the questions ‘Does a moon, or does night, exist or not?’
Hence it follows that in every enquiry we really ask if a middle term for the subject in question is or else what this middle term is. The reason is that the middle term contains the cause, and it is the cause that we look for in all cases. For instance we ask first ‘Is the moon eclipsed?’ Then, ‘Is there any cause of the eclipse or not?’ Next, on learning that some cause of it is known we enquire what the cause is. Now the cause of a thing’s being, (not of its being this or that, but of its being absolutely) or again the cause why a thing has no absolute existence but is an essential or accidental attribute of something else, is nothing but the middle term. When speaking of absolute existence I refer to the existence of the subject, whether it be moon, earth, sun, or triangle; examples of attributes would be eclipse, equality, inequality, interposition or non-interposition of the earth.
In all these cases it is clear that the nature of the thing and its cause are the same. To the question ‘What is an eclipse?’ the answer is ‘An exclusion of light from the moon owing to the interposition of the earth.’ ‘Why does an eclipse take place, or why is the moon eclipsed?’ ‘Because the light fails when the earth excludes it.’ ‘What is harmony?’ ‘An arithmetical proportion between sharp and flat.’ ‘Why does sharp harmonize with flat?’ ‘Because they are in a certain arithmetical proportion.’ Thus the question ‘Can sharp and flat harmonize?’ is equivalent to ‘Is there an arithmetical proportion between them?’ On learning that there is we proceed to ask, ‘What then is the proportion?’ That the object of our enquiry is really the middle term is clearly displayed by those cases in which the middle term is perceptible to the senses. We make an enquiry about it only when we have not perceived it. Thus, in the case of eclipse, we ask whether there is such a thing or not. If, however, we were on the moon we should not enquire whether an eclipse does occur, nor yet why it occurs, for the answers to both these questions would become visible simultaneously. We should in fact have learned the universal as a result of sense perception. Sense perception would shew that the earth was at a particular moment excluding the sun’s light; and since it would also be obvious that the moon was then being eclipsed, knowledge of the universal would have been attained immediately. Thus, as we have said, knowing the nature of a thing is the same as knowing its cause. The former of these may either have or not have an independent existence. E.g. ‘One thing is larger, or smaller, than another.’ ‘The three angles of a triangle are equal to two right angles.’
It has now been made clear that every kind of enquiry involves a search after the middle term.
Definitions and demonstration are not identical. Denfiitions are always general and affirmative, while some syllogisms may be particular or negative. Even universal affirmative syllogisms cannot always be replaced by definitions. The principles of demonstration, which are themselves indemonstrable, may be definitions, but the two processes differ. Definition states a thing’s essence, Demonstration presupposes it.
We may now state in what ways the essential nature of a thing may be proved, and also what definition is and what are its objects; and we may first mention the difficulties connected with these subjects. We will begin with a point closely connected with the matters last treated of, namely the question which might be raised as to whether it is possible to know the same thing and know it in the same way by means of Definition and by means of Demonstration. Ought not this to be held impossible? Definition would seem to express a thing’s essential nature, which is invariably universal and affirmative. Some syllogisms however are negative, others not universal; for instance all in the second figure are negative, those in the third are other than universal. Then too definition is not invariably practicable even in the case of the affirmative syllogisms in the first figure; e.g. the proposition ‘Every triangle has its angles equal to two right angles,’ cannot be arranged as a definition. The reason of this is that knowing a thing demonstratively is equivalent to having a demonstration. Hence if such cases are capable of demonstration they clearly cannot admit of definition as well. Otherwise one would acquire knowledge by means of the definition without possessing any demonstration; for it is quite possible to have a definition without drawing any demonstration from it. An inductive proof will lead to the same conclusion. We never know anything either of the essential or accidental attributes of a thing from merely defining it. Moreover definition is a method of making known substances, while propositions like the above concerning the triangle clearly do not contain the substance of the subject. It is clear then that not everything which is capable of demonstration also admits of being defined; but then the further question arises:—When a thing is definable is it invariably capable of demonstration or not?
One argument against the possibility of this latter suggestion has already been mentioned, namely, that a single subject is, as such, treated of by a single science. Hence if demonstrative knowledge of a thing consists in having a demonstration of it we are placed in a dilemma, as one who possesses a definition without demonstration will have real knowledge.
Further, the elementary principles of demonstration are definitions, and it has been shewn before that these principles admit of no demonstration. Either then these principles must be demonstrable and also the principles of the principles, and the like process will go on to infinity; or else the primary principles will be indemonstrable definitions.
But if the objects of definition and demonstration be not entirely the same, may they not be partly the same? Or is that impossible, nothing which can be defined being capable of demonstration? Definition expresses the nature of a thing and its substance, but demonstrations all clearly assume the nature of a thing as a hypothesis, as, e.g. mathematical demonstrations assume the nature of Unit or Odd, and so with other demonstrations. Further, every demonstration proves something of a subject: e.g. that it exists or does not exist; but in a definition no one thing is predicated of another: e.g. animal is not predicated of biped nor biped of animal; nor figure of superficies; for superficies is not what figure is nor is figure what superficies is.
By this I mean, e.g. that we have already proved that an isosceles triangle has its angles equal to two right angles if we have proved that every triangle has that quality, for isosceles triangle is a part, triangle in general a whole. But a thing’s Nature and its Existence are not thus related to one another, since neither is a part of the other. It is clear then that a demonstration is not invariably attainable in cases which admit of definition, and that definition does not invariably accompany demonstration.
Hence, generally speaking, one cannot have both for any one subject. It is therefore clear that definition and demonstration cannot be identical, nor can one be part of another, for then their objects would have borne a like relation to one another.
This may be regarded as the answer to the present difficulties.
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