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, (40) i. e. axioms which are premisses of demonstration; (3) the subject-genus whose attributes, i. e. essential properties, are revealed by the demonstration. [75b] 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.8 (5) How in certain cases transference is possible I will explain later.9
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. (10) 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, (15) 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, (20) but through some property which it shares with other genera.
8 It is also clear that if the premisses from which the syllogism proceeds are commensurately universal, the conclusion of such demonstration—demonstration, 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. (25) 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. (30) 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. (35)
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. (40) 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. [76a] 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, (5) 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, (10) 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. (15)
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, (20) 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, (25) 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. (30)
10 I call the basic truths of every genus those elements 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, (35) 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, (40) 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. [76b] 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. (5) 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. (10) 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, (15) which are primary premisses of its demonstration; (3) the attributes, the meaning of which it assumes. Yet some sciences m
ay 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 same way the meaning of axioms, (20) 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,10 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,11 and though we can always raise objections to the spoken word, (25) 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, (30) 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 definitions—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. (35) 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, (40) 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. [77a] A further distinction is that all hypotheses and illegitimate postulates are either universal or particular, whereas a definition is neither.
11 So demonstration does not necessarily imply the being of Forms nor a One beside a Many, (5) 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 with 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. (10) 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, (15) 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, (20) 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.12
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, (25) I mean (as I have already explained13), 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 or 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, (30) 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.14
12 If a syllogistic question15 is equivalent to a proposition embodying one of the two sides of a contradiction, (35) 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. (40) 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. [77b] 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. (5) 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, (10) 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. (15)
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, (20) 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, (25) 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, (30) 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. (35) For since every premiss must be applicable to a number of cases (ot
herwise 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, (40) 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. [78a] 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. (5) Sometimes, no doubt, it is impossible to reason from premisses predicating mere attributes: but sometimes it is possible, though the possibility is over-looked. 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, (10) 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, (15) 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, (20) and E even number. Then A is predicable of E.
The Basic Works of Aristotle (Modern Library Classics) Page 19