It is also clear that when A inheres in B, this can be demonstrated if there is a middle term. (20) 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, (25) 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’, (30) 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, (35) 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. [85a] 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. (5) Then if it has to be proved that no C is A, a middle must be found between A 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,41 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. (10)
24 Since demonstrations may be either commensurately universal or particular,42 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. (15) 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. (20)
(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, (25) 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 form so demonstrates, particular demonstration is superior. (30)
(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, (35) 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. [85b] 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. (5) 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. (10) 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. (15) (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, (20) 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, (25) and the commensurate universal is primary;43 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. (30) 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, (35) 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.’ [86a] If rectilinear figure possesses the property for no further reason,44 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, (5) 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, (10) 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, (15) 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, (20) 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, (25) 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. (30)
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, (35) 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. [86b] 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. (5) 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 proved45 that no conclusion follows if both premisses are negative, (10) 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.46 (15) 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, (20) 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, (25) 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, (30) 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, (35) just as being is prior to not-being). It follows that the basic premiss of affirmative demonstration is superior to that of negative demonstration, and the demonstration which uses superior basic premisses is superior.
(4) Affirmative demonstration is more of the nature of a basic form of proof, because it is a sine qua non of negative demonstration.
26 [87a] 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, (5) 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 A 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. (10) 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. (15) When the falsity of the conclusion47 is the better known, we use reductio ad impossibile; 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. (20) 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. (25) 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, (30) will consequently be superior also to reductio ad impossibile.
27 The science which is knowledge at once of the fact and of the reasoned fact, not of the fact by itself without the reasoned fact, is the more exact and the prior science.
A science such as arithmetic, which is not a science of properties qua inhering in a substratum, is more exact than and prior to a science like harmonics, which is a science of properties 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, (35) while a point is substance with position; the latter contains an additional element.
28 A single science is one whose domain is a single genus, viz. all the subjects constituted out of the primary entities of the genus—i. e. the parts of this total subject—and their essential properties.
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