SOURCE: "Greek Mathematics and Greek Logic," in Ancient Logic and Its Modern Interpretations, edited by John Corcoran, D. Reidel Publishing Company, 1974, pp. 35-70.

[In the following essay, delivered as a paper in 1972 and published in 1974, Mueller examines the nature of Euclidean reasoning (as evidenced in Elements), and its relationship to Aristotle's syllogistic logic (a type of logical argument). Mueller concludes that Euclid demonstrates no awareness of syllogistic logic or of the basic concept of logic—that is, that an argument's validity depends on its form.]

Introduction

By 'logic' I mean 'the analysis of argument or proof in terms of form'. The two main examples of Greek logic are, then, Aristotle's syllogistic developed in the first twenty-two chapters of the Prior Analytics and Stoic propositional logic as reconstructed in the twentieth century. The topic I shall consider in this paper is the relation between Greek logic in this sense and Greek mathematics. I have resolved the topic into two questions: (1) To what extent do the principles of Greek logic derive from the forms of proof characteristic of Greek mathematics? and (2) To what extent do the Greek mathematicians show an awareness of Greek logic?

Before answering these questions it is necessary to clear up two preliminaries. The first is chronological. The Prior Analytics probably predates any surviving Greek mathematical text. There is, therefore, no possibility of checking Aristotle's syllogistic against the actual mathematics which he knew. On the other hand, there is no reason to suppose that the mathematics which he knew differs in any essential way, at least with respect to proof techniques, from the mathematics which has come down to us.

The major works of Greek mathematics date from the third century B.C. For determining the role of logic in Greek mathematics it seems sufficient to consider only Euclid's Elements. It is the closest thing to a foundational work in the subject. The surviving works of the other great mathematicians of the period, Archimedes and Apollonius, are more advanced and therefore more compressed in their proofs. The absence of signs of the influence of logic in them is not surprising. The evidence is too obscure to assign a date to the development of Stoic propositional logic, but I shall take as a date the floruit of its major creator, Chrysippus (280-207). Doing so means denying any influence of Stoic logic on the Elements and, tacitly, on Greek mathematics in general. I hope that the over-all plausibility of my reconstruction in this paper will provide a sufficient justification for the denial. But now I wish to discuss, as the second preliminary, a question relevant to the issue: How does one decide whether a given mathematical argument or work is influenced by a given logic?

In Elements I,19 Euclid proves that, given two unequal angles of a triangle, the side opposite the greater angle is greater than the side opposite the lesser. He proceeds as follows:¹

(1) Let ABC be a triangle having the angle ABC greater than the angle BCA; I say that the side AC is also greater than the side AB. (2) For, if not, AC is either equal to AB or less. Now AC is not equal to AB; (3) for then the angle ABC would also have been equal to the angle ACB; (4) but it is not; therefore (5) AC is not equal to AB. Neither is AC less than AB; (6) for then the angle ABC would also have been less than the angle ACB; (7) but it is not; therefore (8) AC is not less than AB. And it was proved that it is not equal either. Therefore (9) AC is greater than AB. Therefore in any triangle the greater angle is subtended by the greater side.

Q.E.D.

Much of the argument here can be analyzed in terms of Chrysippus's anapodeiktoi logoi. Thus (5) follows from (3) (an instance of a previously proved proposition, I.5) and (4)(a 'trivial consequence' of (1)) by the second anapodeiktos. And (8) is related similarly to (6) and (7). If (2) is taken as an expression of trichotomy, then (9) follows from (2), (5), and (8) by two applications of the fifth anapodeiktos.²

There are many other cases in the Elements which could be analyzed similarly. But since reasoning in accordance with the rules of a logic does not in itself imply knowledge of the logic, the possibility of analyzing a Euclidean proof in terms of Stoic propositional logic does not justify attributing to Euclid a knowledge of Stoic logic. Justification of such an attribution requires, at the very least, clear terminological parallels. However, there are none.

The paper which follows has three main sections. In the first I discuss the character of Euclidean reasoning and its relation to Aristotle's syllogistic. In the second I consider the passages in the Prior Analytics in which Aristotle refers to mathematics; my purpose here is to determine whether reflection on mathematics influenced his formulation of syllogistic. In both sections my conclusions are mainly negative. Euclid shows no awareness of syllogistic or even of the basic idea of logic, that validity of an argument depends on its form. And Aristotle's references to mathematics seem to be either supportive of general points about deductive reasoning or, when they relate specifically to syllogistic, false because based on syllogistic itself rather than on an independent analysis of mathematical proof.

In the third main section of the paper I consider the influence of mathematics on Stoic logic. As far as Chrysippean propositional logic is concerned, my conclusions are again negative. However, it is clear that at some time logicians, probably Stoic, began to consider mathematical proof on its own terms. Although they never developed what I would call a logic to cover mathematical proof, they at least realized the difference between it and the logical rules formulated in antiquity. Much of the third section is devoted to an attempt to reconstruct in outline the history of logical reflections on mathematics in the last two centuries B.C. In conclusion I recapitulate briefly my conclusions about the relation between Greek mathematics and logic.

Euclid's Elements and Logic

One still reads that Euclid's logic is Aristotelian syllogistic.³ But one need only try to carry out a single proof in the Elements by means of categorical syllogisms to see that this claim is false. If Euclid has any logic at all, it is some variant of the first order predicate calculus. In order to bring out the specific character of Euclidean reasoning, I reproduce the first proposition of the Elements together with an indication of the customary Greek divisions of a proposition.⁴

protasis: On a given finite straight line to construct an equilateral triangle.

ekthesis: Let AB be the given finite straight line….

diorismos: Thus it is required to construct an equilateral triangle on the straight line AB.

kataskeuē: With center A and distance AB let the circle BCD be described; again, with center B and distance BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined.

apodeixis: Now, since the point A is the center of the circle CDB, AC is equal to AB. Again, since the point B is the center of the circle CAE, BC is equal to BA. But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another.

sumperasma: Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. Quod erat faciendum.

In modern terms all of this proposition except the protasis and diorismos would be considered proof. But, as the terminology suggests, only the apodeixis was considered proof by the Greeks. I shall here analyze proposition I primarily in terms of Gentzen's system of natural deduction for the predicate calculus.⁵ This analysis presupposes a somewhat artificial reformulation of portions of the text. For example, the protasis is not an assertion at all and hence can not be proved in the strict sense. I shall discuss the character of the protasis briefly below. Here I shall take it as a general statement: On any straight line an equilateral triangle can be constructed.

The ekthesis is, then, a particular assumption ('AB is a straight line') from which a conclusion ('An equilateral triangle can be constructed on AB') will be derived. In the kataskeuē the drawing of the two circles and of the lines CA and CB is justified by the postulates 1 and 3:

Let it be postulated to draw a straight line from any point to any point;

and to describe a circle with any center and distance.

I know of no logic which accounts for this inference in its Euclidean formulation. One 'postulates' that a certain action is permissible and 'infers' the doing of it, i.e., does it. An obvious analogue of the procedure here is provided by the relation between rules of inference and a deduction. Rules of inference permit certain moves described in a general way, e.g., the inferring of a formula of the form A ∨ B from a formula of the form A. And in a deduction one may in fact carry out such a move, e.g., write '(P & Q) ∨ R' after writing 'P & Q'. The carrying out of a deductive step on the basis of a rule of inference is certainly not itself an inference. For neither the rule nor the step is a statement capable of truth and falsehood. And if the analogy is correct, Euclid's constructions are not inferences from his constructional postulates; they are actions done in accord with them.

There is a further correspondence between constructions and inferences which lends support to the analogy. If one wants to study inference with mathematical precision, one treats deductions as fixed objects, sequences of formulas satisfying conditions specified on the basis of the rules of inference. In other words, when inference is studied mathematically, acts of inference are dropped from consideration and replaced by objects which could have been created by a series of inferences but for which the question of creation is irrelevant; objects satisfying the conditions are simply assumed to exist. The analogy with geometry should be clear. In the modern formulation of Euclid's geometry⁶ there are no constructions of straight lines or circles. The axioms are stated in such a way as to guarantee the existence of these objects. Rather than construct the circle with center A and distance AB, the modern geometer simply derives the theorem asserting the existence of such a circle.

The analogy proposed here is easily extended to explain the character of the protasis of proposition 1. The Greeks called proposition 1 a problem, a construction to be carried out, and opposed problems to theorems, assertions to be proved.⁷ The analogy suggests that proposition I be likened to a short-cut rule of inference justified by showing that application of it is tantamount to a series of applications of the original rules. And, of course, Euclid does use the construction of an equilateral triangle on a given line directly in subsequent proofs (e.g., in I,2).

The apodeixis is on the surface very simple, very easy to understand, but logically it is fairly complex. The inferences to the equality of AC with AB and of BC with BA are based on definitions 15 and 16 of book I:

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; and the point is called the center of the circle.

It is clear Euclid is making some kind of deductive argument at the beginning of the apodeixis. But it is not at all clear that he thinks of it as a formal argument, an argument based on formal logical laws. In modern notation the definition of 'circle' may be represented as follows:

(1) x is a circle ↔ (i) x is a plane figure &
(ii) (Ely) [y is aline containing x &
(iii) (Elz) (z is a point within x &
(u) (v) (u is a straight line from z to y & v is a straight line from z to y u equals v))].

From (1) and 'CDB is a circle' one can infer the definiens of (1) with 'CDB' substituted for 'x'. Such an inference could be referred to Aristotle's syllogistic if one were willing to allow singular terms in syllogisms⁸ and to treat the complex term corresponding to the definiens as a term in a categorical proposition. But doing these two things will not suffice to recover the whole argument. As a next step we need to apply a propositional rule, &-elimination, to get

(2) (Ely)[y is a line containing CDB &
(Ezl) (z is a point within CDB &
(u (v) (u is a straingt line from z to y &
v is a straing line from z to y→ u equals v))].

Reconstructing the next piece of Euclid's argument seems to be impossible. For in proposition I Euclid makes no reference to the distinction between the circle and its circumference, a distinction which is expressed in the definition of circle. I shall pass over the difficulty here by dropping cause (ii) and identifying the circle with its circumference. As a result we hlave

(3) (Elz) (z is a point within CDB &
(u) (v) (u is a straing line from z to CDB &
v is a straight line from z to CDB→ u equals v)).

We wish to infer from (3) and 'A is the center of CDB'

(4) (A) is a point within CDB &
(u) (v) (u is a straing line from A to CDB &
v is a straight line from A to CDB→ u equals v).

Obviously the definition of 'center' is being invoked for this step, and the move is logically sound. However, the apparatus involved in justifying the step goes beyond any Greek logical theory known. Since Euclid seems to treat his geometric definitions as concrete specifications of intuitive objects rather than as abstract characterizations,⁹ he would probably not recognize that any step of inference at all is involved here.

From (4) by &-elimination we obtain that any two straight lines from A to CDB are equal. The inference from this assertion and 'AB and AC' are straight lines from A to CDB' to 'AB equals AC' is an example of the most common form of explicit inference in the Elements. The form recurs in the apodeixis of I,1 when Euclid establishes the equality of CA and CB using the first common notion, 'Things equal to the same thing are also equal to one another'. In modem notation this argument runs

(5) (u)(v)(w)(u equals w & v equals w→ u equals v);

(6) CA equals AB;

(7) CB equals AB;

(8) therefore CA equals CB;

In later antiquity this argument became the paradigm of a mathematical argument.¹⁰ The Peripatetics, intent upon defending Aristotle, claimed that the argument is really a categorical syllogism:

(A) Things equal to the same thing are also equal to one another;
CA and CB are things equal to the same thing;
therefore CA and CB are equal to each other.¹¹

What is the minor term of this 'syllogism'? Presumably 'CA and CB', i.e., the pair (CA, CB). The modern analysis, according to which the the minor premiss and the conclusion each assert that a certain relation holds between two subjects CA and CB, seems more natural than one according to which the premiss and the conclusion each assert a property of a pair taken as a single thing. But so long as the inference from (5), (6) and (7) to (8) is treated in isolation, there is no way to refute the Peripatetic analysis. Yet the context of the inference makes clear why the Peripatetics were wrong. The following represent plausible renderings of the proofs of (6) and (7) as categorical syllogisms:

(B) Straight lines from A to CDB are equal to each other;
CA and AB are straight lines from A to CDB; therefore CA and AB are equal to each other.

(C) Straight lines from B to ACE are equal to each other;
CB and AB are straight lines from B to ACE; therefore CB and AB are equal to each other.

The minor premiss of (A) is presumably to be inferred directly from the conclusions of (B) and (C). Clearly it cannot be inferred by a categorical syllogism since such a syllogism will require five terms, 'CA and AB', 'CB and AB', 'CA and CB', 'equal to each other', and 'equal to the same thing'. Thus although (A), (B), and (C) can be construed as categorical syllogisms, they cannot be combined to yield a syllogistic reconstruction of Euclid's apodeixis. For it depends on the relations among the three straight lines and not on properties of them taken as pairs.

In ancient logic the sumperasma is the conclusion inferred from the premisses of an argument. In the Elements, however, the sumperasma is not so much a result of inference as a summing up of what has been established. This summarizing character is made clearer in the case of theorems for which the sumperasma consists of the word 'therefore', followed by a repetition of the protasis, followed by 'Q.E.D.' (See the proof of I,19 quoted above.) From the modern point of view the apodeixis ends with a particular conclusion reached from particular assumptions; tacit in the sumperasma are steps of conditionalization to get rid of the assumptions and of quantifier introduction or generalization. Throughout antiquity, indeed down into the nineteenth century, the latter step was not seen as a matter of logic.¹² The inference was brought into the domain of logic only with the invention of the quantifier and the discovery of the rules governing it.

I have analyzed Elements I,1 in order to show that Euclid's tacit logic is at least the first order predicate calculus, nothing less. His logic may even be more than that, since representing his reasoning in the first order predicate calculus would seem to require reformulations foreign to the spirit of the Elements. I hope I have also sufficiently emphasized that in antiquity only the apodeixis would have been thought of as possibly subject to logical rules, and it is often a very small portion of a Euclidean proposition. I would now like to argue that Euclid does not show an awareness of one of the most basic ideas of logic, logical form. Characteristically logicians make clear the importance of form for determining the validity of an argument by obvious artificial devices. When Aristotle writes, "If A is predicated of all B and B of all C, necessarily A is predicated of all C", he uses the letters 'A,' 'B,' 'C' to indicate the truth of the assertion (or correctness of the inference), no matter what terms are put in their place. The Stoics make a similar claim when they call "If the first then the second; but the first; therefore the second" valid: any substitution of sentences for ordinal number words produces a correct inference.

Of course, artificial indications of form are not likely to occur in applications of logic, but a series of correct deductive arguments cannot be said to show a sense of logic unless it shows a sense of form. But Greek mathematics does not show this sense. In it one finds parallel proofs of separate cases which could be treated simultaneously with only slight generalization. In the Elements there are separate proofs of properties of tangent and cutting circles when only the points of contact are relevant.¹³ Better known in Euclid's separate treatment of one and the other numbers¹⁴ and of square and cube numbers when all that is relevant is one number's being multiplied by itself some number of times.¹⁵ Similar examples can be found in Archimedes and Apollonius. The usual explanation of this proliferation of cases invokes the concreteness of Greek mathematics. What is insufficiently stressed is how a sense of derivation according to logical rules, had it existed, would have undercut this concreteness. Greek geometers obviously trusted their geometric intuition much more strongly than any set of logical principles with which they may have been familiar.

The proof of I,19 presented above is logically very elementary. One has a set of alternatives all but one of which imply an absurdity, and so one infers the remaining alternative. A person with a sense of logic probably would not bother to carry out such a proof with Euclid's detail even once. But he certainly would not repeat the same proof with different subject matter several times. Euclid repeats the proof exactly in deriving I,25 from I,4 and 24, and V,10 from V,8 and 7. Another example is perhaps even more surprising. Euclid repeatedly moves from a proof of a proposition of the form (x)(Fx→Gx) to an explicit proof of (x)(-Gx→ -Fx):: assume -Ga and Fa; then, since all F are G, Ga, contradicting -Ga. I have noticed five cases in which such an argument is carried out and two others in which the stylized argument is avoided.¹⁶

One of the main themes of nineteenth-century mathematics was the demand for complete axiomatization, and one of the main charges levelled against Euclid was his failure to make explicit all of the assumptions on which his proofs relied—in particular, assumptions about continuity or betweenness.¹⁷ The absence from the Elements of first principles covering these assumptions is another indication of the intuitive character of their work, but it does not seem to me to throw light on the question whether Euclid wished to axiomatize his subject completely. I do not know what Euclid would have said if challenged to establish the existence of the point C in which the two circles of the proof of proposition I cut each other. But I do believe that he intended to make explicit in the postulates of book I all geometric assumptions to be used in book I. I stress 'in book I' because there is no reason to suppose that Euclid intended his postulates to suffice for the whole of the Elements, since they do not in fact suffice, since they are stated within book I, and since the Elements include the theory of ratios, arithmetic, and solid geometry. I stress 'geometric' because Euclid's proofs depend on other more general assumptions, some of which are stated in the common notions but most of which are not.

Discussion of the common notions is complicated by the issue of interpolation. I shall here simply state my view that only the first three are due to Euclid.¹⁸ At the end of the paper I shall suggest why the other common notions were added. In any case even the most extensive list of common notions in the manuscripts is inadequate to cover all of Euclid's inferences. I illustrate this point by reproducing in outline a segment of the apodeixis (a reductio) of I,7.

1. angle ACD equals angle ADC;
2. therefore angle ADC is greater than angle DCB;
3. therefore angle CDB is 'much' greater than angle DCB.

In this argument, (i) is properly derived from earlier assumptions, (ii) would seem to be derived from (i) plus

• (iv) angle ACD is greater than angle DCB, and the general principle
• (v) (u)(v)(w) (u equals v & v is greater than w→ u is greater than w).

(iv) may be justified by reference to the common notion numbered 8 by Heiberg,¹⁹ which asserts that the whole is greater than the part; more probably it is simply a truth made obvious by the diagram. The principle (v) is nowhere stated explicitly by Euclid, although it would seem to be neither more nor less obvious than the first common notion. Approximately the same thing can be said about the inference to (iii), which follows from (ii) plus

• (vi) angle CDB is greater than angle ADC,

and the principle of transitivity for 'greater than', again a principle equally as obvious as the first common notion. I mention these tacit principles to show that the deductive gaps in the Elements occur at a much more rudimentary level than the level of continuity or betweenness. But more important, this example, which could be buttressed with many others, seems to me to shift the burden of proof to those who claim that Euclid intended to produce a complete axiomatization of even book I.

I have so far concentrated primarily on book I of the Elements because I believe that, at least as far as logic is concerned, it is Greek mathematics par excellence and because it seems to be the main contact point between later Greek logic and mathematics. However, I would like now to consider book V of the Elements, which has been described by some scholars as (more or less) formal in the logical sense.²⁰ There is no question that the theory of proportion of book V isosceles in a way abstract; but, as I hope to make clear, the abstraction involved does not yield a theory based on logic. Rather it yields a theory only slightly less concrete than Greek geometry or arithmetic.

The theory of book V represents Eudoxus's solution to the problem of dealing mathematically with the relation of one quantity to another when the relation cannot be represented as a ratio between two integers. Aristotle apparently refers to this theory and praises it for a kind of abstraction.

Another case is the theorem about proportion, that you can take the terms alternately; this theorem used at one time to be proved separately for numbers, for lines, for solids, and for times, though it admitted of proof by one demonstration. But because there was no name comprehending all these things as one—I mean numbers, lengths, times, and solids, which differ in species from one another—they were treated separately. Now however, the proposition is proved universally; for the property did not belong to the subjects qua lines or qua numbers, but qua having a particular character which they are assumed to possess universally. (Posterior Analytics, 1.5.74a17-25, transi, by T. Heath)

Aristotle here writes as if the whole matter were terminological, as if separate proofs of the law

A is to B as C is to D→ A is to C as B is to D

were given for different kinds of objects simply because no one term covered them all. But it is generally agreed that Eudoxus did not just supply a new term, 'magnitude' (megethos), in the Elements; he provided a new foundation for the theory of proportion. This foundation survives in Definitions 5 and 7 of book V.

DEFINITION 5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third and any equimultiples whatever of the second and the fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively, taken in corresponding order.

DEFINITION 7. When of equimultiples the multiple of the first magnitude exceeds the multiple of the second but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

In modern notation:²¹

(5) (A, B)=(C, D)↔
(m)(n)[(m·A>n·B→ m·C>n·D) &
(m·A=n·B→ m·C=n·D) & (m·A<n·B→ m·C<n·D)].

(7) (A, B)>(C, D)↔(Em)(En)(m·A>n·B &-(m·C>n·D)).

In these definitions comparisons of size between ratios are reduced to comparison of size between multiples of magnitudes. To see what the definitions mean, one need only think of A, B, C, D as real numbers, (X, Y) as X/Y, 'm' and 'n' as ranging over integers, and give ' ', '>', '<', and '=' their standard meanings. Definition 5 is then equivalent to

But A/B and C/D may be thought of as arbitrary real numbers, since any real can be represented as a ratio of two reals and any such ratio represents a real. Thus, Definition 5 can be thought of as saying that two reals are equal if they make the same cut in the system of rationals—Dedekind's account of equality for reals.²² If the same interpretation is applied to Definition 7, it becomes

i.e., a first real is greater than a second if and only if there is a rational n/m separating them.

In terms of Greek mathematics one remarkable feature of Definitions 5 and 7 is that they attach relatively abstract explanations to the relatively intuitive notions of equality and inequality of ratio. And the explanations are the basis for proving some intuitively obvious facts, e.g.,

(V.7) A=B→(A, C)=(B, C);

(V.11) (A, B)=(C, D) & (E, F)=(C, D)→(A, B)=(E, F).

Intuitions concerning ratios are undoubtedly intended to play no role in the derivations of book V. However, the derivations are not purely logical. Euclid makes constant use of addition, subtraction, multiplication, and division of magnitudes—operations which are characterized nowhere in Greek mathematics. He also assumes laws governing the performance of these operations and laws governing comparisons of size.²³

The tacit assumptions in book V should probably not be attributed to intutions about magnitudes and operations on them. For Aristotle's remarks show that 'magnitude' is intended in a general sense. And there is no single intuitive notion of, say, addition for all the different kinds of objects to which the word is supposed to apply. Moreover, in other parts of Greek mathematics which are either Eudoxus's work or stem from it, the operations in question are performed on geometric objects (e.g., circles in Elements XII,2; parabolic segments in Archimedes's Quadrature of the Parabola) for which the operations could not be given a precise intuitive (i.e., constructive) sense. This deviation from the generally constructive tendency of Greek mathematics is probably not an oversight. Rather, the deviation represents the only available means of solving certain problems. So too in the theory of proportion Eudoxus deviates from the generally intuitive character of Greek mathematics, reducing the theory to generalized notions of magnitude, addition, multiplication, etc. But these notions remain informal. No attempt is made to characterize them by means of first principles. Hence the underpinning of the theory of proportion is the theory of magnitudes rather than logic.

Mathematics in the Prior Analytics

In his systematic presentation of the categorical syllogism in the first twenty-two chapters of the Prior Analytics, Aristotle never invokes mathematics. His examples are always of the 'white'-'man'-'animal' variety, and they suggest a close connection between Aristotle's logic and the somewhat mysterious dialectical activities associated with Plato's Academy.²⁴ The difficulty of fitting mathematical argument into syllogistic form may explain the absence of mathematical references in these chapters. But even in later chapters where Aristotle does invoke mathematics to support some points, a substantial majority of his considerations are either directly pointed at dialectical argument or more obviously relevant to it than to anything else. It seems clear to me that mathematics could not have played in the development of Aristotle's syllogistic anything like the role it played in the development of modern quantification theory. However, it is perhaps worthwhile to examine the mathematical references in the Prior Analytics to determine what role mathematics did play. I first describe references which have no special relevance to the categorical syllogism.

(A) I.30.46a19-22. Aristotle illustrates the empirical basis of our knowledge of the first principles of a deductive science by reference to astronomy, presumably of the kind found in Euclid's Phaenomena and Autolycus's On the Moving Sphere and On Risings and Settings.

(B) 1.31.46b26-35. Aristotle invokes the incommensurability of the side of a square with its diagonal to illustrate the impossibility of establishing an unknown fact by means of Platonic division.

(C) I.41.49b33-50a4 is a difficult passage to interpret. Aristotle compares his use of ekthesis to the geometer's calling 'this line a foot long and that line straight and breadthless when it is not.'²⁵ Apparently Aristotle is thinking of the ekthesis of a geometric proposition and pointing out that the diagram to which the geometer seems to be referring may not satisfy the description he gives and yet does not affect the correctness of his argument. Ross²⁶ points out the different ways in which Aristotle uses the word ekthesis. None of them provide a satisfactory basis for interpreting Aristotle's remark here. Yet, whatever Aristotle means, he is clearly only making an analogy between his use of ekthesis and geometric ekthesis. His point would apply equally well whatever logical principles are taken to be involved in mathematical argument.

(D) II.16.65a4-7. Aristotle illustrates 'begging the question' with a brief reference to "those who think they draw parallel lines". A satisfactory explanation of this passage would throw light on the history of mathematics but not on syllogistic. For the illustration occurs in a general description of 'begging the question' and would be compatible with any deductive logic.

(E) II. 17.65b 16-21 and 66a11-15 are equally general. In the former Aristotle gives a presumably fictitious example of a reductio ad absurdum in which the absurdity is not attributable to the hypothesis refuted, namely, an attempt to derive a Zenonian paradox from the hypothesis of the commensurability of the side of a square with its diagonal. In the second he illustrates that a falsehood may follow from more than one set of premisses by means of another mathematically fascinating example: 'Parallels meet' follows from 'The interior [angle] is greater than the external' and from 'The angles of a triangle are greater than two right angles'. Since what Aristotle says does not depend in either case on the form of derivation involved, there is no reason to connect these passages with the categorical syllogism.

The remaining references to mathematics in the Prior Analytics have a much more obvious connection with syllogistic. The first is perhaps the most important. Having run through the various figures of the various forms of syllogism, Aristotle turns in 1.23 to establishing a very general claim: every syllogism in the general sense (i.e., every deductive proof) is a syllogism in the technical sense (i.e., a categorical syllogism). He repeats this claim more than once in the Prior Analytics, and there can be no doubt that Aristotle includes mathematical proofs among syllogisms in the general sense. His first step in establishing the claim is to assert, without justification, that the conclusion of every proof is a categorical proposition.

Necessarily every proof and every syllogism proves that something belongs [to something] or does not belong, and either universally or in part. (40b23-25)

It is easy enough from our standpoint to produce counter-instances to this assertion, but from Aristotle's it is not. Consider an example he uses commonly, the proposition which Euclid states as "the three interior angles of any triangle are equal to two right angles" (Elements I,32, second part). Aristotle renders this proposition rather succinctly as 'Every triangle has two right angles'.²⁷ A more precise rendering would be 'Every triangle has its interior angles equal to two right angles'. The imprecision is indicative of Aristotle's casual attitude toward translation into categorical form. Even more significant is his casual attitude toward the analysis of categorical propositions into terms. According to him, the terms in 'Every triangle has two right angles' are 'triangle' and 'two right angles'. It seems clear, however, that the verb 'have' must be included in the predicate of the proposition, since what is predicated of every triangle in 1,32 is having two right angles, not being two right angles. Aristotle apparently considers such distinctions irrelevant as far as deduction is concerned. In Prior Analytics I.38 he considers a number of valid arguments which, according to him, differ from categorical syllogisms only because of the grammatical case of one of the terms, e.g., "If wisdom is knowledge and wisdom is of the good, the conclusion is that knowledge is of the good" and "Opportunity is not the right time because opportunity is god's, but the right time is not". For Aristotle these arguments are syllogisms with the terms 'wisdom', 'knowledge', 'good' and 'opportunity', 'right time', 'god' respectively.

We say generally about all instances that the terms are always to be set out in the nominative case, e.g., 'man' or 'good' or 'opposites', not 'of man' or 'of good' or 'of opposites', but the premisses are to be taken with the appropriate case, e.g., 'equal' with the dative, 'double' with the genitive, 'striking' or 'seeing' with the accusative, or in the nominative, e.g., 'man' or 'animal', or if the noun occurs in the premiss in some other way. (I.36.48b39-49a5)

As Lukasiewicz has pointed out, "Aristotelian logic is formal without being formalistic."²⁸ That is to say, Aristotle is throughly aware that the validity of an argument depends on its form, but he is not very strict in his determination of the form of a statement in an argument. The freedom of paraphrase which he allows himself in representing statements may well have been a major factor in his conclusion that a proof is always of a categorical statement. Certainly, given Aristotle's liberal standards, all the theorems in Euclid could be transformed into categorical statements. When Aristotle wrote the Prior Analytics probably no one was aware of the possibility of a formalistic logic. But the Stoics apparently did move toward one.²⁹ Unfortunately the idea does not seem to have spread outside Stoic circles. Alexander of Aphrodisias, commenting on Aristotle's remark that words and phrases with the same meaning may be interchanged in arguments, asserts: "The syllogism does not have its being in the words but in what they signify".³⁰ Even if one believes this assertion, one cannot deny that the insistence on strict formalization characteristic of modern logic has made clear a number of things which reliance on meaning obscures. As we shall see, later Peripatetics were able to defend Aristotle's claim of universality for the categorical syllogism because they were content with rather loose formulations of arguments.

It would be impossible to refute Aristotle's liberal attitude toward translation into categorical form, although the success of modern logic surely shows the attitude to be unfortunate. However, one might even concede that only categorical propositions are proved in mathematics without admitting the syllogistic character of mathematical proof. The analysis of Elements 1,1 was intended to show how far from the categorical syllogism Euclidean reasoning is. Aristotle, however, produces in Prior Analytics 1.23 a general argument for the universality of the categorical syllogism. The main point of the argument is the need for a middle term to establish a categorical proposition. There is no reason to examine the argument in detail, since it presupposes the universality of reasoning based on the predicational relation of terms. The important point is that no thorough investigation of mathematical proof would support Aristotle's claim.

Aristotle's own mathematical examples are consistently vague. In 1.35 he writes as though the proof that the angles of a triangle are equal to two right angles requires only the proper specification of a middle term. Almost certainly the proof he has in mind involves the drawing of a parallel line, as in the first or second diagram, and arguing that angle Β = angle B',

angle C = angle C, and angle A + angle B + angle C = two right angles. In such a proof the terms 'triangle' and 'two right angles' cannot function as categorical terms because the proof involves breaking the triangle and the two right angles into parts, and the spatial relations of the parts are crucial. Elsewhere Aristotle simply asserts that categorical syllogisms are used in the derivation of a contradiction from the assumption of the commensurability of side and diagonal (1.23.41 a21-37 and I.44.50a29-38). And, to take the most extreme case of all, he is content to describe a very elaborate attempt of Hippocrates to square the circle³¹ with the following cryptic remark:

If D is 'to be squared', E 'rectilinear', F 'circle', if there be only one middle for the [proposition] EF, the circle with lunes becoming equal to a rectilineal [figure], we should be close to knowledge. (II.25.69a30-34)

Here Aristotle apparently thinks of Hippocrates's quad rature of a circle plus a lune as the insertion of a middle term between 'rectilinear' and 'circle'. In itself this interpretation of the quadrature is dubious, but the crucial point is that no concern is shown for the details of Hippocrates's reasoning. Aristotle is contented with a vague statement of the general result.

The closest Aristotle comes in the Prior Analytics to considering a mathematical proof in detail is in 1.24 where he wishes to show that at least one premiss of a valid syllogism must be universal. This wish is somewhat strange, since a simple survey of the detailed presentation in the first twenty-two chapters would suffice to establish the point. Aristotle uses examples to make it plausible. The first is non-mathematical.

For let it be put forward that musical pleasure is worthwhile. If pleasure is assumed to be worthwhile but 'all' is not added, there won't be a syllogism. And if it is taken to be some pleasure, then, if it is a different pleasure [than musical pleasure], it does not help for the thesis, and if it is the same, the question is begged. (41b9-13)

Here Aristotle seems to lose sight completely of the notion of formal validity which is so crucial in his original presentation. He could have simply pointed out that the argument with 'some' is invalid because it is of a form already shown to be invalid, or, more directly, because there are interpretations which make the premisses true and the conclusion false. In any case, Aristotle continues:

This is made clearer in geometrical propositions, e.g., that the angles at the base of an isosceles triangle are equal. Let the straight lines A and B be drawn to the center. Then if one takes (1) the angle AC to be equal to the angle BD without assuming (Al) the angles of a semicircle to be equal in general, and again that (2) C is equal to D without adding that (A2) all angles of the segment are equal, and further that since the whole angles are equal and the subtracted angles are equal, (3) the remainders E, F are equal without assuming that (A3) if equals are subtracted from equals the results are equal, he will beg the question. (41b13-22)

Aristotle's presentation here is somewhat obscure and hardly rigorous by Euclidean standards. But the drift of the proof which he describes is clear. In the diagram, bcdef is a circle with center a. According to Aristotle,

the following argument involves petitio principii:

1. mixed angle ade = mixed angle afe;
2. mixed angle fde = mixed angle dfe;
3. therefore, rectilineal angle adf = rectilineal angle afd.

The addition of three general premisses is required to correct the reasoning:

1. The angles made by diameters and circumferences of circles are always equal.
2. The two angles made by a chord and the circumference of a circle and on the same side of the chord are equal.
3. If equals are subtracted from equals, the results are equal.

Quite clearly the proof which Aristotle has in mind here is logically very similar to the apodeixis of Elements 1,1. This proof is slightly more complicated (and less syllogistic) because there is a subtraction involved between steps (2) and (3). Exactly how Aristotle would have tried to syllogize the proof is anybody's guess. There is no evidence that he ever did try, and I suspect that he never considered the problem of reducing mathematical proof to syllogistic form in a systematic way. In the present passage he is simply using a mathematical example as inductive evidence for his claim that a valid syllogism requires a universal premiss. And perhaps Aristotle is here using the word 'syllogism' in the broader rather than the narrower sense. His failure to refer to the earlier chapters of the Prior Analytics for a clear substantiation of his claim, his inconclusive treatment of the argument yielding 'Musical enjoyment is worthwhile', and the vagueness of his discussion of the mathematical proof incline me to think so. I would be certain except for Aristotle's references to the modes and figures of the syllogism at the end of 1.24.

It looks, then, as though Aristotle did not study mathematical proof carefully or make any detailed attempt to vindicate his claims for the universality of syllogistic. A general argument based on a rather superficial analysis of mathematical theorems was sufficient for his purposes. This point of view is confirmed by the semi-mathematical arguments in other Aristotelian and pseudo-Aristotelian works. None of them show any closer relation to syllogistic than the main texts of Greek mathematics do. Further evidence is provided by Eudemus's presentation of Hippocrates's quadratures of lunes and circles plus lunes.³² Eudemus was a pupil of Aristotle with at least some interest in logic,³³ but nothing in his presentation suggests an interest in connecting mathematics with syllogistic. Alexander of Aphrodisias is too late a figure to serve as a direct indicator of Aristotle's own ideas, but the surviving parts of his commentaries on the Organon are our best source of information on what became of those ideas among the later Peripatetics. Alexander makes clear in many passages that, for him, the doctrine of the universality of the categorical syllogism has the status of a dogma. In one such passage he discusses Aristotle's claim that the derivation of a contradiction from the assumption of the commensurability of the side of a square with its diagonal is syllogistic.³⁴ Alexander reproduces a protracted but essentially correct derivation that is no more syllogistic in style than any proof in the Elements. He simply asserts that the derivation is syllogistic. For him any interesting conclusive argu ment must be a categorical syllogism.

Thus far I have argued as if Aristotle acknowledged no form of conclusive argument other than the categorical syllogism. In fact he does acknowledge a general class of non-syllogistic argument which he calls argument from a hypothesis.³⁵ An especially important member of the class is the reductio ad absurdum. However, Aristotle always treats the general class and its most important member separately, and I shall follow him in my discussion. Argument from a hypothesis is for Aristotle basically modus ponendo ponens. Wishing to prove Q, one adds P → Q as a hypothesis and proves P. Aristotle represents argument from a hypothesis as a form of dialectical reasoning. The hypothesis P → Q is a matter of agreement between two opponents. The opponent who denies P but concedes P → Q is declaring a proof of Q unnecessary once a proof of P has been found; he is not providing a premiss which might be used in a proof of Q. Thus Aristotle does not conceive of modus ponens as a rule of logical inference. As far as he is concerned, the proof in an argument from a hypothesis is the proof of P. Since he assumes that P will be categorical, he assumes that the proof of P will be a series of categorical syllogisms. Lukasiewicz argued that Aristotle was oblivious to the use of rules of propositional logic in his own development of syllogistic.³⁶ His obliviousness to their use in mathematics seems at least as clear.

On the other hand, reductio arguments are an obvious feature of mathematics. And Aristotle's standard example of a reductio proof is the indirect derivation of the incommensurability of the side of a square and its diagonal. Aristotle's analysis of reductio is obviously intended to be like his analysis of argument from a hypothesis, but the details of the analysis of reductio are less clear. Prima facie, one would expect the hypothesis of a reductio to be the assumption refuted; but, if it is, the analogy with argument from a hypothesis breaks down. Unfortunately Aristotle contents himself with saying that the hypothesis in a reductio is not agreed to in advance "because the falsehood is obvious" (I.44.50a35-38). The obvious falsehood would seem to be the contradiction derived from the assumption refuted. In saying that no advance agreement is made, Aristotle is apparently again envisaging a dialectical situation: one person claims P; the other derives a contradiction from P; the falsehood is so blatant that no explicit agreement is needed to get the first person to abandon P. One might then consider the hypothesis of a reductio to be the law of propositional logic '(P → (Q & -Q))→ -P', but there is no evidence that Aristotle even tried to reformulate it. For him the crucial points are (1) the reductio part of an indirect proof is syllogistic, and (2) the nonsyllogistic part is a matter of tacit agreement rather than logic.

However, reductio is a part of mathematics and is recognized as such by Aristotle. Was he then forced to recognize a non-syllogistic feature of mathematics? Apparently not, for Aristotle also realized that "everything which can be inferred directly (deiktikos) can be inferred by reductio and vice versa, and by the same terms" (II.14.62b38-40). In other words, A & B) → C is a valid categorical syllogism if and only if (A & -C) → -B is (with negated statements properly formulated). Thus any theory whose logic is syllogistic has no need of reductio proof. It is unfortunate that no one ever tried to illustrate this truth about the categorical syllogism by recasting indirect proofs from mathematics into direct ones. An attempt to do so would have made the limitations to the categorical syllogism obvious.

Aristotle seems, then, to have had a largely a priori conception of the relation between his logic and mathematical proof. He may have taken the formulation of mathematical theorems into account in trying to justify his estimation of the significance of the categorical proposition in demonstrative science, but his notion of the categorical proposition was so broad that virtually any general statement would satisfy it. On the other hand, Aristotle does not seem to have looked at mathematical proof in any detail, at least as far as its logic is concerned. He recognizes some common features of mathematical proof, e.g., the use of reductio ad absurdum and the reliance on universal assumptions, but he is apparently content to rely on the abstract argument of 1.23 to establish the adequacy of syllogistic for mathematics. His Peripatetic successors do not seem to have gone much beyond him either in logic or in the logical analysis of mathematical proof.

Stoic Logic and Greek Mathematics

Some of the Stoics do seem to have shown an awareness of the complexity of mathematical proof. Unfortunately the scatteredness and scantiness of the evidence makes it difficult to determine the details of Stoic logical theory and, in particular, to assign a chronology to its development. Recent interpreters of Stoic logic have disagreed sharply with their predecessors on questions of analysis and evaluation, but both have forsworn the attempt to provide a chronology. And certainly there is little hope of reconstructing a precise and detailed chronology, since probably the majority of sources describe only "what the Stoics (or dogmatists or recent philosophers) say" about some question. On the other hand, some sources attribute particular doctrines to particular people. The material quoted by Diogenes Laertius from Diodes Magnes is especially rich in these attributions, and they are almost certainly reliable. Of course, when a doctrine is assigned to a person we cannot be sure that he was the first person to espouse it, but it seems to me we should assume he was in the absence of other negative evidence or of countervailing systematic considerations. Almost equal strength, I think, should be assigned to associations of doctrines with students or followers of a person, usually referred to as "those about" (hoi peri) him. Normally there are no grounds for distinguishing the views of "those about a person" from the views of the person himself.

What I have said so far about scholarly methodology is uncontroversial. The crucial issue arises with respect to ascriptions to "the more recent philosophers" (hoi neōteroi). The more recent philosophers are almost always Stoics, but it is difficult to determine the chronological boundary between more recent philosophers and others. In some authors the neōteroi seem to be Stoics in general or at least to include Chrysippus. Iamblichus³⁷ speaks generally of the original philosophers and more recent ones and goes on to discuss the views of Plato, Aristotle, and Chrysippus. Galen associates with the more recent philosophers two terms (diezeugmenon axiōma, sunēmmenon axiōma³⁸) which are certainly Chrysippean, as Galen himself says elsewhere in the case of one of them.³⁹ However, the important source to be evaluated is Alexander of Aphrodisias, who uses the phrase hoi neōteroi more often than anyone else. As far as I have been able to determine, the following characterization holds for his usage. On occasion Alexander does contrast the neōteroi with the older Peripatetics (rather than the older Stoics).⁴⁰ He also sometimes uses the word neōteroi interchangeably with 'Stoics'⁴¹ and sometimes associates with neōteroi doctrines or practices common in the Stoic school.⁴² But he never ascribes to the neōteroi terminology or doctrine elsewhere attributed explicitly to Chrysippus. And in some cases terminology or doctrine associated with the neōteroi by Alexander can be determined with reasonable plausibility to be post-Chrysippean.

The most certain case is the idea of the argument with one premiss, e.g., 'You breathe; therefore you are alive',⁴³ which Sextus Empiricus explicitly dissociates from Chrysippus and attributes to Antipater (flor. 2nd cent. B.C.).⁴⁴ Another almost equally certain case is the use of the word proslambamenon or proslēpsis⁴⁵ for the 'minor premiss' of a hypothetical syllogism. At least Diodes Magnes ascribes to those about Crinis, a contemporary of Antipater, the description of an argument as consisting of lēmma, proslēpsis, and epiphora.⁴⁶ In his commentary on the Topics Alexander says that the neoteroi call a certain kind of question a pusma, a word used for questions requiring more than a 'yes' or 'no' answer.⁴⁷ There is some reason to regard this word as post-Chrysippean, since from the book titles in Diogenes Laertius it appears that Chrysippus used the word peusis with the same meaning.⁴⁸ The ground, however, is not very firm because peusis and pusma seem to have been used interchangeably in later antiquity.

In the matter of arguments, what can be attached most firmly to Chrysippus are the five anapodeiktoi.⁴⁹ None of the obscure four themata are ever ascribed explicitly to him, nor does the word thema occur in the list of his works given by Diogenes Laertius. Alexander attributes a second and a third thema to the neōteroi.⁵⁰ Perhaps Chrysippus did put forward some themata for reducing arguments to his five anapodeiktoi. But Alexander's ascription of the second and third themata to the neōteroi, combined with the absence of any clear presentation of the themata in surviving discussions of Stoic logic, suggests at least that the themata never became fixed in the way in which the anapodeiktoi more or less were. The other arguments which Alexander attributes to the neōteroi are, according to him, useless. They are the diphoroumenoi (e.g., 'If it is day, it is day; but it is day; therefore it is day'), the adiaphoros perainontes ('Either it is day or it is night; but it is day; therefore it is day'), the so-called infinite matter,⁵¹ arguments semantically but not formally equivalent to categorical syllogisms and called hyposyllogisms,⁵² and correct arguments which are not formally valid—called amethodōs perainontes, unsystematically conclusive ('The first is greater than the second; the second is greater than the third; therefore the first is greater than the third').⁵³ None of these arguments is ever associated with a specific person. To dissociate them from Chrysippus there is only Alexander's apparently consistent use of the word neōteroi and the absence of any titles containing the words diphoroumenoi, adiaphorōs perainontes, 'infinite matter', 'hyposyllogism,' or 'unsystematically conclusive' in Diogenes Laertius's long list of the works of Chrysippus. If the arguments are dissociated from Chrysippus, a rather clear picture of one aspect of the development of Stoic logic emerges. In the mid-third century B.C. Chrysippus developed or codified the propositional logic which became the core of Stoic logic. After him, in the period of transition from the old to the middle Stoa, other Stoics introduced into consideration certain curious propositional arguments and other apparently valid arguments not satisfying either Stoic or Peripatetic accounts of validity.

With this rough chronological framework it is possible to investigate the relation between Stoic logic and Greek mathematics somewhat more precisely. I shall consider propositional logic first. I have already given an example of a propositional argument in the Elements. Familiarity with modern logic makes it easy to find many more, both explicit and implicit. However, the evidence indicates rather strongly that no Stoic ever conceived of propositional logic as a basic tool of mathematics. Mathematical illustrations of propositional arguments are practically non-existent. There are none in Sextus Empiricus or Diogenes Laertius or Alexander, for example. Indeed, the only extended illustrations are given in the sixth century A.D. by John Philoponus in his discussion of Aristotle's treatment of argument from a hypothesis.⁵⁴ The most interesting part of the discussion for my purposes is Philoponus's claim that reductio ad absurdum involves application of two Stoic anapodeiktoi, the second and the fifth. He illustrates his claim in terms of Aristotle's example, the proof that the side and the diagonal of a square are incommensurable.

Fifth anapodeiktos:

1. The diagonal is either commensurate or incommensurate with the side;
2. But it is not commensurate (as I will show);
3. Therefore it is incommensurate.

Second anapodeiktos:

• (4) If the diagonal is commensurate with the side, the same number will be even and odd;
• (5) But the same number is not even and odd;
• (6) Therefore the diagonal is not commensurate with the side.

Philoponus presumably thinks of (1) and (5) as immediate truths, and, like Aristotle and Alexander, he insists that (4) requires a proof by categorical syllogism. Thus, although Philoponus grants Stoic propositional logic more status than Alexander does, he still maintains the false Peripatetic view of the dominance of the categorical syllogism.

It is, of course, possible that the propositional part of Philoponus's analysis ultimately derives from an early Stoic source. But such a derivation seems unlikely. For Philoponus does not formulate arguments in the Stoic manner. He does not place the word 'not' at the front of the sentence in (2), (5), and (6); he does not formulate (1) as a disjunction but as a simple sentence with a disjunctive predicate; and he formulates (4) artificially, perhaps to make it seem more categorical. (Literally (4) runs: The diameter with the side, if it is commensurate, the same number will be even and odd.') There are similar features of Philoponus's whole discussion of hypothetical syllogisms which indicate that its origin is in later eclectic thinking. However, the exact origin is not known to me. I have traced it back as far as Proclus who, in commenting on proposition 6 of book I of the Elements, refers to the role of the second anapodeiktos in indirect proofs:

In reductions to impossibility the construction corresponds to the second of the hypotheticals. For example, if in triangles having equal angles the sides subtending the equal angles are not equal, the whole is equal to the part; but this is impossible; therefore, in triangles having equal angles the sides subtending the equal angles are themselves equal.⁵⁵

Proclus, of course, taught Ammonius on whose lectures Philoponus's commentary on the Prior Analytics seems to have been based.

Thus, Chrysippean propositional logic would not seem to have been developed out of reflection on mathematics. Any connection between Stoic logic and Greek mathematics must be sought in the later refinements already mentioned. And among these there is one obvious candidate for consideration, the unsystematically conclusive argument. The example given above is clearly mathematical. So is another, also due to Alexander, the inference to the equality of CA and CB in Elements I, 1.⁵⁶ But the following fairly common example shows that the domain of unsystematically conclusive argument extends beyond mathematics: 'It is day; but you say that it is day; therefore you speak the truth'⁵⁷

The first question I wish to consider is how the conception of these arguments arose. After discussing categorical and hypothetical syllogisms, Galen introduces in chapter xvi of his Institutio Logica a third form of syllogism, namely, the relational (kata to pros ti genesthai). He gives examples analogous to the unsystematically conclusive arguments above and mentions the frequency of relational syllogisms in mathematics. Galen apparently takes credit for the name 'relational' and for recognizing that relational syllogisms depend for their validity on some axiom, by which he means a self-evident proposition. There is no reason to deny Galen's origination of the term 'relational', since it is used in this way only in the Institutio. However, it is important not to read too many modern connotations into the term. For there is no evidence that Galen made any attempt to explain the validity of a relational syllogism by reference to what are now called the logical properties of a relation, such as transitivity or asymmetry, or to classify relations in terms of such properties. Indeed, there is no general treatment of relations at all. Each relational argument is to be examined in isolation to determine if there is an axiom which makes it valid.⁵⁸ Moreover, many of Galen's examples do not depend on logical properties of relations but on mathematical or semantic truths, e.g., '(a=2b & b=2c)→ a=4c' or "'son' is the converse of 'father'". It seems fair to say that Galen calls the arguments he is considering relational because they contain a relation word. He does not conceive of the idea of a logic of relations. And his account of the validity of relational syllogisms as deriving from an axiom would apply to any argument turning on the meaning of some of its terms, even if the terms were not relation words.

In the last chapter of the Institutio Galen dismisses from consideration several kinds of argument as being redundant in his presentation of logic. One is "called unsystematic, with which one must syllogize when there is no systematic argument at all".⁵⁹ There is no reason to doubt that Galen is referring to unsystematically conclusive arguments and classing them with his own relational arguments. It is not clear, however, in what way relational syllogisms constitute a broader class than unsystematically conclusive arguments. Perhaps all Galen did was to produce a few new examples of such arguments and provide a new label for them. A more important question concerns Galen's claim to originality in his account of the validity of relational syllogisms. At the end of his discussion of the relational syllogism⁶⁰ he admits that the Stoic Posidonius (ca. 135-ca. 50 B.C.) called such arguments "valid on the strength of an axiom". It looks, then, as though the fundamental idea of Galen's account was put forward more than two centuries before him.

Moreover, it looks as though the Peripatetics held the same view of unsystematically conclusive arguments as Galen, but in a more specific form. Galen criticizes the Aristotelians for trying by force to count relational syllogisms as categorical.⁶¹ The subsequent discussion in the Institutio, supplemented with Alexander's logical commentaries, makes it virtually certain that the Peripatetic way of treating Galen's relational syllogisms was to add a universal premiss corresponding to Galen's axiom and to reformulate the argument as a 'categorical syllogism'. To give one example, Alexander transforms the unsystematically conclusive argument 'A is greater than B; B is greater than C; therefore A is greater than C' into

Everything greater than a greater is greater than what is less than the latter;

A is greater than B which is greater than C;

Therefore A is greater than C.⁶²

Galen's criticism of such transformations as forced is mild, to say the least. The transformations make no logical sense. Alexander makes them only because he is bent on defending Aristotle's general claims about the universality of the categorical syllogism.

Galen's own attitude toward the added axiom and the resulting argument is harder to figure out. After criticizing the Aristotelians for forcing relational syllogisms into an arbitrary mold, he goes on to propose reducing the syllogisms to categorical form.⁶³ But shortly thereafter, he considers a 'reduction' apparently as ridiculous as the one just given and clearly prefers a 'reduction' to a propositional argument by adding a conditional premiss.⁶⁴ Galen is so antiformalistic that it is impossible to tell how serious he is about reduction to categorical form. His main stress is on the tacit assumption in relational syllogisms of an axiom, which he usually describes as universal. But one cannot tell whether for him the result of adding the axiom is always a categorical syllogism, always either a categorical or hypothetical syllogism, or sometimes neither. I am inclined to accept the last alternative, but with inconclusive reasons. Galen introduces the relational syllogism as a third form or species of syllogism, and if he believed it was really an enthymemic form of the first two, he could easily have said so. Alexander accepts the first alternative for unsystematically conclusive arguments and is very explicit about it. In any case, Galen and probably every other logician in antiquity showed no interest in developing a special logic to account for relational arguments.

In describing what Galen calls relational syllogisms as valid on the strength of an axiom, Posidonius was probably offering an explanation of the conclusiveness of unsystematically conclusive arguments, which, as Galen's description of unsystematic arguments suggests, were regarded as simply unsystematic—i.e., incapable of analysis. It is uncertain when the amethodōs perainontes arguments were first introduced, or in what connection. But the evidence I have given suggests dating their introduction in the second century B.C., that is, between Chrysippus and Posidonius. Probably the connection between these arguments and mathematical proof was not at first recognized or, at least, emphasized. I have already pointed out that some of the acknowledged unsystematically conclusive arguments were not mathematical. However, even the mathematical argument 'A is equal to B; C is equal to B; therefore A is equal to C' cannot have originally been considered in a context like Elements I, 1. For there the role of the axiom (common notion) 'Things equal to the same thing are equal to each other' is clear. But apparently amethodōs perainontes arguments were thought of as containing no general premisses of this kind.⁶⁵ Perhaps, then, Posidonius used mathematical examples like the proof of I, 1 to explain the unsystematically conclusive arguments as valid on the strength of an axiom.⁶⁶ Subsequently the Peripatetics claimed that the axiom was always a universal statement which, when added to the argument, turned it into a categorical syllogism.

Posidonius's use of the word 'axiom' (axioma) is curious. For the Stoics any any proposition is an axiom.⁶⁷ Galen's use of 'axiom' to mean 'self-evident proposition' is derived ultimately from Aristotle.⁶⁸ Posidonius could, of course, have been using 'axiom' in the standard Stoic sense. He could have been pointing out the possibility of turning any amethodōs perainōn argument into a valid propositional argument by adding as an additional premiss the so-called corresponding conditional: the conditional with the conjunction of the original premisses as antecedent and the conclusion as consequent. But to suppose he did this is to accuse Galen of misrepresentation or misunderstanding. Moreover, Posidonius is known to have been a philosophical eclectic. There is no great surprise in his using a Stoic word with a Peripatetic sense.

Proclus's commentary on book I of the Elements contains enough references to Posidonius and to his pupil Geminus to confirm Posidonius's interest in the fundamentals of Greek mathematics. Particularly interesting in connection with logic are Proclus's references to Posidonius's replies to an attack on geometry by the Epicurean Zeno of Sidon. Zeno's motivation was probably destructive skepticism,⁶⁹ although Vlastos has tried to represent Zeno as a 'not unfriendly' and 'constructive' critic of Euclid's Elements.⁷⁰ I shall not pursue the question of motivation here because the crucial thing for my purposes is the form of Zeno's criticism. He is classed by Proclus as one who concedes the truth of geometric first principles but insists on the need for further assumptions in order to complete the proofs.⁷¹

According to Proclus, Posidonius wrote a 'whole book' refuting Zeno's attack on geometry.⁷² Unfortunately Proclus refers to this controversy in an explicit way only in connection with Elements I, 1.⁷³ He reproduces only two replies by Posidonius to Zeno. In both Posidonius denies the need for the additional assumption which Zeno claims is required. Nevertheless it seems quite possible that the controversy with Zeno is the source of Posidonius's account of relational arguments as valid on the strength of an axiom. The evidence which I have given for this possibility is sparse and basically circumstantial. To this evidence I would like to add one more consideration. In discussing Elements I, 10, the bisection of a straight line,⁷⁴ Proclus refers to 'some' who say that "this appears to be an agreed principle in geometry, that a magnitude consists of parts infinitely divisible". In reply Proclus invokes Geminus's statement that the geometers do assume, "in accordance with a common notion", that the continuous is divisible. Later Proclus refers to this assumption as an axiom. Crōnert has identified Zeno with the 'some' referred to by Proclus.⁷⁵ Perhaps Crönert is right, but in any case replies like the one ascribed to Geminus in the passage under consideration would have to be attributed to Posidonius if a connection is to be made between his controversy with Zeno and his analysis of Galen's relational arguments. The hypothesis I propose is the following: Posidonius may have been unable to fill some of Zeno's alleged gaps in mathematical proofs and may have noticed the correspondence between Stoic unsystematically conclusive arguments and the proofs with gaps. Obviously it is no reply to a critic to call a proof unsystematically conclusive. Nor will it do to invoke the corresponding conditional, since establishing that is tantamount to establishing the correctness of the conclusion directly.⁷⁶ Hence Posidonius may have invoked self-evident principles—axioms—to fill the gaps he could not analyze away. And he may have described the proofs with gaps, and unsystematically conclusive arguments in general, as valid on the strength of an axiom.

After the composition of the Elements the common notions or axioms were a matter of great controversy, which centered on the need or lack of need for more axioms than the first three.⁷⁷ The result of this controversy was the incorporation of a total of ten axioms into the main texts of the Elements. The additions are undoubtedly due to a desire to fill alleged gaps in Euclid's argumentation. The date of the inception of this controversy is uncertain. I would like to suggest that it begins with the skeptical attack of Zeno and the more positive reply of Posidonius. The earliest person mentioned by Proclus in connection with the controversy is Heron, who attempted to limit the axioms to three, apparently the first three.⁷⁸ It would seem that by Heron's time the list of axioms had already been expanded. Unfortunately Heron's dates are uncertain; scholars have placed him everywhere between 200 B.C. and 300 A.D. Neugebauer's dating of Heron's floruit in the first century A.D.⁷⁹ seems now to have won general acceptance. We do not know who added to the Elements the common notions rejected by Heron. Proclus never mentions Posidonius in connection with the axioms and postulates but does mention Geminus, who wrote extensively on mathematics,⁸⁰ several times. Geminus seems to be a plausible but by no means certain candidate.

Recapitulation

(1) Aristotle's formulation of syllogistic in the fourth century is basically independent of Greek mathematics. There is no evidence that he or his Peripatetic successors did careful study of mathematical proof.

(2) Similarly, the codification of elementary mathematics by Euclid and the rich development of Greek mathematics in the third century are independent of logical theory.

(3) Likewise, Stoic propositional logic, investigated most thoroughly by Chrysippus in the third century, shows no real connection with mathematical proof.

(4) Subsequent to Chrysippus, hoi neōteroi considered various new forms of argument, including the unsystematically conclusive. Some of these new forms of argument may have come from mathematics. However, as the name 'unsystematically conclusive' suggests, no attempt was made to provide a logic for these arguments.

(5) Around the end of the second century B.C. Zeno of Sidon (and perhaps other skeptics and Epicureans) tried to undermine mathematics by pointing out gaps in proofs. Posidonius replied to Zeno, in many cases denying the existence of the gaps. But Posidonius also recognized that some geometric arguments, which resemble unsystematically conclusive arguments, depended on unstated principles. He considered the unstated principles self-evident and therefore called the arguments valid on the strength of an axiom. However, he made no progress in developing a logic to apply to these arguments. The debate over the need for further axioms in geometry continued for centuries and affected the text of the Elements itself.

(6) The reawakening of interest in Aristotle's works in the first century B.C.⁸¹ produced a Peripatetic reaction to Posidonius's analysis of ordinary mathematical argument. Aristotle's general remarks about the universality of the categorical syllogism became a dogma to be defended at all costs. Unsystematically conclusive arguments were made systematic by adding a universal premiss and attempting to transform the result into a categorical syllogism. The attempt was uniformly a failure.

(7) In Galen's Institutio Logica there is a more balanced view of unsystematically conclusive arguments, which Galen calls relational. Relational arguments depend for their validity on an additional axiom which is usually universal and usually categorical, but relational syllogisms are distinct from both categorical and hypothetical syllogisms. However, there is no evidence that Galen made any attempt to formulate a logic of relational syllogisms.

Notes

¹ The translations of the Elements are by T. Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, England, 1925.

² John Philoponus, In Aristotelis Analytica Priora Commentaria (ed. by M. Wallies), Berlin, 1905, 246.3-4, gives a similar illustration of the fifth anapodeiktos: 'The side is either equal to or greater than or less than the side; but it is neither greater nor less; therefore it is equal'. For details on the anapodeiktoi and other aspects of Stoic logic, see B. Mates, Stoic Logic, Berkeley 1961.

³ For example, one reads in Studies in History and Philosophy of Science 1 (1970), 372: "And what of Greek geometry? What are its characteristics? It employs no symbols, for it is concerned not with structures formed by relations between mathematical objects, but with the objects themselves and their essential properties. It is not operational, but contemplative; its logic is the predicate logic of Aristotle's Organon". A footnote adds: "Indeed, the Organon includes, with one or two rare exceptions, no elements of relational logic".

⁴ These divisions and their names are taken from Proclus, In Primum Euclidis Elementorum Librum Commentarii (ed. by G. Friedlein), Leipzig, 1873, 203.1-210.16. The rigidity which they suggest is fully confirmed by Euclid's Elements; and the terms themselves, or forms of them, can all be found in third-century mathematical works. For references, see C. Mugler, Dictionnaire Historique de la terminologie géométrique des grecs, Paris 1958.

⁵ See 'Investigations into Logical Deduction', in The Collected Papers of Gerhard Gentzen (ed. by M. E. Szabo), Amsterdam 1969, pp. 68-81.

⁶ As in D. Hubert, Foundations of Geometry (transi, by L. Unger), La Salle, 111., 10th ed., 1971.

⁷ Proclus, In Primum Elementorum, 77.7-81.22.

⁸ J. Łukasiewicz, Aristotle's Syllogistic, Oxford, 2nd ed., 1957, p. 1, asserts that Aristotle does not allow singular terms in syllogisms. If Łukasiewicz is right, then no Euclidean argument would be an Aristotelian syllogism.

⁹ See, for example, H. Zeuthen, Geschichte der Mathematik im Altertum und Mittelalter, Copenhagen 1896, p. 117.

¹⁰ At the beginning of Galen's Institutio Logica (ed. by C. Kalbfleisch), Leipzig 1896, 1.2, the reader is introduced to the idea of proof by means of the following example: 'Theon is equal to Dion; Philon is equal to Dion; things equal to the same thing are also equal to one another; therefore Theon is equal to Philon'.

¹¹ See Alexander of Aphrodisias, In Aristotelis Analyticorum Priorum Librum I Commentarium (ed. by M. Wallies), Berlin 1883, 344.13-20.

¹² For inadequate attempts to explain the move, see Proclus, In Primum Elementorum, 49.4-57.8; and J. S. Mill, A System of Logic, London, 9th ed., 1875, III.ii.2.

¹³ E.g., in III, 5, 6.

¹⁴ E.g., in VII, 9, 15.

¹⁵ E.g., in VIII, 11, 12.

¹⁶ The cases with the stylized proof are VIII, 16, 17 and X, 7, 8, 9. The cases where the possibility of the stylized proof is apparently overlooked are X, 16, 18. Each of these examples except X, 7, 8 actually contains two instances of failure to recognize the elementary logical equivalence.

¹⁷ See, for example, F. Klein, Elementary Mathematics from an Advanced Standpoint (transi, by E. R. Hedrick and C. A. Noble), New York 1939, II, pp. 196-202.

¹⁸ Heath gives five common notions in his translation of the Elements, but in discussing the fourth and fifth (I, 225, 232) he admits that they are probably interpolations.

¹⁹ In the standard edition of the Elements (Leipzig 1883), I, 10, now reissued under the direction of E. S. Stamatis (Leipzig 1969).

²⁰ H. Hasse and H. Scholz, 'Die Grundlagenkrisis in der griechischen Mathematik', Kant-Studien XXXIII (1928), 17, call it a first attempt at an axiomatization in the modern sense.

²¹ The notation '(X,Y)' for 'the ratio of X to Y' is taken from E. J. Dijksterhuis. See his Archimedes (transi, by C. Dikshoorn), Copenhagen 1956, p. 51. The symbols '·', '<', '>', and '=' do not have their usual numerical sense, since '.' designates an operation on magnitudes, and the other three symbols designate relations of size holding between either magnitudes or ratios of magnitudes to one another.

²² See 'Continuity and Irrational Numbers' in R. Dede-kind, Essays on the Theory of Numbers (transi, by W. Beman), Chicago 1924, pp. 15-17.

²³ F. Beckmann, in 'Neue Gesichtspunkte zum 5. Buch Euklids', Archive for History of Exact Sciences IV (1967/8), 106-107, lists twenty-four 'tacit assumptions' of Book V.

²⁴ Łukasiewicz, Aristotle's Syllogistic, p. 6, denies Platonic influence. But see W. and M. Kneale, The Development of Logic, Oxford 1962, pp. 44, 67-68.

²⁵ Translation by T. Heath, Mathematics in Aristotle, Oxford 1949, p. 26. This is an excellent work to consult for details of the mathematical aspects of the passages I am discussing.

²⁶ W. D. Ross, Aristotle's Prior and Posterior Analytics, Oxford 1949, pp. 412-414.

²⁷ See, for example, Prior Analytics, II.21.67a12-16, or I.25.48a33-37.

²⁸Aristotle's Syllogistic, p. 15.

²⁹ See, for example, Alexander, In Analyticorum Priorum, 373.29-31, or Galen, Institutio, IV.6.

³⁰In Analyticorum Priorum, 372.29-30.

³¹ For the details, see Simplicius, In Aristotelis Physicorum Libros Quattuor Priores Commentaria (ed. by H. Diels), Berlin 1882, 60.22-68.32.

³² See the passage cited in n. 31.

³³ The evidence is collected in F. Wehrli, Die Schule des Aristoteles, Basel/Stuttgart, 2nd ed., 1969, VIII, 11-20.

³⁴In Analyticorum Priorum, 260.9-261.28.

³⁵ Aristotle discusses argument from a hypothesis briefly in I.23.41a22-41b5 and in somewhat more detail in 1.44.

³⁶Aristotle's Syllogistic, pp. 49, 74.

³⁷ Quoted by Simplicius, In Aristotelis Categorias Commentarium (ed. by C. Kalbfleisch), Berlin 1907, 394.13-395.31.

³⁸Institutio, III.3, 4.

³⁹Institutio, V.5 (said of those about Chrysippus). Diogenes Laertius, Vitae Philosophorum (ed. by H. S. Long), Oxford 1964, VII. 190, lists among Chrysippus's works 'On a true diezeugmenon' and 'On a true sunēmmenon'.

⁴⁰ See In Analyticorum Priorum, 262.28-32.

⁴¹ See, for example, In Analyticorum Priorum, 21.30-31; 22.18.

⁴² Following the words, not their meanings (In Analyticorum Priorum, 373.29-30); espousing the hypothetical syllogism (ibid., 262.28-29); using the words adiaphora and protēgmena (In Aristotelis Topicorum Libros Octo Commentarium (ed. by M. Wallies), Berlin 1891, 211.9-10).

⁴³ Attributed to the neōteroi by Alexander, In Analyticorum Priorum, 17.11-12.

⁴⁴ Sextus Empiricus, Adversus Mathematicos, VIII.443, in Opera II (ed. by H. Mutschmann and J. Mau), Leipzig 1914. Alexander associates the one-premissed arguments with 'those about Antipater' (In Topicorum, 8.16-19). Other relevant passages are collected in C. Prantl, Geschichte der Logik im Abendlande, Leipzig 1855, I, 477-478.

⁴⁵ Alexander, In Analyticorum Priorum, 19.4-6, 262.9, 263.31-32, 324.17-18.

⁴⁶ Diogenes Laertius, Vitae Philosophorum, VII.76.

⁴⁷In Topicorum, 539.18.

⁴⁸Vitae Philosophorum, VII. 191.

⁴⁹Vitae Philosophorum, VII.79-81; Sextus Empiricus, Adversus Mathematicos, VIII.223-226; Galen, Institutio, VI.6.

⁵⁰In Analyticorum Priorum, 164.30-31, 278.6-14. At 284.13-17 Alexander ascribes to hoi apo tēs Stoas the second, third, and fourth themata. On the themata, see O. Becker, Über die vier Themata des stoischen Logik, in Zwei Untersuchungen zur antiken Logik, Klassischphilologische Studien XVII (1957), 27-49.

⁵¹ Explicit attribution of these three arguments to the neōteroi is found at In Anaylticorum Priorum, 164.28-30. The examples of the first two are taken from Alexander's commentary on the Topics, 10.8-12. Nothing is known about the infinite matter argument. For a guess as to its character, see O. Becker, Über die vier Themata, 38.

⁵²In Analyticorum Priorum, 84.12-15.

⁵³ Ascribed by Alexander to the neōteroi (In Analyticorum Priorum, 22.18; 345.13), but elsewhere simply to the Stoics. However, these arguments are discussed in close conjunction with the one-premissed arguments at 21.10-23.2, the source of the example in the text. This example and others are discussed below, p. 27ff.

⁵⁴In Analytica Priora, 245.24-246.32.

⁵⁵ In Primum Elementorum, 256.1-8.

⁵⁶In Analyticorum Priorum, 22.3-7.

⁵⁷In Analyticorum Priorum, 22.17-19.

⁵⁸Institutio, XVII.7. Alexander takes the same approach (In Analyticorum Priorum, 344.9-345.12).

⁵⁹Institutio, XIX.6.

⁶⁰Institutio, XVIII.8.

⁶¹Institutio, XVI. 1.

⁶²In Analyticorum Priorum, 344.23-27.

⁶³Institutio, XVI.5.

⁶⁴Institutio, XVI. 10-11. The argument in question is of the form 'a is the son (father) of b; therefore b is the father (son) of a'. The conditional premiss to be added is, of course, 'If a is the son (father) of b, then b is the father (son) of a'. The categorical premiss is unfortunately lacking in the manuscript.

⁶⁵ Alexander's and Galen's discussions would seem to presuppose this. See especially Alexander, In Analyticorum Priorum, 68.21-69.1; 345.13-346.6.

⁶⁶ In his article 'Posidonius d'Apamée, théoricien de la géométrie', Revue des études grecques XXVII (1914), 44-45 (reprinted in Études de philosophie antique), E. Bréhier argues that Posidonius was the first (and also the last) Stoic with a "theory of the logic of geometry".

⁶⁷ See, for example, Diogenes Laertius, Vitae Philosophorum, VII.65. Other references are given in Mates, Stoic Logic, pp. 132-133.

⁶⁸ See the passages in H. Bonitz, 'Index Aristotelicus', in Aristotelis Opera (ed. by I. Bekker), Berlin 1831-70, V, 70b4-13.

⁶⁹ According to Cicero's Academica, I.xii.46 (ed. by O. Plasberg), Leipzig 1922, Zeno attended lectures by the skeptic Carneades and admired him very much.

⁷⁰ G. Vlastos, 'Zen o of Sidon as a Critic of Euclid', in The Classical Tradition (ed. by L. Wallach), Ithaca, N.Y., 1966, pp. 154-155.

⁷¹ Proclus, In Primum Elementorum, 199.11-200.1.

⁷²In Primum Elementorum, 200.1-3.

⁷³In Primum Elementorum, 214.15-218.11. I shall dis-cuss the details of this passage in another paper.

⁷⁴In Primum Elementorum, 277.25-279.11.

⁷⁵ W. Crönert, Kolotes und Menedemos, Studien zur Palaeographie und Papyruskunde VI (1906), 109.

⁷⁶ This is a very common ancient criticism of the first anapodeiktos. See, for example, Sextus Empiricus, Adversus Mathematicos, VIII.440-442.

⁷⁷ See Proclus, In Primum Elementorum, 193.10-198.15.

⁷⁸In Primum Elementorum, 196.15-18.

⁷⁹ O. Neugebauer, Über eine Methode zur Distanzbestimmung Alexandria-Rom bei Heron, Det Kgl. Danske Videnskabernes Selskab XXVI (1938), 21-24.

⁸⁰ See. L. Tittel, De Gemini Stoici Studiis Mathematicis Quaestiones Philologae, Leipzig 1895. Bréhier ('Posidonius d'Apamée', pp. 46-49) thinks that Geminus's work on mathematics is derived entirely from Posidonius.

⁸¹ See E. Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung, Leipzig, 5th ed., 1923, pt. 3, sec. 2, pp. 642-645. The same material is found in E. Zeller, A History of Eclecticism in Greek Philosophy (transi, by S. F. Alleyne), London 1883, pp. 113-117. The importance of the reawakening of inter est in Aristotle's work for the history of logic is stressed by J. Mau, 'Stoische Logik', Hermes LXXXV (1957), 147-158. The historical reconstruction of the present paper seems to provide support for Mau's views.