This <sequential resolution into primitives> is how, among mathe maticians, theoretical theorems and practical rules are reduced by analysis to

definitions, axioms, and postulates.

34. C'est ainsi que chez Mathématiciens, les Théorèmes de speculation et les Canons de practique sont réduits par l'Analyse aux Definitions, Axiomes et Demandes.

(NE, p. 360.) You must understand that geometers do not derive their proofs from diagrams. . . . It is universal propositions, i.e. definitions and axioms and theorems which have already been demonstrated, that make up the reasoning, and they would sustain it even if there were no diagram.

(NE, pp. 361-62.) The primary truths which we know by "intuition" are of two sorts, as are the derivative ones. They are either truths of reason or truths of fact. Truths of reason are necessary, and those of fact are contingent. The primary truths of reason are the ones to which I give the general name "iden- tities," because they seem to do nothing but repeat the same thing without telling us anything. They are either affirmative or negative. Examples of affir- mative ones are: What is, is; Each thing is what it is, and as many others as you want: A is A; B is B. . . . An equilateral rectangle is a rectangle. . . . If a

regular four-sided figure is an equilateral rectangle then this figure is a rec- tangle. ...IfAis non-B it follows that A is non-B.

(G VII, 300; "Principles" [ca. 1696].) Identical propositions are the primary propositions of all, and are incapable of proof and thus true per se, for of course nothing can be found to serve as a middle term to connect something with itself, so as a result, truths are virtually identical which can be reduced to for- mal or explicit identities through an analysis of their terms, if we substitute for the original term either an equivalent concept or a concept included in it. It is obvious that all necessary propositions, or propositions which have eternal truth, are virtual identities and can be demonstrated or reduced to primary truths by ideas or definitions alone, that is, by the analysis of terms, so that it is made clear that their opposite implies a contradiction and conflicts with some identity or primary truth. Hence the Scholastics also observed that truths which are absolute or have metaphysical necessity can be proved by their terms alone, since the opposite involves a contradiction.

In general, every true proposition which is not identical or true in itself can be proved a priori with the help of axioms or propositions that are true in them- selves and with the help of definitions or ideas. For no matter how often a pred-

icate is truly affirmed of a subject, there must be some real connection between subject and predicate, such that in every proposition whatever, such as A is B (or B is truly predicated of A), it is true that B is contained in A, or its concept is in some way contained in the concept of A itself. And this must be either by absolute necessity, in propositions which contain eternal truth; or by a kind of certainty which depends upon the supposed decree of a free substance in con- tingent matters, a decree, however, which is never entirely arbitrary and iiee from foundation, but for which some reason can always be given. This reason, however, merely inclines and does not truly necessitate. Such truth could itself be deduced from the analysis of concepts, if this were always within human power, and will certainly not escape the analysis of an omniscient substance who sees everything a priori from ideas themselves and from his decrees. It is certain, therefore, that all truths, even contingent ones, have a proof a priori or some reason why they are rather than are not. And this is what is commonly asserted: that nothing happens without a cause, or there is nothing without a reason. Yet however strong this reason may be—though whatever kind it is, it is enough to effect a greater inclination in one direction or the other—even if it establishes certainty in a predicting being, it does not place necessity in the thing itself, because its contrary would still remain possible per se and implies no contradiction. Otherwise what we call contingent would rather be necessary or of eternal truth.

(Couturat, Opuscules, pp. 518-19; Loemker, pp. 267-68; Ariew &. Garber, pp. 30-31; "Primary Truths" [ca. 1685].) Primary Truths are those which pred- icate something of itself or deny the opposite of its opposite. For example, A is

A, or A is not non-A,- if it is true that A is B or that A is non-B. Likewise,

everything is what it is; everything is similar or equal to itself; nothing is

greater or less than itself. These and other truths of this kind though they may have various degrees of priority, can nevertheless all be grouped under the one name of identities.

All other truths are reduced to primary truths with the aid of definitions or by the analysis of concepts; in this consists proof a priori which is independent of experience. I shall give as example this proposition which is accepted as an axiom by mathematicians and all other people alike: the whole is greater than its part, or the part is less than the whole. This is very easily demonstrated from the definition of less or greater, with the addition of a primitive axiom or iden- tity. For that is less which is equal to a part of another thing (the greater). This definition is very easily understood and is consistent with the general practice of men, when they compare things with each other and measure the excess by subtracting an amount equal to the smaller from the greater. Hence one may reason as follows. A part is equal to a part of the whole (namely, to itself, by the axiom of identity, according to which each thing is equal to itself). But what is equal to a part of a whole is less than the whole (by the definition of less). Therefore the part is less than the whole.

The predicate or consequent therefore always inheres in the subject or an- tecedent. And as Aristotle, too, observed, the nature of truth in general or the connection between the terms of a proposition consists in this fact. In identities this connection and the inclusion of the predicate in the subject are explicit; in all other propositions that are implied and must be revealed through the analysis of the concepts, which constitutes a demonstration a priori.