In his most generally known works Leibnitz contents himself with expounding the conse quences of this train of thought without clearly indicating the ultimate logical premises upon which it is based. What these premises are may best be learned from some of the papers recently printed by M. Couturat. In particular the little tract headed (Prima Veritates> (utti mate truths) contains a deduction of all the leading doctrines of the Leibnitian system, with one exception, from one or two ultimate logical theories. This sketch will be closely followed in the next few paragraphs.
The fundamental assumption from which Leibnitz starts is the doctrine (1) that all true propositions are analytical, i.e., the predicate of every true proposition is part of the meaning of its subject term. This is, in fact, the very meaning of the word truth. All ultimate truths are therefore identical and can be reduced to one of the two forms, A is A (Law of Iden tity), A is not non-A (Law of Contradiction, which Leibnitz wrongly regards as a mere ver bal transformation of the Law of Identity). What we call an a priori or deductive proof is merely the reduction of a derivative proposition to a form in which this identity between sub ject and predicate is made explicit by means of logical analysis. So-called synthetic proposi tions, in which the predicate appears to involve a new determination of the subject, are merely propositions for which we have not succeeded in performing this analysis. It is this denial that any true proposition can be really synthetic which constitutes the fundamental difference between Leibnitz and Kant.
It is now further assumed (2) that every proposition is one which affirms or denies an attribute of a substance or subject, and (3) that there is an ultimate plurality of such logical subjects or substances. Of these assumptions (2), which is, in fact, logically a mere trans formation of (1), is the source of Leibnitz's worst logical difficulties, as it forces him to hold that no proposition asserting a relation between two substances can be an ultimate truth. Any statement of the form, has the relation x to B," must be decomposable into a pair of statements of the form UA has an at tribute xi" "B has an attribute (3) Con stitutes the only real fundamental divergence between Leibnitz and Spinoza, since according to the latter the subject of every true proposi tion must ultimately be the one real substance viz., God. Leibnitz's reason for assuming a plurality of substances seems to have been his strong conviction that the human self is a real and ultimate individual.
Leibnitz goes on to observe that the whole difference between the necessary truths of science and the contingent truths of every-day experience is explained by the distinction be tween explicitly and implicitly analytical propo sitions. When writing for others, Leibnitz habitually spoke of this difference between truths of reason and truths of fact as if it were an ultimate difference in kind. Truths of rea son, he usually says, depend upon the Law of Contradiction, and are therefore demonstrable; truths of fact depend on the Principle of Suffi cient Reason, and are, with the one exception of the existence of God, contingent and in demonstrable. From the 'Prima Veritates) it is clear that his real meaning was merely that truths of fact (propositions involving the as sertion of existence) would require an infinitely prolonged analysis before they could be re solved into identities. Leibnitz s Law of Suffi cient Reason, in fact, turns out to mean simply the doctrine that every truth is analytical and therefore capable of a priori proof, though the production of the proof may demand an in finitelyprolonged analysis and thus be prac tically impossible for us. It now follows (1) that no two things can be exactly alike; nu merical diversity always implies difference in attributes, since there must be a reason why the things are counted as two and not as one, and this reason must, on the principles just laid down, lie in some difference in their attributes.
(Principle of the Identity of Indiscernibles). Hence Leibnitz denies that space is a reality, for if it were, there would be infinitely numer ous real things (the points of space), which are all exactly alike. (2) There are no relations between things, i.e., no relations which do not arise out of the attributes of the terms related, since every truth, reduced to its simplest terms, consists in the ascription of an 'attribute to a subject, and no proposition asserting re lation between two things can therefore be an ultimate truth. Hence, since geometry and me thanks consist precisely in the study of such "external° relations, the whole world of ex tended and moving objects must be merely ap parent or phenomenal, not real. (3) The true concept of any individual substance logically involves the whole series of its past, present and future states. Everything that can happen to such a substance Is logically a consequence of the character of that substance, each substance is causa sui, the reason for the succession of its own states. Hence God, being omniscient, can deduce the whole history, e.g., of Saint Peter or Alexander the Great, from contemplation of the concept of Peter or Alexander. (4) Every individual substance involves in its concept the whole structure of the universe, so that from complete knowledge of any one substance we could deduce a priori, if our power of analysis were infinite, the whole history of the universe. This follows from the consideration that every substance stands in some relation to while, according to the logic of Leibnitz, every relation presupposes as its foundation a corre sponding attribute in each of its terms. Hence all the substances are different expressions of the same fundamental system. They ((mirror') or /represent') the same structure but with vary ing degrees of clearness, like perspective draw ings, taken from different points, of the same fort or city. The connection between different substances is thus ideal or metaphysical, not real or causal. Strictly speaking, no finite substance exercises a real influence on another, since every, substance is the sufficient reason for the suc cession of its own states, and thepassing.over of a state or predicate from one thing into another is unintelligible. Thus what we com monly call external causes are, in truth, merely occasions or conditions of the occurrence of a change of which the real cause is always the nature of the substance in which the change takes place. But, since all substances "mirror° the same system, the result is that there ap pears to be causal interconnection between the states of all the things in the universe. Each thing develops from within, entirely unaffected by any other, and yet the result is the same as It would be if everything were casually affected at every moment by all the rest. This is Leib nitz's most famous theory, the doctrine of the Pre-established Harmony between all sub stances. A particular case of it is the psycho physical correspondence between the soul and the body. The soul develops independently ac cording to the law of its own nature, and the same is true of each of the substances which, as an aggregate, compose the body, yet the cor respondence between the psychical and the boctity series of changes is as complete as it would he if each series were at every step con ditioned by the other, nor is there any need, after the fashion of the Occasionalists, to in yoke the supernatural interference of God to effect the correspondence. The point is well illustrated by the analogy of a band of musi cians who keep perfect time and tune simply because each of them is playing correctly from his own score without waiting for his cue from any of the others.