Leibniz had a vision of a universal ambiguity-free language based on a new symbol set, a characterica universalis, and a machine-like calculus ratiocinator that would automatically prove all necessary truths, true in "all possible worlds." Gottlob Frege called the idea "a system of notation directly appropriate to objects." In the three hundred years since Leibniz had this vision, logical philosophers and linguistic analysts have sought those truths in the form of "truth-functional" propositions and statements formulated in words, but they have failed to find a necessarily "true" connection between words and objects.

Information philosophy uses such system of notation, not in words, but in bits of digital information. And the interconnected computers of the Internet are not only Leibniz's calculus ratiocinator, but humanity's storehouse of shared experiences and accumulated knowledge. Like the individual Experience Recorder and Reproducer (ERR) in each human mind, the World Wide Web is our shared Knowledge Recorder and Reproducer. Computer simulations of physical and biological processes are the best representations of human knowledge about the external world of objects.

Leibniz's Principle of Sufficient Reason says that every event has a reason or cause in the prior state of the world. This appears to commit him to a necessary determinism, but like the ancient compatibilist Chrysippus, Leibniz argues that some empirical things are contingent.

Another of his great principles is his Principle of Contradiction (Aristotle's Principle of Non-Contradiction), a proposition cannot be true and false at the same time, and that therefore A is A and cannot be not A.

That "A is A" follows from what Leibniz called the Identity of Indiscernibles, which came to be known as Leibniz's Law. In modern times, philosophers use both the Identity of Indiscernibles and the converse Indiscernibility of Identicals, often calling either of these Leibniz's Law.

We must distinguish between these concepts, first by noting that given two numerically distinct things that are "identical," by definition there can be no discernible differences between them. This Indiscernibility of Identicals may be simply an ideal concept.

Leibniz says that given the apparent indiscernibility of two things, they are nevertheless not identical if there are differences between them beyond the reach of our senses, differences too small to be discernible.

Leibniz himself described this possibility. And he was very clear that even if minute differences are indiscernible so that two distinct things appear to be identical, there are simply no two things perfectly identical that only differ in number (solo numero).

Leibniz thought that perfect identity can only be the identity of one thing with itself. Two distinct things that are indiscernible must be same thing under two names, he said (anticipating the modern discussions starting with Gottlob Frege about the Morning Star and Evening Star).

Information philosophy now shows that two concrete objects or abstract entities can be perfectly identical if and only if it is their intrinsic properties are compared. If their relational properties with other objects and position in space and time are ignored, two objects may be intrinsic information identicals.

Despite his denial of necessarily identical things in the world, Leibniz was a great mathematician and articulated a Principle of Substitutivity, that identical things can be substituted for one another without changing the meaning or truth value of a statement. Gottlob Frege, Bertrand Russell, and Ludwig Wittgenstein all followed Leibniz and accepted this principle of substitutivity.

Primary truths are those which either state a term of itself or deny an opposite of its opposite. For example, 'A is A', or 'A is not not-A'; If it is true that A is B, it is false that A is not B, or that A is not-B'; again, 'Each thing is what it is', 'Each thing is like itself, or is equal to itself, 'Nothing is greater or less than itself—and others of this sort which, though they may have their own grades of priority, can all be included under the one name of 'identities'.

("Primary Truths," in Leibniz: Philosophical Writings, ed. G. H. R. Parkinson, p.87)

4. There are no two individuals indiscernible from one another... Two drops of water or milk looked at under the microscope will be found to be discernible. This is an argument against atoms, which, like the void, are opposed to the principles of a true metaphysic.

The Identity of Indiscernibles is known as Leibniz's Law

Cf., Hesperus and Phosphorus as identical to Venus

5. These great principles of a Sufficient Reason and of the Identity of Indiscernibles change the state of metaphysics, which by their means becomes real and demonstrative; whereas formerly it practically consisted of nothing but empty terms.

6. To suppose two things indiscernible is to suppose the same thing under two names. [Cf. Frege]

("Correspondence with Clarke," in Leibniz: Philosophical Writings, ed. G. H. R. Parkinson, p.216)

In a word, the insensible perception is of as much use in pneumatics as is the insensible corpuscle in physics; and it is equally unreasonable to reject the one or the other on the pretext that it is beyond the reach of our senses...

To think otherwise is to have but little knowledge of the immensely subtle composition of things, which always and everywhere include an actual infinity.

I have also noted that, by virtue of insensible variations, two individual things can never be perfectly alike, and that they must always differ more than numero...

if we thought in good earnest that the things we do not apperceive are not there in the soul or in the body, we should fail in philosophy as in politics, by neglecting το μικρóν insensible progressions; whereas an abstraction is not an error, provided we know that what we are ignoring is really there. This is the use made of abstractions by mathematicians when they speak of the perfect lines they ask us to consider, and of uniform motions and other regular effects, although matter (that is to say the mixture of the effects of the surrounding infinite) is always providing some exception. We proceed in this way so as to distinguish the various considerations from one another, and to reduce the effects to their reasons as far as is possible to us, and to foresee some consequences; for the more careful we are to neglect no consideration which we can regulate, the more does practice correspond to theory.

This knowledge of insensible perceptions serves also to explain why and how two souls, whether human or of some other identical species, never come perfectly alike from the Creator's hands, but each has always from the beginning its own relation to the point of view it will have in the universe. But this follows from what I pointed out previously about two individuals, namely that their difference is always more than a numerical one.

(New Essays on the HumanUnderstanding, p.56)

In his Monadology, Leibniz maintains that even though his Monads are the simplest of all substance, with no internal parts, they are nevertheless numerically distinct individuals, because there are never in nature two things exactly alike.

1. The Monad, of which we will speak here, is nothing else than a simple substance, which goes to make up composites; by simple, we mean without parts.

2. There must be simple substances because there are composites; for a composite is nothing else than a collection or aggregatum of simple substances.

3. Now, where there are no constituent parts there is possible neither extension, nor form, nor divisibility. These Monads are the true Atoms of nature, and, in fact, the Elements of things...

9. Each Monad, indeed, must be different from every other. For there are never in nature two beings which are exactly alike, and in which it is not possible to find a difference either internal or based on an intrinsic property.

(Monadology)

Leibniz's emphasis on intrinsic, internal properties is very insightful. In his posthumous New Essays responding to Locke, he clearly says that extrinsic properties like space and time, and the relations to other things outside allow us to distinguish things we might not be able to tell apart by reference to themselves alone.

He also says two things cannot occupy the same place and time.

Chapter xxvii
What identity or diversity is

PHILALETHES. §1. A relative idea of the greatest importance is that of identity or of diversity. We never find, nor can we conceive it possible, that two things of the same kind should exist in the same place at the same time. That is why, when we demand, whether any thing be the same or no, it refers always to something that existed such a time in such a place . . . . From whence it follows, that one thing cannot have two beginnings of existence, nor two things one beginning. . .in time and place'.

THEOPHILUS. In addition to the difference of time or of place there must always be an internal principle of distinction: although there can be many things of the same kind, it is still the case that none of them are ever exactly alike. Thus, although time and place (i.e. the relations to what lies outside) do distinguish for us things which we could not easily tell apart by reference to themselves alone, things are nevertheless distinguishable in themselves. So time and place do not constitute the core of identity and diversity, despite the fact that diversity in time or place brings with it differences in the states that are impressed upon a thing, and thus goes hand in hand with diversity of things. To which it can be added that it is by means of things that we must distinguish one time or place from another, rather than vice versa; for times and places are in themselves perfectly alike, and in any case they are not substances or complete realities. The method which you seem to be offering here as the only one for distinguishing among things of the same kind, is founded on the assumption that interpenetration is contrary to nature. This is a reasonable assumption; but experience itself shows that we are not bound to it when it comes to distinguishing things. For instance, we find that two shadows or two rays of light interpenetrate, and we could devise an imaginary world where bodies did the same. Yet we can still distinguish one ray from the other just by the direction of their paths, even when they intersect.

P H I L . §3. What is called the principle of individuation in the Schools, where it is so much inquired after,... is existence it self, which determines a being. . .to a particular time and place incommunicable to two beings of the same kind.'

THEO. The 'principle of individuation' reduces, in the case of individuals, to the principle of distinction of which I have just been speaking. If two individuals were perfectly similar and equal and, in short, indistinguishable in themselves, there would be no principle of individuation. I would even venture to say that in such a case there would be no individual distinctness, no separate individuals. That is why the notion of atoms is chimerical and arises only from men's incomplete conceptions. For if there were atoms, i.e. perfectly hard and perfectly unalterable bodies which were incapable of internal change and could differ from one another only in size and in shape, it is obvious that since they could have the same size and shape they would then be indistinguishable in themselves and discernible only by means of external denominations with no internal foundation; which is contrary to the greatest principles of reason. In fact, however, every body is changeable and indeed is actually changing all the time, so that it differs in itself from every other. I remember a great princess [*Sophie], of lofty intelligence, saying one day while walking in her garden that she did not believe there were two leaves perfectly alike. A clever gentleman who was walking with her believed that it would be easy to find some, but search as he might he became convinced by his own eyes that a difference could always be found. One can see from these considerations, which have until now been overlooked, how far people have strayed in philosophy from the most natural notions, and at what a distance from the great principles of true metaphysics they have come to be.

(New Essays on the HumanUnderstanding, pp.229-231)

True in all possible worlds simply means that their truth is independent of the physical world. Kant will describe such statements as analytic.

An affirmative truth is one whose predicate is in the subject; and so in every true affirmative proposition, necessary or contingent, universal or particular, the notion of the predicate is in some way contained in the notion of the subject, in such a way that if anyone were to understand perfectly each of the two notions just as God understands it, he would by that very fact perceive that the predicate is in the subject. From this it follows that all the knowledge of propositions which is in God, whether this is of the simple intelligence, concerning the essence of things, or of vision, concerning the existence of things, or mediate knowledge concerning conditioned existences, results immediately from the perfect understanding of each term which can be the subject or predicate of any proposition. That is, the a priori knowledge of complexes arises from the understanding of that which is not complex.

An absolutely necessary proposition is one which can be resolved into identical propositions, or, whose opposite implies a contradiction... This type of necessity, therefore, I call metaphysical or geometrical. That which lacks such necessity I call contingent, but that which implies a contradiction, or whose opposite is necessary, is called impossible. The rest are called possible.

In the case of a contingent truth, even though the predicate is really in the subject, yet one never arrives at a demonstration or an identity, even though the resolution of each term is continued indefinitely. In such cases it is only God, who comprehends the infinite at once, who can see how the one is in the other, and can understand a priori the perfect reason for contingency; in creatures this is supplied a posteriori, by experience. So the relation of contingent to necessary truths is somewhat like the relation of surd ratios (namely, the ratios of incommensurable numbers) to the expressible ratios of commensurable numbers.

(Necessary and Contingent Truths)

Leibniz knew that our ideas, even those we think necessary, began with our sensory experiences. His famous idea that some "necessary truths" are "true in all possible worlds," was more fundamentally his insight that the necessary truths do not depend in any way on the physical world or our senses.

From this arises another question, whether all truths depend on experience, that is to say on induction and on instances, or whether there are some which have another basis also. For if certain events can be foreseen before we have made any trial of them, it is clear that we contribute in those cases something of our own. The senses, although they are necessary for all our actual knowledge, are not sufficient to give us the whole of it, since the senses never give anything but instances, that is to say particular or individual truths. Now all the instances which confirm a general truth, however numerous they may be, are not sufficient to establish the universal necessity of this same truth, for it does not follow that what happened before will happen in the same way again... And any one who believed that [day must follow night] is a necessary and eternal truth which will last for ever, would likewise be wrong, since we must hold that the earth and even the sun do not exist of necessity, and that there may perhaps come a time when that beautiful star and its whole system will exist no longer, at least in its present form.

The idea of truths necessary "in all possible worlds" is really that "proofs" do not depend on instances, sensory evidence. Necessity is independent of the physical world

From which it appears that necessary truths, such as we find in pure mathematics, and particularly in arithmetic and geometry, must have principles whose proof does not depend on instances, nor consequently on the testimony of the senses, although without the senses it would never have occurred to us to think of them.

(New Essays on the Human Understanding, Preface, p.49)

11. That the meditations of the so-called Scholastic philosophers and theologians are not to be entirely despised.

I know that I am putting forward a great paradox in claiming to rehabilitate ancient philosophy to some extent, and to restore the rights of citizenship to substantial forms, which have practically been banished. But perhaps I shall not readily be condemned when it is known that I have thought carefully about modern philosophy, and that I have devoted much time to physical experiments and to geometrical demonstrations. I was for a long time persuaded of the emptiness of these entities, and was finally obliged to take them up again despite myself, and as it were by force. This was after I had myself conducted some researches which made me recognise that our modern philosophers do not do enough justice to St. Thomas and to other great men of that era, and that the views of the Scholastic philosophers and theologians have much mere soundness than is imagined, provided that one uses them in a proper way and in their right place. I am even persuaded that if some, precise and thoughtful mind were to take the trouble of clarifying and setting in order their thoughts, in the manner of analytic geometry, he would find in them a treasury of truths which are extremely important and wholly demonstrative.

12.That the notions which consist in extension include something imaginary, and cannot constitute the substance of a body.

But to take up again the thread of our discussion: I believe that anyone who will meditate on the nature of substance, as I have explained it above, will find that the entire nature of body does not consist in extension alone, that is to say in size, shape and motion. Rather, he will find that it is necessary to recognise in it something which has some relation to souls, and which is comrnonly called a substantial form, although this changes nothing in phenomena, any more than the soul of the lower animals does, if they have one. It can even be demonstrated that the notion of size, shape and motion is not as distinct as is imagined, and that it contains something that is imaginary and relative to our perceptions, just as is the case (though even more so) with colour, heat and other similar qualities, of which it may be doubted whether they are really found in the nature of things outside us. This is why qualities of these kinds cannot constitute any substance. And if there is no other principle of identity in bodies besides that which we have just mentioned, no body will ever last longer than a moment. However, the souls and substantial forms of other bodies are very different from intelligent souls. Only the latter know their actions, and not only do not perish naturally, but even retain perpetually the basis of the knowledge of what they are. It is this which brings it about that they alone are capable of punishment and reward, and makes them citizens of the commonwealth of the universe, of which God is the monarch. It also follows that all other creatures must serve them, of which we shall speak at greater length presently.

("Discourse on Metaphysics," in Leibniz: Philosophical Writings, ed. G. H. R. Parkinson, p.21-23)

It is a very old doubt of mankind, how freedom and contingency can be reconciled with the series of causes and with providence. The difficulty of the matter has been increased by the dissertations of Christian authors on God's justice in procuring the salvation of men.

For my part, I used to consider that nothing happens by chance or by accident, except with respect to certain particular substances; that fortune, as distinct from fate, is an empty word; and that nothing exists unless its individual requisites are given, and that from all these taken together it follows that the thing exists. So I was not far from the view of those who think that all things are absolutely necessary; who think that security from compulsion is enough for freedom, even though it is under the rule of necessity, and who do not distinguish the infallible—that is, a truth which is certainly known—from the necessary.

But I was dragged back from this precipice by a consideration of those possibles which neither do exist, nor will exist, nor have existed. For if certain possibles never exist, then existing things are not always necessary; otherwise it would be impossible for other things to exist instead of them, and so all things that never exist would be impossible. For it cannot be denied that many stories, especially those which are called 'romances', are possible, even if they do not find any place in this series of the universe, which God has chosen—unless someone supposes that in the vast magnitude of space and time there exist the regions of the poets, where you could see wandering through the world King Arthur of Britain, Amadis of Gaul, and Dietrich von Bern, famed in the stories of the Germans. A certain distinguished philosopher of our century [Spinoza] seems to have been close to this opinion, for he says expressly somewhere that matter takes on successively all the forms of which it is capable (Principles of Philosophy, Part III, art. 47). This view is indefensible, for it would remove all the beauty of the universe and all choice, to say nothing here of other arguments by which the contrary can be shown.

Once I had recognised the contingency of things, I then began to consider what a clear notion of truth would be; for I hoped, not unreasonably, to derive from this some light on the problem of distinguishing necessary from contingent truths. However, I saw that it is common to every true affirmative proposition—universal and particular, necessary or contingent—that the predicate is in the subject, or that the notion of the predicate is in some way involved in the notion of the subject, and that this is the principle of infallibility in every kind of truth for him who knows everything a priori. But this seemed to increase the difficulty. For if, at a given time, the notion of the predicate is in the notion of the subject, then how, without contradiction and impossibility, can the predicate not be in the subject at that time, without destroying the notion of the subject?

A new and unexpected light finally arose in a quarter where I least hoped for it—namely, out of mathematical considerations of the nature of the infinite. There are two labyrinths of the human mind: one concerns the composition of the continuum, and the other the nature of freedom, and both spring from the same source—the infinite. That distinguished philosopher whom I mentioned above could not unravel these knots, or at any rate was unwilling to make his opinion known, but preferred to cut them with a sword. For he says (Principles of Philosophy, Part I, arts. 40 and 41) that we can easily involve ourselves in great difficulties if we try to reconcile God's preordination with the freedom of the will, and that we must abstain from discussing them, since God's nature cannot be comprehended by us. He also says (Part II. art. 35) that we ought not to doubt that matter is divided ad infinitum, even though we cannot understand this. But tl-.b is not enough: for it is one thing for us not to understand a thing, and another for us to understand its contradictory.

So it is at all events necessary to be able to answer those arguments which seem to imply that freedom or the division of matter imply a contradiction.

It must be known, therefore, that all creatures have impressed on them a certain mark of the divine infinity, and that this is the source of many wonders which amaze the human mind.

For example, there is no portion of matter so small that there does not exist in it a world of creatures, infinite in number. Again?"every individual created substance, however imperfect, acts on all others and is acted on by all others, and contains in its complete notion (as this exists in the mind of God) the whole universe, and whatever is, was or will be. Further, every truth of fact or of individual things depends on a series of infinite reasons, and all that is in this series can be seen by God alone. This is also the reason why God alone knows contingent truths a priori, and sees their infallibility in another way than by experience.

When I had considered these more attentively, a profound difference between necessary and contingent truths came to light. Every truth is either original or derivative. Original truths are those of which a reason cannot be given; such truths are identical or immediate, and they affirm a term of itself or deny a contradictory of its contradictory."1 Derivative truths are again of two sorts: some are analysed into original truths, others admit of an infinite process of analysis. The former are necessary, the latter contingent. A necessary proposition is one whose contrary implies a contradiction, such as all identical propositions and all derivative propositions which are analysable into identical propositions. These are the truths which are said to be of metaphysical or geometrical necessity. For demonstration consists simply in this: by the analysis of the terms of a proposition, and by substituting for a defined term a definition or part of a definition, one shows a certain equation or coincidence of predicate with subject in a reciprocal proposition, or in other cases at least the inclusion of the predicate in the subject, in such a way that what was latent in the proposition and as it were contained in it virtually is rendered evident and express by the demonstration...

But in the case of contingent truths, even though the predicate is in the subject, this can never be demonstrated of it, nor can the proposition ever be reduced to an equation or identity. Instead, the analysis proceeds to infinity, God alone seeing—not, indeed, the end of the analysis, since it has no end—but the connexion of terms or the inclusion of the predicate in the subject, for he sees whatever is in the series; indeed, this very same truth has arisen in part from his own intellect and in part from his will, and expresses in its own way his infinite perfection and the harmony of the whole series of things.

Here we see Leibniz may be as empiricist as Locke and Hume

However, there have been left to us two ways of knowing contingent truths; one is the way of experience and the other the way of reason. The way of experience is when we perceive a thing clearly enough by our senses; the way of reason is derived from the general principle that nothing happens without a reason, or, that the predicate is always in some way in the subject.

("On Freedom," in Leibniz: Philosophical Writings, ed. G. H. R. Parkinson, pp.106-109)

Leibniz imagined a scientist who could see the events of all times and predict the future, just as all times are thought to be present to the mind of God.

"Everything proceeds mathematically...if someone could have a sufficient insight into the inner parts of things, and in addition had remembrance and intelligence enough to consider all the circumstances and take them into account, he would be a prophet and see the future in the present as in a mirror."

Pierre-Simon Laplace particularized this Leibniz vision as an intelligent being who knows the positions and velocities of all the atoms in the universe and uses Newton's equations of motion to predict the future. Laplace's Demon has become a cliché for physical determinism.

Beyond the world, i.e. beyond the collection of finite things, there is some one being who rules, not only as the soul is the ruler in me (or, to put it better, as the self is the ruler in my body), but also in a much higher way. For the one being who rules the universe doesn’t just •govern the world but also •builds or makes it. He is above the world and outside it, so to speak, and therefore he is the ultimate reason for things. ·That follows because

•he is the only extramundane thing, i.e. the only thing that exists out of the world; and •nothing in the world could be the ultimate reason for things.

I now explain that second premise·. We can’t find in any individual thing, or even in the entire collection and series of things, a sufficient reason why they exist. Suppose that a book on the elements of geometry has always existed, each copy made from an earlier one, ·with no first copy·. We can explain any given copy of the book in terms of the previous book from which it was copied; but this will never lead us to a complete explanation, no matter how far back we go in the series of books. For we can always ask:

Why have there always been such books?
Why were these books written?
Why were they written in the way they were?

The different states of the world are like that series of books: each state is in a way copied from the preceding state—though here ·the ‘copying’ isn’t an exact transcription, but happens· in accordance with certain laws of change. And so, ·with the world as with the books·, however far back we might go into earlier and earlier states we’ll never find in them a complete explanation for •why there is any world at all, and •why the world is as it is

(Jonathan Bennett's edited version of Leibniz's The Ultimate Origin of Things)

The article "A Logical Calculus of the Ideas Immanent in Nervous Activity," was published in Nicholas Rashevsky's new Journal of Mathematical Biophysics. "Logical Calculus" is a direct reference to Leibniz' calculus ratiocinator.

In the 1930's McCulloch had studied logic at Yale in Frederic Fitch's course on propositional logic, based on Leibniz and the great Principia Mathematica of Bertrand Russell and Alfred North Whitehead.

Fitch also was the thesis adviser for Ruth Barcan Marcus, whose work on Leibniz' Law, the "identity of indiscernables" and the "necessity of identity" influenced Saul Kripke.

Despite the fact that information flows along neurons in the brain, the neural network is not a computer network, brain processes are not algorithms, and there is no central processing unit (CPU) or even distributed parallel processing.

Very simply, man is not a machine and the brain is not a computer.

Nevertheless, we can regard McCulloch as the first thinker to offer a solution to the mind-body problem that "embodies" an immaterial logical software mind in a material mechanical hardware computer, as the title of his 1965 book, "Embodiments of Mind, suggests.

And we can look closely at Leibniz's original words in the Monadology to see how much they anticipate the work of today's computational neuroscientists to buid a machine that can see, think, and feel like a human being.

Suppose that there were a machine so constructed as to produce thought, feeling, and perception, we could imagine it increased in size while retaining the same prportions, so that one could enter as one might a mill [e.g., a flour mill]. On going inside we should only see the parts impinging on one another; we should not see anything which would explain a perception.

Monadology, in Leibniz Philosophical Writingd, Dent & Sonns, London, 1973, p.181

Necessary and Contingent Truths
On Freedom
Résumé on Metaphysics
The Ultimate Origin of Things( From Early Modern Texts)