a) The Platonic Heritage

Plato’s world consists of two realms, the realm of being and the realm of becoming. It consists of the realm of unchanging forms and the realm of changing individuals. There is the realm of atemporal forms and the realm of temporal things. But Plato’s world threatens to split apart into these two realms, for the connection between them remains a mystery. In our world, no threat exists, no split is immanent. Our world is a fact, not a collection of properties to one side and a collection of individuals to the other. The world as a fact divides into further facts, and these contain, ultimately, both Plato’s forms and his individuals. What binds properties to individual things is the undefinable nexus of exemplification. What is thus created is a simple fact. And this fact is part of the larger fact which is the world. Plato saw the distinction between properties (forms) and individuals. What he did not see is that these two kinds exist only as parts of a third kind, namely, as parts of facts. What he did not realize is that the fundamental category of ontology is the category of fact. The problem is not how properties and individuals can come together to form facts, but how facts can contain properties and individuals. Facts guarantee that all of the diverse categories in the end form just one world, for all of them are merely constituents of the one over-arching fact which is the world.

Corresponding to the two realms of Plato’s world, there exist in his epistemology two “eyes”: One eye sees the ever-changing individuals of the realm of becoming; the other eye is fixed on the eternal forms. The first eye can only see individuals; the second can only contemplate forms. How, then, is knowledge possible? Just as Plato’s world is in danger of splitting into two unconnected parts, so is his knowledge in danger of breaking into two unconnected faculties: intuition of individual things and conception of properties. Intuition, the “eye” of the senses, acquaints us with the ever-changing realm of individuals. Conception, the “eye” of the mind, contemplates the atemporal forms. How, then, do we come to know that this individual has that property? Knowledge cannot be a matter of either intuition alone or of conception alone. This much the Tradition cannot help but admit. The two faculties must somehow combine. This combination is called “judgment.” What is a judgment? This is the question to which the Platonic tradition has no reasonable answer. It is assumed that a judgment somehow results when the two “eyes” work together. But how, precisely, do they work together? How can they possibly work together if one of them is condemned to be fixed on the eternal forms, the other is doomed to be blind to everything but changing individuals? The important dogma of the Tradition, to enlarge our metaphor, is that there is no third “eye,” that there is no further unique faculty in addition to intuition and conception. Just as the Platonic ontology lacks the uniting category of fact, so does its epistemology lack the unifying faculty of judgment. Obviously, these two fundamental shortcomings go hand in hand. Without facts, there is no need for judgments; and without judgments, there is no call for facts. But what holds the world together are facts. And what constitutes knowledge of the world are judgments.

However, the Platonic tradition does not only neglect the importance of judgments, it also embraces a most pernicious dogma, namely, the dogma that the senses can acquaint us with nothing but individuals. This dogma makes rationalism all but inevitable. For, if the senses acquaint us with nothing but individuals, then there must be another faculty that acquaints us with forms. Empiricism, as it turns out, has not just one but two strikes against it. Firstly, true knowledge cannot be concerned with the buzzing, booming confusion of the realm of becoming, but must be a matter of unchanging forms. Thus it cannot be acquired through the senses. Secondly, judgment, insofar as it plays a role in the Tradition, cannot involve the senses. Add to this that knowledge is contained only in judgments, and the conclusion is inevitable that the senses know nothing.

What chance does empiricism have against a view so heavily fortified with rationalistic premises? Very little. History shows that empiricism knows only one way out of its corner: It attacks the realm of being. Empiricism thus gets settled with nominalism. Quite obviously, if there is no realm of timeless entities, and if there is no special contemplation of the denizens of this realm, then knowledge, whatever it may turn out to be, can concern only the realm of temporal individuals. Two unholy alliances are formed: one, between empiricism and nominalism; the other, between rationalism and realism. But both combinations are doomed. Rationalism must fail, not because it embraces the realm of forms, but because it has no room for empirical knowledge. Empiricism must fail, not because it takes care of individuals, but because it rejects the realm of universals.

As far as the theory of knowledge is concerned, empiricists have one decisive argument: Rationalists never explain the mysterious faculty of conception; and, indeed, there is ample evidence that it does not exist. On the other hand, rationalists, too, can produce a powerful argument: Empiricists cannot explain the nature of mathematical knowledge.

The fight between rationalism and empiricism has an air of unreality about it because it takes place without regard for the existence of facts and judgments. Knowledge consists, as I mentioned earlier, neither in intuition nor in conception, but in judgment (assertion). And what judgment intends is neither a mere individual nor a property, but a state of affairs.

The two opponents, ironically, share a common dogma; a dogma that determines the traditional dialectic. According to this presupposition, the senses cannot acquaint us with anything but individuals. This dogma, in turn, rests on two unquestioned beliefs. Firstly, one takes for granted, as Kant succinctly puts it, that space and time are the forms of sensibility. Secondly, one assumes that properties (forms, universals), if there are such things, are not located in space and/or time. This assumption is almost a matter of terminology: what is not spatial and/or temporal is not called an “individual.” It is the first of these two beliefs that is crucial to the dialectic. Let me formulate it differently: Perception (“the senses”) cannot acquaint us with anything but individual things. Since I believe that this proposition is false, I believe that both rationalists and empiricists start out with a false premise. Perception acquaints us not only with individual things, but also with properties. It acquaints us not only with spatio-temporal entities, but also with non-spatial and atemporal things. The senses are a window through which we see not only the realm of becoming, but also the realm of being. The senses put us in touch not only with changing things, but also with timeless universals. No special faculty, no mysterious power of conception, contemplation, eidetic intuition, or what-have-you, is needed to explain how we are acquainted with universals.

The key to getting rid of the common dogma is the insight that perception is propositional. What we perceive are not individual things by themselves, but states of affairs. The eye of the senses sees states of affairs rather than isolated individuals. States of affairs, of course, have many kinds of ingredients. They contain individual things; but they also contain properties and relations. They even contain, as I shall argue, sets and numbers. Thus there really is only one eye, the eye of perception; and it acquaints us with temporal (“concrete”) as well as atemporal (“abstract”) things, for both are conjoined in states of affairs.

The Platonic tradition is partially correct: There do exist entities other than individual things; there do exist abstract entities. But the Platonist is mistaken when he thinks that knowledge consists in the contemplation of such abstract things. The empiricist is also partially correct: All of our knowledge (of the external world) comes ultimately through the senses. But he is mistaken when he goes on to deny the existence of the realm of being. One can be a realist and reject rationalism. One can be an empiricist and reject nominalism. This is the position I take: I am a realist when it comes to the existence of abstract entities, but I am an empiricist in regard to the nature of knowledge, for I hold that we perceive abstract things. I have chosen the “fourth way.”

This position, I submit, opens up a new approach to the problem of mathematical knowledge. We can combine the view that numbers and sets are abstract entities with the contention that we perceive some numbers and sets. Not only do we perceive some numbers and sets, we also perceive some relations among them. Mathematical knowledge is therefore “empirical” knowledge, but “empirical” knowledge about structures of abstract entities.

b) Kant’s Philosophical Situation

Kant’s epistemological framework is on the whole Platonistic. But he adds an idealistic twist and makes some minor improvements. Temporal (and spatial) individuals are represented by (acts of) intuition; atemporal properties, by conception (concepts). Of equal importance is the fact that Kant subscribes to the Platonic dogma that the senses are confined to individual things. In other words, intuitions and only intuitions are sensible. This, as I said a moment ago, is the fatal dogma that makes a correct theory of knowledge impossible. According to the dogma, there are two basic faculties, intuition and conception, and each of these has its specific kind of object, intuitions (individuals) and concepts (properties). Kant’s basic distinction between intuitions and concepts serves a dual purpose. It serves an ontological purpose, for it distinguishes between the ontological categories of individual and property. And it also serves an epistemological purpose, for it distinguishes between the two faculties of sensibility and conception.

But Kant also talks about judgments. Knowledge, he clearly sees, is neither a matter of blind intuition nor of empty conception. Knowledge only results when these two faculties work together harmoniously. Knowledge consists in seeing with both eyes simultaneously. But judgment, in Kant’s opinion, does not constitute a third and equally unique faculty. It must be emphasized that Brentano’s insight that judgments form a unique and indefinable kind of mental act eludes Kant. He defines judgment as a kind of presentation: “A judgment is the presentation of the unity of the consciousness of different presentations or the presentation of their relationship, insofar as they constitute a concept” (I. Kant, Logik, Gesammelte Schriften, Akademie Edition, 9:101). From our point of view, Kant’s notion of judgment is doubly defective. He does not see, as Brentano later does, that judgments form an irreducible kind of mental act. Nor does he realize, as Bolzano later does, that their objects belong neither to the category of individual nor to the category of property, but belong to a third, irreducible, kind, namely, to state of affairs.

Knowledge resides in judgments. This much Kant clearly sees. What kinds of judgment are there? Kant’s most important distinction, and a distinction he was very proud of, is between analytic and synthetic judgments. We shall have a close look at this distinction in a moment, but let us get clear, first, about the context. Leibniz had maintained that in all true judgments of the form All F’s are G’s, the predicate concept G is contained in the subject concept F. We need not go deeply into the reasons for Leibniz’s view, for it is clear that he thought the only intelligible notion of predication is that of concepts being contained in concepts:

We must consider, then, what it means to be truly attributed to a certain subject. Now it is certain that every true predication has some basis in the nature of things, and when a proposition is not an identity, that is to say, when the predicate is not expressly contained in the subject, it must be included in it virtually. This is what the philosophers call in-esse, when they say that the predicate is in the subject. So the subject term must always include the predicate term in such a way that anyone who understands perfectly the concept of the subject will also know that the predicate pertains to it.

(G. W. Leibniz, Discourse on Metaphysics, pp. 471-72)

Leibniz speaks here of subject and predicate, but I think that we can substitute ‘subject concept’ and ‘predicate concept’ for these expressions. Of course, I would rather speak of properties: In all true judgments of the form All F’s are G’s, the property G is part of the property F. One of the most pernicious legacies of the Kantian school is the idealistic slant in thought and terminology that substitutes concepts for properties. Not even Bolzano and Frege were untouched by this curse.

Arnauld, in his correspondence with Leibniz, objects to Leibniz’s view. The predicate concept, he maintains, is only contained in the subject concept if the judgment is not only true, but necessary. All necessary propositions, in other words, are of the “analytic” sort described by Leibniz. But there are also, contrary to Leibniz, true propositions in which the predicate concept is not part of the subject concept. Kant, viewed against the background of this controversy between Leibniz and Arnauld, sides neither with Leibniz nor with Arnauld, but claims that they are both mistaken. Leibniz is mistaken when he thinks that all true judgments are necessary; and Arnauld is mistaken when he holds that all necessary judgments are analytic as explained by Leibniz. There are, Kant objects, necessary judgments which are not analytic. It therefore becomes imperative to distinguish between analytic judgments and a priori judgments: The latter are necessary without being analytic. Looked at from another angle, it becomes imperative to distinguish between two kinds of necessity: the necessity of analytic judgments and the necessity of a certain kind of non-analytic judgement. Kant discovers a new kind of necessity, a necessity which is not based on the fact that complex concepts contain other concepts, and this discovery demands an answer to the question: Whence the necessity of non-analytic judgments?

Arithmetic (and mathematics as a whole) illustrates Kant’s contention that neither Leibniz nor Arnauld grasped the philosophical situation. The judgment that 7 + 5=12, Kant claims, is necessary. Yet it is not analytic: the concept 12 is not somehow composed of the concepts 7, 5, and plus. Thus Arnauld is mistaken. (But notice that this is not a categorial judgment like All F’s are G’s.) There is necessity without analyticity. Where does this necessity come from? To repeat, this is the crucial question with which Kant challenges all later philosophers. As we know, he gives a most original answer. Alas, his answer has so many problems that its popularity can only be explained by the fact that nobody proposed a better answer.

In order to clarify the context of Kant’s answer, we must look at the nature of synthetic judgments. Knowledge can only be based on intuitions and concepts. This is a Kantian axiom. Well, if analytic judgments involve the analysis of concepts, then synthetic judgments must somehow be a matter of intuition. In a synthetic judgment of the form All F’s are G’s, where G is not a part of F, F and G must be connected in some other way:

In all judgments in which the relation of a subject to a predicate is thought . . . this relation is possible in two different ways. Either the predicate B belongs to the subject A, as something which is (covertly) contained in this concept A; or B lies outside the concept A, although it does indeed stand in connexion with it. In the one case I call the judgment analytic, and the other synthetic.

(I. Kant, Critique of Pure Reason. B 10)

Kant here formulates the problem but does not solve it: What, precisely, is the connection between A and B when B lies outside of A? One thing is clear, though, the connection must somehow be based on intuition. How else, a Kantian would ask, could it possibly be established?

I do not fully understand Kant’s explanation of how intuition connects concepts with each other, but I shall bravely describe a view which, if not Kant’s, is at least in the Kantian spirit. Consider the judgment that all men are mortal. This is a synthetic judgment: the concept mortal is not contained in, is not part of, the concept man. Yet, since it is a true judgment, the two concepts must somehow be connected with each other. Well, the concept man applies to certain intuitions; certain intuitions fall under the concept man. Also, the concept mortal applies to certain intuitions. The first concept applies, for example, to Plato and to Aristotle. Under the second fall not only Plato and Aristotle, but also my neighbor’s dog Honey. Now, in every case in which an intuition falls under the concept of man, it also falls under the concept mortal (but not conversely). Plato is a man, and he is also mortal; Aristotle is a man, and he is also mortal; and so on. By induction from these individual cases, in each of which the respective concepts are indirectly connected by being directly attached to the same intuition, we establish a general connection between the two concepts; and we express this connection by means of the judgment All men are mortal. According to this interpretation, two things are essential to synthetic judgments. Firstly, there must be an intuition which connects the two concepts by virtue of the fact that both are attached to that intuition. You may think of F and G as being connected by the relation of being-jointly-exemplified. Secondly, there must occur induction in order to get from individual cases to the general judgment.

Synthetic judgments, to say it again, must be based on intuition. This follows from two Kantian axioms. According to the first, knowledge can only be based on intuitions and concepts; nothing else can be of importance for knowledge. According to the second, all conceptual knowledge is analytic; as long as nothing but concepts are involved, no synthetic judgment is possible. It follows that even in synthetic a priori judgments the connection between F and G must be established through intuitions. On the other hand, and here we touch upon a crucial point, such judgments cannot result from induction; for induction could never guarantee the necessity of a synthetic a priori judgment. The inductive leap precludes necessity. Somewhere, therefore, there must be an intuition which is such that (a) it serves to connect the respective concepts, and (b) by doing so in an individual case, it guarantees that the connection holds universally. This, I believe, is the mysterious pure intuition on which Kant’s theory of the synthetic a priori rests.

c) An Explication of the Analytic-Synthetic
Distinction

Kant’s example of an analytic judgment is All bodies are extended: “I do not require to go beyond the concept which I connect with ‘body’ in order to find the extension as bound up with it. To meet with this predicate, I have merely to analyze the concept, that is, to become conscious to myself of the manifold which I always think in that concept.” (Kant, 1965, B 10) There is much that is obscure in this passage. As has often been pointed out, it is not clear whether the analytic judgment is a result of a mental process of analysis, or whether analysis merely brings out the analyticity of a judgment (see, for example, Lewis White Beck, “Can Kant’s Synthetic Judgments Be Made Analytic?”). This kind of ambiguity is the price one has to pay for an idealistic methodology. We can remove at least part of the ambiguity if we adopt the following interpretation. Whatever the psychological state of a person may be, the state of affairs before her mind, represented by the sentence ‘All bodies are extended’, either contains the simple property body or it does not. If it does, then the corresponding judgment cannot be analytic, for the simple property cannot contain any other property as a part. On the other hand, if body is not a simple property, then the property of being extended may or may not be a part of it. If extended is a part of the complex property body, then the corresponding judgment is analytic. In this case, ‘body’ is a mere abbreviation for a longer expression which, among other words, contains ‘extended’. Assume, for example, that ‘body’ is merely short for ‘extended individual thing’. The sentence ‘All bodies are extended’ is then merely short for ‘All extended individual things are extended’, and the judgment expressed by this sentence is analytic.

We have to distinguish between (1) (true) analytic sentences, (2) (true) analytic judgments, and analytic states of affairs (facts). The basic notion is that of an analytic state of affairs (which, if it obtains, is of course an analytic fact). So-called analytic judgments, it is clear, are judgments of the form All things which are F, G, H, etc., are F. In other words, what is judged is always a state of affairs of this form. A state of affairs of this form is an instance of the logical law: All properties f, g, h, etc. are such that: if anything is f, g, h, etc., then it is f. An analytic state of affairs, therefore, is an instance of this law of logic. An analytic judgment is a judgment that intends such an instance of the law. And an analytic sentence is a sentence that represents an instance of the logical law.

It follows from our explication that the notion of analyticity serves at best a minor function, and that in connection with sentences. It may be the case that a simple expression is used to represent a complex property. Then it is not obvious from the sentence that we are dealing with a logical truth. And then the expression ‘analytic sentence’ may be useful. An analytic sentence is a sentence which represents an instance of the logical law mentioned earlier, but which does not do so in an obvious way. We can make it perspicuous by replacing all abbreviations in it by their abbreviants. The analytic, therefore, plays a minor role in our philosophy, and we can easily do without it. It is “reduced,” to speak for once in this misleading way, to the logical. An analytic sentence is simply a sentence that represents an instance of a certain logical law, that is, that represents a logical fact or logical truth. Is the judgment that all bodies are extended analytic? Well, what state of affairs is its intention? Equivalently, what state of affairs is represented by the “text” of the judgment, namely, the sentence ‘All bodies are extended’? The answer to this question depends on the answer to another question: Is the word ‘body’ used as an abbreviation for a longer expression, an expression that represents a “conjunctive property” involving the property of being extended? What I wish to stress at this point is the fact that this last question may be difficult to answer, but that the difficulty is not of a philosophical nature. In order to answer the question, we may have to inquire into what a particular person had in mind, or into how a group of people commonly use certain words, or into whether or not anyone ever laid down an abbreviation proposal, and so on. But all of these inquiries are philosophically unproblematic, no matter how difficult they may be from a practical point of view.

There exists a well-known objection to the analytic-synthetic distinction which is based on such practical difficulties. Generally speaking, one tries to discredit the distinction by discrediting the linguistic notion of synonymity. But this seems to me to be the wrong approach. The notion of synonymity is perfectly clear, even though it may be hard if not impossible to decide in a given case whether or not two English expressions are synonymous. Our approach, by way of contrast, is to question specific claims about synonymity, for example, Hempel’s assertion that ‘5 + 7’ is synonymous with ‘12’. We understand the notion of synonymity perfectly well, and because we understand it, we are convinced that these two expressions are not synonymous.

Is ‘bachelor’ synonymous with ‘unmarried male of marriageable age’? If the shorter expression is a mere abbreviation for the longer one, then the judgment that all bachelors are unmarried is analytic. We assume that the logical law mentioned earlier and its instances are (trivially) analytic, and then we ask whether or not the sentence ‘Bachelors are unmarried’ represents an analytic state of affairs. If I understand him correctly, Quine, in order to discredit the analytic-synthetic distinction, argues at this point that we cannot use without circularity the interchangeability of expressions salva veritate in all contexts as the criterion of synonymity. There is circularity because we must ask for “necessary” or “analytic” interchangeability, that is, the resulting expressions must be analytically equivalent. In our case, it is not enough that it is true that bachelors are unmarried if and only if unmarried males of marriageable age are unmarried. Rather, this equivalence must be analytic. But now we have to ask whether it, in turn, represents a logical law or an instance of one. And in order to answer this question, we must already know whether or not ‘bachelor’ is synonymous to ‘unmarried male of marriageable age’. We are back to the original question and the circle is closed (W. V. Quine, “Two Dogmas of Empiricism”; and “Carnap and Logical Truth”).

Quine’s argument is pertinent as long as one thinks of synonymity in terms of substitutibility salva veritate. Unfortunately, this conception is so popular that I have little hope to change it. But I shall try. I contend that in our example the question is not whether or not ‘bachelor’ and ‘unmarried male of marriageable age’ are interchangeable salva veritate, but whether or not the first expression is an abbreviation for the second. The question is whether or not ‘bachelor’ is merely short for the longer expression. But this question is not a philosophical question. Nor is it one that raises philosophical problems. It is the question of whether or not certain people have agreed, explicitly or merely tacitly, to use ‘bachelor’ as an abbreviation for the longer phrase. It may well be that this question has no clear answer, even though it is a perfectly clear question. In that case, we shall not be able to decide whether or not the statement that all bachelors are unmarried is analytic. But this result does not imply that the notion of analyticity is vague, unclear, imprecise, etc. Let ‘ground’ be short for ‘green and round’. Is the sentence ‘All ground things are green’ analytic? It is, for it is merely short for ‘All green and round things are green’, and this sentence represents an instance of the logical law mentioned earlier. There is nothing unclear about this notion of analyticity.

Of course, our notion presupposes that we have a firm understanding of logical laws. As long as we are content with Kant’s narrow conception of logic, the problem of explicating what a logical law is, is not very pressing. We can just single out the law mentioned earlier and call it “the logical law.” But if there are other logical laws, as we know there are, then the task of describing the nature of logic becomes urgent. Our generalized notion of analyticity then becomes:

As I said a moment ago, this explication is only as clear as the notion of logical law is. But it shows, and this is my main point, how unimportant, from a philosophical perspective, this notion of analyticity is. Does this meager notion suffice to tackle the urgent philosophical task before us? Does it suffice to answer the Kantian challenge? The history of philosophy of the last hundred years indicates that few if any philosophers agreed with us that the notion of analyticity is not equal to the task.

Our notion of analyticity was anticipated by Frege:

The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths. If, in carrying out this process, we come only on general logical laws and definitions, then the truth is an analytic one, bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depends

(G. Frege, The Foundations of Arithmetic, p. 4).

But there is a difference between Frege’s formulation and ours: he speaks of definitions, while I mention abbreviations. And thereby hangs a tale. No, not a tale, but a cluster of tales; for nothing less is at stake than the success or failure of logicism. In order to prove that arithmetic propositions are analytic, contrary to Kant’s claim, Frege defines certain crucial terms. But these “definitions” turn out to be no mere linguistic conveniences, no mere abbreviations, no mere stipulations. Quite to the contrary, they turn out to be assertions among assertions. And since these propositions are not logical laws or instances of logical laws, the so-called “reduction” of arithmetic to logic fails. As a consequence, the “reduced” sentences of arithmetic do not deserve to be called “analytic.” This, in a nutshell, is my objection to Frege’s logicism.

d) One Kind of Necessity: Lawfulness

Necessity pervades the world on many layers. I shall argue that there is ontological necessity, logical necessity, set-theoretical necessity, arithmetic necessity, even biological and chemical necessity. What all of these necessities have in common is that they emanate from lawfulness. In this respect, too, Frege anticipated our view. He describes this kind of necessity in these words: “The apodictic judgment differs from the assertoric in that it suggests the existence of universal judgments from which the proposition can be inferred, while in the case of the assertoric one such a suggestion is lacking. By saying that a proposition is necessary I give a hint about the grounds for my judgment” (G. Frege, Begriffsschrift, in Translations from the Philosophical Writings of Gottlob Frege, p. 4). I would put the same idea as follows. The sentence ‘It is necessary that P’ is merely short for the sentence ‘P follows (logically) from some (unspecified) laws’. Similarly, ‘It is possible that P’ is an abbreviation for ‘P is consistent with some laws’. Laws themselves are supposed to be nonmodal and, of course, trivially necessary. Necessity, so conceived, may be said to be a “property” of facts, namely, the “property of following from laws.” It is a “property” which pervades the world.

This kind of necessity is part of the furniture of the world, insofar as some facts do indeed follow from certain laws; insofar as they have this feature. To have this feature amounts, roughly speaking, to standing in the logical implication relation to laws. And this relation dissolves, in turn, into a general conditional. To say that P follows from L is to say that the fact If L, then P is an instance of a logical law. From an ontological point of view, therefore, this kind of necessity is doubly founded on lawfulness. There is, firstly, the lawfulness inherent in L; and there is, secondly, the lawfulness of logic. This kind of necessity is not an unanalyzable, not an irreducible, quality of facts. (It should be mentioned in passing that this kind of necessity can also be conceived of in terms of consistency between possible worlds. See my The Categorial Structure of the World, pp. 372-374, and J. M. Dunn, “A Truth Value Semantics for Modal Logic.”)

It will be objected by some that our explication has things upside down: lawfulness must be explicated in terms of necessity, and not the other way around. Necessity, it may be claimed, is precisely that feature which distinguishes between laws and other (accidental) generalities. I must confess that I do not know how to resolve the issue, and to discuss it in detail would, at any rate, lead us too far astray. I shall therefore rest content with the confession that it is one of my basic assumptions in this book that necessity rests on lawfulness, and not the other way around.

It is a consequence of our analysis that there are many kinds of necessity (and of possibility), namely, as many as there are different fields of inquiry. For example, it is an ontological necessity that the pen before me on the desk is not exemplified by anything. This follows from the ontological law that no individual is exemplified by anything. It is also true that the pen is not both blue and also not blue (at the same time, all over). But this fact does not follow from any ontological law or laws. It is ontologically possible that the pen is both blue and not blue. However, there exists a law of logic from which it follows that the pen is not both blue and also not blue. This fact is therefore a logical necessity. Contradictions, in short, are ontologically possible, but logically impossible. The so-called formation rules for logical systems reflect not logical but ontological possibilities. What is ontologically possible may not be logically possibly. Similarly, what is logically possible may not be biologically possible. Certain states of affairs may not clash with the laws of logic, but may clash with biological laws. There exists, therefore, a hierarchy of necessities. On the very top are the necessities of ontology. On the next layer appear the necessities of logic, set-theory, and arithmetic. And on lower levels we find the necessities of science. However, this picture gets more complicated if we add that there also exist “bridge laws” which establish connections among some disciplines. As a result, the necessities of one field of inquiry spill over into the necessities of another. We shall return to this point later on.

I have called attention to the necessity which flows from lawfulness because it sheds some light on Kant’s discovery of synthetic a priori truths. In Arnauld and Leibniz, as we have seen, the necessary is the analytic, and what is not necessary is synthetic. But consider the proposition that the sun will rise tomorrow. It is (a) not analytic and hence (b) synthetic; but it is also (c) necessary, for it follows from the laws of planetary motion. Thus there are synthetic propositions which are necessary, namely, all of those propositions which follow from laws other than the laws of logic, as understood by Arnauld and Leibniz. What happens to Kant’s newly discovered kind of necessity? Arithmetic propositions, Kant claims, are synthetic and necessary. Is this kind of necessity lawfulness? Does Kant mean to assert that arithmetic propositions, even though they are not logical truths, follow from some other kind of law? I do not think so. Otherwise, he would have to hold that the statement about the sun’s rising is just as synthetic a priori (necessary) as the propositions of arithmetic. No, Kant must have had some other kind of necessity in mind. And it is our task to hunt this sort of necessity down.

e) An Explication of the A Priori-A Posteriori
Distinction

A priori knowledge, Kant tells us, is knowledge absolutely independent of all experience (Kant, 1965, B 3). This is the bone of contention between us. We wish to maintain that a certain kind of necessary knowledge, a priori knowledge, depends on experience. There are at least two ways in which we may proceed. We could either try to show directly, as it were, that examples of Kant’s a priori knowledge do depend on experience, or else we could prove this indirectly by attacking Kant’s axiom that a truth which is necessary and universal cannot be built on experience. As I see it, Kitcher adopts the first strategy (P. Kitcher, The Nature of Mathematical Knowledge). I shall adopt the second. I shall attack Kant’s axiom by elucidating the twin notions of necessity and universality in such a way that it becomes clear that necessary and universal propositions rest on experience.

A priori knowledge is opposed to empirical knowledge, that is, to a posteriori knowledge. Kant states two criteria for a priori judgments:

Experience teaches us that a thing is so and so, but not that it cannot be otherwise. First, then, if we have a proposition which in being thought is thought as necessary, it is an a priori judgment; and if, besides, it is not derived from any proposition except one which also has the validity of a necessary judgment, it is an absolutely a priori judgment. Secondly, experience never confers on its judgments true or strict, but only assumed and comparative universality, through induction . . . If, then, a judgment is thought with strict universality, that is, in such a manner that no exception is allowed as possible, it is not derived from experience, but is valid absolutely a priori.

(Kant, 1965, B3-B4)

Necessity and universality are the characteristics of a priori judgments. A posteriori judgments are neither necessary nor universal. But what is this necessity and this universality that defines the a priori? We cannot possibly take a journey through the jungle of Kantian scholarship and Kant interpretations. But our goal is not to vindicate or condemn Kant, but to understand the true nature of our knowledge of arithmetic; and for this purpose, a concise nonhistorical discussion must suffice.

Let us look at the alleged necessity and universality of the judgment that 5 + 7 = 12. The first thing to notice is that this is not, as it stands, a general truth. It resembles the proposition that point B is between points A and C more than it resembles the paradigm that all bodies are extended. But we can supplant this arithmetic proposition by the general truth that five things plus seven things are twelve things. Kant, I shall assume, claims that this general judgment is not gotten by induction. To put it differently, Kant holds that this is a universal judgment, not just a general judgment. This judgment is not arrived at by induction from such individual instances as that these five oranges plus those seven oranges are twelve oranges, and that these five pencils plus those seven pencils are twelve pencils, etc., etc. Since it is not arrived at by induction, so Kant seems to argue, and since it is nevertheless “general,” it is universal.

Well, is it or is it not arrived at by induction? Consider the truth that all midnight blue things are darker (in color) than all lemon yellow things. Is this true generalization arrived at by induction? It seems to be quite obvious that it is not. It would not occur to us to confirm this law by looking at more and more midnight blue and lemon yellow objects. What we see is that midnight blue, this color shade, is darker than lemon yellow, that color shade. What we recognize is that the first shade stands in the relation of being darker-than to the second. There exists a relation between color shades, and this relation holds between midnight blue and lemon yellow. Since this relation holds between the two properties, it follows logically that anything which has the first property has a property which stands in this relation to the property which anything has that has the second property. More elegantly expressed, it follows from the fact that P stands to Q in the relation R, that anything which has P and anything which has Q are such that the properties they have stand in the relation R. Thus the truth of the general judgment is implied by the truth of the relational proposition. No induction is necessary or even possible. Yet, the truth of the general law rests on experience; for it rests on the truth of the relational judgment, and that proposition is known by experience: we see, with our very eyes, that midnight blue is darker than lemon yellow.

Now if we think of the arithmetic general truth in analogy to this color shade example, then we see immediately why the arithmetic law is universal and not just general. We understand why it holds for all things, even though it is not arrived at by means of induction from individual instances. Here, too, we have a relation among things: the sum relation among numbers. In this case, the relation holds not between properties, but between entities of a different category. But the principle is the same: Since this relation holds between these entities, it is true that all things are such that the relation holds between the numbers of these things. The proposition that five things and seven things are twelve things follows from the proposition that five plus seven is twelve. The general proposition is thus universal and not merely general. But, as in the first case, its truth derives from the truth of the proposition that five plus seven is twelve and is not arrived at by induction.

There is a difference between the color case and the number case. That midnight blue things are darker in color than lemon yellow things follows logically from the fact that midnight blue is darker than lemon yellow, for the following is a law of logic (of the general theory of properties and relations, of the “functional calculus”):

All properties of properties, all properties of individuals, and all relations between properties are such that: If a relation r² holds between two properties f¹ and g¹ which share a property h², then all individuals x⁰ and y⁰ are such that if x⁰ is f¹ and y⁰ is g¹, then there exist two properties i¹ and j¹ such that x⁰ is i¹ and i¹ is h², and y⁰ is j¹ and j¹ is h², and r² holds between i¹ and j¹.

There exists, to be precise, no such logical law for the number case, because numbers, as I shall argue, are not properties (or relations). But there exists a law that corresponds to this logical law:

All numbers and relations among numbers are such that: If a relation r holds between the numbers n₁, n₂, and n₃, then all things, e₁ e₂, and e₃ are such that if n₁ is the number of e₁ n₂ is the number of e₂, and n₃ is the number of e₃, then there exist the numbers m₁, m₂, and m₃, such that r holds between m₁, m₂, and m₃, and m₁ is the number of e₁, m₂ is the number of e₂, and m₃ is the number of e₃.

We see why a well-known argument for the analyticity of the arithmetic proposition fails. According to this argument, the proposition must be analytic because it is not an empirical generalization. It is not an empirical generalization because there can be no disconfirming instances (see, for example, Carl G. Hempel, “On the Nature of Mathematical Truth,” pp. 367-68). Of course, there can be no disconfirming instances. Nobody in his right mind would try to disprove the law that midnight blue things are darker in color than lemon yellow things by trying to find a midnight blue thing and a lemon yellow thing such that the first is not darker in color than the second. But it is also true that nobody would try to confirm this law by citing more and more instances of midnight blue and lemon yellow individuals. No, this law is neither established nor can it be discredited by individual cases. In this regard, it is indeed fundamentally different from “ordinary, inductive” laws. This law follows logically from a nongeneral fact, namely, from the fact that a certain relation holds between the two colors midnight blue and lemon yellow. And the same is true for the arithmetic law under discussion. It, too, follows from a nongeneral relational fact about the numbers five, seven, and twelve. Hempel makes the simple mistake, a mistake quite common among logical positivists, of assuming that a law which is not established by induction (and therefore cannot be discredited by a counter instance) must be analytic.

We now face the crucial question: Is the arithmetic proposition that five plus seven is twelve based on experience? I shall argue that it is, thus extending the analogy between the color case and the number case. But let us first return to Kant. We have just seen that the arithmetic law, just as the law about colored objects, is indeed universal. Kant is quite correct in his claim that there are general propositions which are not arrived at by induction. But we have also seen that universality can be based on experience. Kant is mistaken about the source of this universality. Since he did not realize that relational facts can yield the universality of general propositions, he searched for a different source for universality. He found it, as we know, in the pure intuitions of space and time. Since this is such a crucial turning point of Kant’s philosophy, let me diagnose the situation once again. From my point of view, Kant saw correctly that certain judgments are universal; that, though general, they are not based on induction. They have a “necessity” about them which induction cannot provide. What Kant did not see was that there are relations among properties and other kinds of nonindividuals, and that these relations account for the universality of those judgments. Put in a more Kantian terminology, his greatest mistake was that he did not realize that there exist relations (other than partial identity) among concepts. Had he realized this fact, there would have been no “Transcendental Turn.”

We have discovered the source of the universality of a priori judgments; what about their necessity? Of course, our arithmetic law is trivially necessary in the sense which we have explicated in the last section. But this is not the necessity which Kant is trying to explain. In his sense, the arithmetic statement is necessary, but a biological law is not. I think that this kind of necessity is a matter of imaginability. We cannot imagine five things and seven things not being twelve things, and that is why we hold it to be necessary that five things and seven things are twelve things. And we cannot imagine midnight blue not being darker than lemon yellow, and that is why we also say that it is necessary that blue things are darker than lemon yellow things. On the other hand, since we can imagine that a horse has a horn, we do not believe it to be necessary that horses are without horns. Imaginability is the second important source of necessity, in addition to lawfulness. If I am correct, then the a priori is characterized by a universality that flows from certain relations among abstract entities, and by a necessity that derives from unimaginability. This is my alternative to Kant’s transcendental idealism.

Synthetic a priori judgments, according to Kant, must be based on intuitions rather than concepts. The connection between the subject concept and the predicate concept must somehow be established through intuitions, that is, through individuals. But the universality of such judgments precludes the possibility that this connection is established by means of induction from a number of individual cases. How, then, can the universality and necessity of this connection be explained? This is Kant’s most pressing problem. His solution is infamous. He postulates a mysterious kind of intuition, pure intuition, which (a) establishes a connection between the respective concepts, and (b) guarantees that this connection holds universally (without induction) and is necessary. This is obviously one of the weakest spots of Kant’s philosophy. It is not surprising that Kant’s great critic Bolzano attacked the notion of pure intuition, as we shall see later on.

f) Kant and the Concept of Number

Arithmetic is about numbers. What are numbers and how are we acquainted with them? Let us start with simple things, the positive integers (natural numbers). What kind of thing is the number two and how do I know it? Kant is primarily interested in the epistemological question. He does not say much about the ontological problem. However, his explanation of how we know numbers is notoriously obscure. Somehow, pure intuition is involved, and so is time. We have already ventured a guess as to why Kant invents pure intuition. He may have reasoned like this. Arithmetic truths are synthetic a priori. Since they are synthetic, they cannot be conceptual truths; for all such truths are analytic. Thus they must rest on intuition. But they cannot be based on ordinary intuition, for such intuition can only yield a posteriori knowledge. Hence there must be an intuition which is unique in that it can deliver universality and necessity. This is pure intuition.

Intuition, as I pointed out, plays a double role in Kant’s philosophy. It is both an epistemological as well as an ontological notion. It is both a kind of mental act, sensibility, and a kind of object (individuals). The riddle of pure intuition, therefore, has two sides to it. What is this special kind of mental act; how is it distinguished from ordinary intuition? And what is this special kind of object; how is it distinguished from ordinary individuals? What, for example, is the object of pure intuition in geometry? I find no plausible answer to this urgent question in Kant. Nor do I find one for the case of arithmetic. Kant’s penchant for symmetry dictates that time must play a role in arithmetic, for does not geometry deal with space? But what kind of temporal object could possibly play a role in the theory of numbers?

What does Kant say about numbers? Here is a famous passage:

We might, indeed, at first suppose that the proposition 7 + 5 = 12 is a merely analytic proposition, and follows by the principle of contradiction from the concept of a sum of 7 and 5. But if we look more closely we find that the concept of the sum of 7 and 5 contains nothing save the union of the two numbers into one, and in this no thought is being taken as to what this single number may be which combines both. The concept of 12 is by no means already thought in merely thinking this union of 7 and 5; and I may analyze my concept of such a possible sum as long as I please, still I shall never find the 12 in it. We have to go outside these concepts, and call in the aid of the intuition which corresponds to one of them, our five fingers, for instance, or, as Segner does in his Arithmetic, five points, adding to the concept of 7, unit by unit, the five given in intuition. For starting with the number 7, and for the concept of 5 calling in the aid of the fingers of my hand as intuition, I now add one by one to the number 7 the units which I previously took together to form the number 5, and with the aid of that figure see the number 12 emerge [Kemp Smith has: “come into being”]. That 5 should be added to 7, I have indeed already thought in the concept of a sum = 7 + 5, but not that this sum is equal [Kemp Smith: “is equivalent to”] to the number 12.

(Kant, 1965, B15-B16)

Kant speaks here of the concept of the sum of 7 and 5. With our realistic bias, ‘concept’ translates into ‘property’: There is a property, in other words, of being the sum of 7 and 5, and a certain number has this property. More accurately, from our point of view, there is a description (to be distinguished from a description expression): The number which is the sum of 7 and 5. This is a relational description: A number is described as the number which stands in the sum relation to the two numbers 7 and 5. We may write this perspicuously as: The n such that: Sum (n, 7, 5). Assume that there are a number of peas lined up on a table, and let us call the peas “a,” “b,” “c,” etc. We can now describe one of the peas, say b, as the pea which lies between the peas a and c: The x such that: Between (x, a, c).

Kant argues that the concept of the sum of 7 and 5 does not contain the concept of 12. Notice that he seems to think of the number 12 as a concept. Extrapolating in our usual uninhibited way, we conclude that for Kant numbers are concepts. (What else could they be in an ontology as poverty stricken as Kant’s? Individuals, located in space and/or time? Judgments?) Removing the idealistic bias: Numbers are properties. But if they are properties, then we must immediately ask with Frege: What are they properties of? We know from Frege’s brilliant analysis that this question is hard to answer. But let us not get ahead of ourselves. Kant maintains, at any rate, that the property of being twelve is not contained in the property of being the sum of seven and five. At this point, our terminological adjustments become crucial. The number twelve, I hold, is not a property. Therefore, it cannot be contained as a part in another property. But we can reformulate Kant’s point within our framework: Does the description the number which is the sum of 7 and 5 describe something in terms of the number 12? Obviously not. Compare this case with the question of whether the description the unmarried cousin of mine describes somebody in terms of the property of being unmarried. In this case, the answer is obviously affirmative. We think therefore of the statement that the unmarried cousin of mine is unmarried as analytic. (Never mind, for the moment, that this statement has “existential import.”)

Is the following statement analytic: The number which is the sum of 7 and 5 is the same as 12? We have here a description of a number in terms of the sum relation which this number has to two other numbers, and we have also a name for this number. This case is precisely like our pea example: Is the statement that pea b lies between peas a and c analytic? Of course, not! Why, then, would anyone think that the arithmetic truth is analytic?

Kant appeals to intuition in order to establish a connection between the concept of the sum and the concept of 12. The reasoning behind this move is clear: Since we cannot establish a connection between these two concepts directly, we have to connect them indirectly by means of a common individual. But even though the principle of the philosophical move is clear, its actualization is shrouded in mystery. Why do we need an intuition for the number 5 (the five fingers or points), but not for the number 7? Or do we also have to look, first, at seven fingers? And how, precisely, is the addition of units to be understood? Do we move fingers over? Or do we open them up, one by one? And how does the number 12 emerge? Do we have to look at twelve fingers? Do we need someone else for this intuition, someone who lends us two fingers? Or can we use two of our toes? But let us give Kant a run for his money and try to give an account which, at least, is in the spirit of Kant’s analysis. Take the simple case of the sum of 2 and 1. In order to realize that 3 is the sum of 2 and 1, we must use intuition, since the concepts are not contained in each other. We turn to two of our fingers. These two fingers, we assume, form some kind of individual which falls under the concept 2 (which has the property 2). Now we add one finger, that is, an individual thing which falls under the concept 1. What results is a third individual, namely, the three fingers in close proximity. And now we notice that this individual thing falls under the concept 3 (that is, has the property 3). Thus we learn that 2 plus 1 is 3. We learn that the three concepts are connected with each other in this fashion.

Even this account has many flaws. Firstly, there is again the Fregean question of what, precisely, falls under the concepts 2 and 3, respectively. Secondly, there is the problem of what the addition of fingers could possibly be. Thirdly, there is the Kantian objection that in this way we can learn, at best, that these two fingers and that one finger are these three fingers, but not that two plus one is three. We have not even learned that any two fingers plus one finger are three fingers. To arrive at this conclusion, we obviously need induction. But arithmetic, according to Kant, is not a matter of induction. The problem is, of course, that in our example we have used “empirical” intuition. Kant knows that this will not do. Our story provides, at best, an “empirical” account of arithmetic, and not the “transcendental account” defended by Kant. To get a transcendental view, pure intuition and time must enter into the picture. But the quotation from Kant gives us no clue whatsoever how they enter.

g) A Preview

According to Kant, arithmetic truths are not analytic and yet necessary. If arithmetic truths were analytic, then their necessity would be explained: their necessity would then be the necessity of logic. But they are not analytic. From where, then, do they get their necessity? What kind of necessity is the necessity of arithmetic? This is the dialectic that has confronted philosophers since Kant.

The most obvious response to Kant, a response made famous by Frege, is to deny Kant’s claim that the necessity of arithmetic is not the necessity of logic. Frege maintains that, contrary to Kant’s view, arithmetic is analytic. Kant thought otherwise, so the objection goes, because of his limited understanding of logic. If one has a firm grasp of the variety of logical truths, then one can show that arithmetic can be “reduced” to logic. Arithmetic, one can show, is merely logic in verbal disguise. But this response fails, I shall argue, because it rests on a tainted notion of definition. The so-called definitions of the reduction are not mere abbreviational conveniences, but bridge laws disguised as conventions.

In addition to the logicist’s response to Kant’s thesis, there is what I shall call the “semantic” theory. The necessity of arithmetic, according to this view, is indeed not based on logic, but is a matter of meaning. Meaning is the source of a special kind of necessity: some statements are necessarily true by virtue of the meaning of their terms. What Kant did not realize, it is said, is that in addition to empirical truths and purely logical truths, there exist also semantic truths. These truths rest not on inclusions among concepts or on pure intuitions, but on relations among meanings. That midnight blue is a color is not analytic, because the concept of color is not a part of the concept of midnight blue, but it is not an empirical truth either. Rather, it belongs to a third and peculiar kind of truth, namely, semantic truths, that is, truths that rest on certain meaning relations among concepts. Actually, there are two versions of this semantic theory. According to an ontological version, meanings are entities dwelling in Plato’s realm of being. This is Husserl’s view. According to a linguistic version, meanings are linguistic practices; they are rules of use. This, of course, is the Wittgensteinian position.

Last, but by no means least in popularity, is the view that the necessity of arithmetic derives from convention. According to the conventionalist theory, truths divide into two large groups, empirical truths and conventional truths. Conventional truths play then the role of analytic truths in other philosophies. The verbal bridge that leads from necessity to conventionality is easy to spot: What is true by convention could not possibly be false; it is true, irrespective of what the world is like. But what could not possibly be false, no matter what the world is like, is necessary. This view has been adopted by many Logical Positivists in the face of the Kantian challenge. When one reads the literature of the Vienna Circle, one finds a curious shift back and forth between some kind of semanticist view and some version of the conventionalist position, and one gets the impression that these philosophers were not really happy with either view, but could find no better response to Kant’s challenge.

These three alternatives to Kant’s explication of arithmetic necessity agree with the traditional wisdom that arithmetic is necessary. While Kant grounds this alleged necessity in pure intuition, they try to ground it in logic, semantics, or convention, respectively. In addition to these alternatives, however, there are two more radical objections to Kant’s analysis.

Firstly, there is the empiricist denial that arithmetic is a priori and hence universal and necessary. Arithmetic truths, according to this view, are not different from other “empirical laws.” They are not, in essence, different from the laws of physics or from the laws of biology. They are justified, like the laws of science, by observation and induction. This is Mill’s answer to the Kantian challenge. Contrary to received opinion, there is much to be said for Mill’s empiricism. The laws of arithmetic (even the laws of logic) are indeed in some sense empirical. But Mill is mistaken, I shall argue, when he puts them side by side with the laws of science. Knowledge of arithmetic, as all knowledge, is a matter of perception and introspection, but not all perceptions are perceptions of individual things and their properties.

Secondly, there is a radical response to Kant associated with Wittgenstein’s early philosophy. According to this view, logical truths are “empty tautologies.” They say nothing about the world. They are necessary in the perverse sense that they are vacuous. To some philosophers, this conception of logic seemed to offer a way out of the problem posed by Kant: if logic is necessary because it is vacuous, perhaps arithmetic is necessary for the same reason. I shall let Carnap describe this line of reasoning:

Wittgenstein formulated this view in the more radical form that all logical truths are tautological, that is, that they hold necessarily in every possible case, therefore do not exclude any case, and do not say anything about the facts of the world.

. . . he did not count the theorems of arithmetic, algebra, etc., among the tautologies. But to the members of the Circle there did not seem to be a fundamental difference between elementary logic and higher logic, including mathematics. Thus we arrived at the conception that all valid statements of mathematics are analytic in the specific sense that they hold in all possible cases and therefore do not have any factual content.

What was important in this conception from our point of view was the fact that it became possible for the first time to combine the basic tenet of empiricism with a satisfactory explanation of the nature of logic and mathematics.

(R. Carnap, “Carnap’s Intellectual Autobiography,” in

The Philosophy of Rudolf Carnap, pp. 46-47)

To us, this is a funny kind of empiricism indeed. The laws of science, upon this view, are empirical. But the statements of logic, arithmetic, etc. are turned into nonstatements, so that the question of how we know these truths simply cannot arise. This is the reason why I called this response to Kant’s challenge “radical.” It solves Kant’s problem by denying that there is a problem; and there is no problem, because there are no statements of arithmetic. Since there are no such statements, no such truths, the question of whether they are a priori or a posteriori cannot arise.

This preview of some of the recent and contemporary attempts to answer Kant reminds us of the fact that his account of arithmetic takes for granted that the necessity peculiar to logic poses no philosophical problem. We now know better, and so do most contemporary philosophers. Our inquiry into the nature of the necessity of arithmetic leads inevitably to an even deeper inquiry into the necessity of logic. It is not simply a question of how we come to know the truths of arithmetic, but we must also answer the question of how we come to know the truths of logic. Rationalism must be overcome, not only in regard to arithmetic, but also in regard to logic. We shall therefore turn next to one of the most astute philosophers of logic and one of the greatest critics of Kant: Bernard Bolzano.