a) The Rejection of Pure Intuition

Earlier, I spoke of escaping from Plato’s cave and compared Reid’s philosophy to a bright spring morning. A similar experience awaits us if we turn from the jungle of Kant’s system to the clear world of Bolzano’s philosophy. To my mind, Bolzano is one of the greatest philosophers of the 19th century. A contemporary of Hegel, Schelling, and Kierkegaard, he was at once one of the greatest mathematicians and one of the most profound metaphysicians of his day.

Bolzano saw clearly that Kant’s explanation of the synthetic a priori character of arithmetic in terms of pure intuition is unacceptable. In a remarkable paper of 1817, he gives a demonstration of how intuition, empirical or pure, can be eliminated from a mathematical proof (B. Bolzano, Rein analytischer Beweis . . .). Bolzano considers the problem of how we know that a continuous function, taking values both below and above zero, must take a zero value in between. The problem is to be solved by appealing only to basic assumptions concerning numbers and functions. In the process of solving the problem, Bolzano gives the first clear explanation, as far as I know, of the epsilon-delta description of continuity. In response to Kant, he raises the same objection we made earlier:

Kant poses the question, “What justifies our understanding in assigning to a subject a predicate which is not contained in the concept (or explanation) of the former?”—And he thought he had discovered that this justification could only be an intuition that we link with the concept of the subject and which also contains the predicate. Thus, for all concepts from which we can construct synthetic judgments there must be corresponding intuitions. If these intuitions were always merely empirical the judgments which they mediate should also be empirical. Since nonetheless there are synthetic a priori judgments (as those undeniably contained in mathematics and pure natural science) there must also be a priori intuitions, however odd this might sound. And once one has decided that there can be such, one will also convince oneself easily that for the purposes of mathematics and pure natural science that time and space are these intuitions.

(Quoted from J. A. Coffa, “To the Vienna Station,”

unpublished manuscript, chap. II)

Bolzano’s opposition to Kant’s doctrine started very early. Here is how he describes it, speaking of himself in the third person:

From very early on he dared to contradict him directly on the theory of time and space, for he did not comprehend or grant that our synthetic a priori judgments must be mediated by intuition and, in particular, he did not believe that the intuition of time lies at the ground of the synthetic judgments of arithmetic, or that in the theorems of geometry it is allowable to rest so much on the mere claim of the visual appearance, as in the Euclidian fashion.

(Quoted from Coffa, chap. II)

In paragraph 305 of his Wissenschaftslehre, Bolzano gives a detailed criticism of Kant’s mysterious notion of pure intuition (B. Bolzano, Wissenschaftslehre). Let us follow him step by step. (1) Bolzano points out that the distinction between two kinds of intuition, empirical and pure intuition, does not by itself explain how pure intuition gives rise to synthetic a priori judgments. And he complains that not a single passage in all of Kant’s work makes this point clear. (2) However, from some passages it is obvious that Kant believes he has discovered an important difference between an intuition which results from the actual drawing of, say, a triangle, on the one hand, and the intuition which occurs in mere imagination (Einbildungskraft), on the other. Bolzano thinks that this distinction is rather unimportant. The only difference between the two is that the former is more vivid and can more easily be fixed and reproduced. (3) Kant himself seems to have noticed this and therefore claims sometimes that it is not just imagination, but pure imagination which plays a role. Bolzano comments that this amounts to playing hide and seek with an obscure term. We must ask: What is pure imagination and how does it differ from empirical imagination? (4) Kant answers that empirical imagination merely produces pictures, while pure imagination produces schemata, and that schemata are not pictures, but rather ideas of methods or rules for the creation of a picture for a given concept. But how, asks Bolzano, can the idea of a method be called an intuition? How, moreover, does the idea of a method to provide a picture for a given concept differ from a so-called genetic definition of the concept? For example, if the schema of a circle is merely the idea of how one provides an object for the concept of a circle, then the schema is nothing else but the idea of how a circle is generated, that is, nothing else but the familiar description of a circle as a line which a point travels if it moves in a plane in such a way that it remains at a constant distance from another point. Bolzano concludes with these words: “If one therefore recognizes the truth of a synthetic judgment from a consideration of the schema of its subject concept, then one recognizes its truth from a consideration of a mere concept.” Never before and never since, I think, has this point been made so concisely and decisively.

b) Bolzano’s Notion of Number and the Beginnings
of Logicism

From geometry, Bolzano turns next to arithmetic. He explains what numbers are and claims that arithmetic is analytic rather than synthetic.

According to Kant, we add the units of the number five to the number seven step by step, and this shows that the addition must be based on an intuition of time. Bolzano exlaims: “What an inference! One may as well reason that every sorites is based on the intuition of time, since we arrive at the conclusion only in the course of time.” But more important than Bolzano’s scorn is his attempt to prove that the addition in question is an analytic truth. His discussion contains the program of logicism in a nutshell. Bolzano claims, firstly, that the equation:

a + (b + c) = (a + b) + c

is analytic and follows from the explanation (Erklaerung) of a sum: “By a sum we think of a totality (Inbegriff) such that the order of its parts are not attended to, and the parts of the parts are considered to be parts of the whole” (cf. G. Frege’s The Foundations of Arithmetic, p. 8, where Frege discusses a similar ploy by Grassmann). Secondly, he maintains that from this analytic equation and the explanation that 7 + 1 =8, 8 + 1 = 9, etc., the proposition that 7 + 5 = 12 emerges as a purely analytic truth. I think we may put it this way: From the definition of a sum and the definitions of the numbers as sums, it follows logically that 7 + 5 = 12.

As always when we consider a version of logicism, the crucial notion is that of a definition. It is clear that if Bolzano’s alleged definitions (Erklaerungen) are anything other than either logical truths or mere linguistic conventions, then the result of employing them cannot be an analytic truth.

Bolzano’s sums are wholes of a certain sort. They are not sets, for the members of the members of a set are not members of the set, as Bolzano stipulates for sums. In order to remove all temptation to think of these wholes as arithmetic sums in the ordinary, pre-philosophical, sense of the term, let me call this kind of whole, the kind for which the associative law is stipulated to hold for its “generating relation,” an “aggregate.” Are there such aggregates? I think that spatial wholes are aggregates if, and this is an important if, we take as the crucial relation the relation of spatially-consisting-of. Consider the patterns of a chessboard. The squares that form the rows of the board are parts of the rows, and since the rows are parts of the board, the squares are also parts of the board. Notice that the squares are not members of the set consisting of rows, even though they are members of the sets consisting of squares from rows. (Nelson Goodman would therefore embrace Bolzano’s aggregates, even though he abhors sets.)

Bolzano’s claim that arithmetic is analytic rests, as we have just seen, on the following assumptions:

In the manuscript Reine Zahlenlehre (Pure Science of Number), Bolzano explains his view further:

If we form a series such that its first member is a unit of an arbitrary kind A, every other member, however, is a sum which appears if we connect an object which is equal to the preceding member with a new unit of kind A, then I call every member of this series insofar a number, as I conceive of this member by means of an idea which shows its manner of originating. In order to distinguish [this series] from other series which appear if one takes instead of things of kind A things of another kind as units, I call the members of the series we considered earlier numbers of kind A or numbers which are based on the unit A. The property through which every one of these members becomes a number (which it therefore retains however one may change the things themselves which one takes as units), I call a number in the abstract meaning of this term, or an abstract number; and in contrast to such abstract numbers (that is, the mere properties), I call the members themselves concrete numbers in the concrete meaning of the term.

(My translation, quoted from the German in J. Berg, Bolzano’s Logic, pp. 165-66)

Consider the kind pea, corresponding to kind A in Bolzano’s explanation, and take a particular pea P₁. P₁ is the first member of the following series. We take another object of the sort pea, equal to P₁, call it “P₂,” and connect it with another pea, say, P₃. This gives us the aggregate consisting of P₂ and P₃. This aggregate is the second member of our series. In order to form the third member, we take an aggregate equal to the previous one, say the aggregate consisting of the two peas P₄ and P₅, and add another pea, so that we get the aggregate consisting of P₄, P₅, and P₆, and so on. Next, we consider the properties which the respective members of different series have in common, namely, such properties as being an aggregate consisting of a member which is equal to the first member of a series and of another member. This property, for example, is the abstract number two.

It is important to realize that we can start with any property in order to form a series of that sort. The things which are A need not even be individual things; they could be properties themselves. Nor, of course, do they have to be spatio-temporal. For example, Bolzano holds that his Saetze an sich (propositions) are not “real,” are not spatio-temporal. Yet they can be numbered. Aggregates, therefore, are not spatio-temporal heaps. The crucial relation, characteristic of aggregates, cannot be the relation of spatially-consisting-of. We must not be misled by Bolzano’s talk about “concrete” numbers. A certain aggregate may consist of three propositions. But aggregates are not sets either, as we noted earlier. This is an important difference between Bolzano’s conception of number and many later ones, especially Cantor’s. Cantor explains his notion of cardinal number in this way: “I call ‘cardinal number [Maechtigkeit] of a multitude [Inbegriff] or set of elements’ (where the latter may be homogenous or not, simple or complex) that general concept [Allgemeinbegriff] under which all and only those sets fall which are similar to the given one” (G. Cantor, Gesammelte Abhandlungen, p. 441). According to this explanation, the number three, for example, is a property (or concept) which all sets share which are similar to a set of three peas.

Aggregates, then, are neither spatio-temporal wholes nor are they sets. What are they? What amounts to the same: What is the characteristic relation of an aggregate?

At this point, we can adopt either one of two attitudes, depending on whether we take seriously Bolzano’s talk of a sum or else his story about what constitutes a concrete number. Let us adopt the latter first. From this point of view, concrete numbers are a certain kind of whole, and the puzzle is what kind they are. They cannot be spatial wholes, nor can they be sets. We are merely told that the “internal arrangement” of their parts does not matter and that the “consists of’ relation is transitive. I cannot think of a kind of whole that fulfills these two conditions (and that consists of peas or of propositions). I do not think, therefore, that there are such wholes. I do not think that there are aggregates of the sort required by Bolzano’s theory of number.

I know the familiar objection to my conclusion. It will be said that an aggregate is simply anything that fulfills the two conditions just mentioned, so that a spatial whole, for example, is an aggregate. But this will not do. My point is not that the alleged aggregate consisting of three propositions is not a spatial whole, but rather that there is no such aggregate at all. To see this, we merely have to ask what kind of whole these three propositions are supposed to form. I can think of no reasonable answer to this question. Nor will it do to “postulate” aggregates for the cases where there are none. Whether or not three propositions form a certain sort of whole is not up to us; it is not a matter of fiat.

On the other hand, our emphasis may be placed on Bolzano’s talk about sums. From this perspective, aggregates are simply sums understood in the familiar arithmetic way. So conceived of, a spatial whole is not a sum of its spatial parts. But then the only sums there are, are numbers. There are no sums of peas or sums of propositions. Consider the assertion that the sum of two peas and one pea is three peas (or that two peas plus one pea are three peas). Now this, as I said earlier, is a true statement. Furthermore, its truth follows from the fact that two plus one is three (that the sum of two and one is three). The sum relation, according to my analysis, holds between these three numbers, not between “aggregates” consisting of peas. To say that the sum of two peas and one pea is three peas, is to say, more perspicuously, that the number of peas which is the sum of the number of two peas and the number of one pea is three. Compare this once again to the color example. To say that midnight blue sweaters are darker than lemon yellow blouses is to say, not that the relation of darker than holds between sweaters and blouses, but rather that midnight blue sweaters have a color that is darker than the color of lemon yellow blouses. Bolzano’s view, transferred to the color example, would come to this: There are certain wholes, for example, blue sweaters, yellow blouses, green apples, red flags, etc., and the relation of darker than holds between these wholes. These wholes are the “concrete colors.” When we abstract from the particular objects that form these wholes, that is, sweaters, blouses, apples, flags, etc., we get “the abstract colors.” Our analysis precisely reverses this process.

That there is something wrong with Bolzano’s notion of a concrete number can be seen when we return to his description of how a series of such numbers is formed. Starting with one pea, the next member of the series consists of an object which is “equal” (gleich) to this one pea and of another pea. What does the word ‘equal’ here mean? It does not mean identical or same. If it did, then Bolzano could have said that the next member of the series consists of the first member (the first pea) and another pea, but he does not. The most plausible interpretation is that ‘equal’ means equal in number. We take an object that is equal in number to the previous object—in our case, one other pea—and add one more thing of the same sort, namely, another pea. But upon this interpretation, we are obviously using the concept of number in order to explain what a number is. There is one other possibility that I can see. ‘Equal’ could mean isomorphic to. We are then instructed to take an object which is isomorphic to the previous object of the series and add one more object of the same sort. We are instructed, in other words, to pick an object whose parts can be coordinated one to one to the parts of the previous object of the series. Put still differently, we are invited to consider a set of things isomorphic to the previous set. Upon this interpretation, Bolzano’s aggregates are really sets (“the order of its parts are not attended to”), and we must disregard his assertion that “the parts of the parts are considered to be parts of the whole.”

Abstract numbers are presumably formed when we abstract from the particular properties (being a pea, being a proposition, etc.) which characterize the parts of the aggregates of the various series, and pay attention only to how these series are created. Consider an aggregate of two peas and an aggregate of two propositions. What do these two aggregates have in common? Well, they both consist of the first member of a series plus one other member. And they share this property with all other two-part aggregates: they all share the property of consisting of one of a kind and one other thing of the same kind. This property, it seems, is the number two. In short, the number two, according to Bolzano, is the property of consisting (of being the sum) of one and one. The number three is the property of consisting of two and one. And so on. It is clear that this conception rests on two notions, namely, on the notion of the (abstract) number one (the “unit” of kind A) and the conception of a consisting-of relation (the sum relation) among numbers. In the end, then, Bolzano’s analysis may perhaps be summarized as follows. Numbers, with the exception of the number one, are properties of certain wholes. The number one is a property of a unit of any sort. The other numbers are sums of numbers. For example, the number two is the sum of the unit and one other thing.

c) Analyticity and Form

Bolzano’s notion of analyticity is at once brilliant and idiosyncratic. I shall concentrate on those parts of it which have endured. Just as Bolzano’s concept of number was to become the first in a line of modern attempts to give an ontology of arithmetic, so has his notion of analyticity turned out to be the forerunner of several contemporary explications of analyticity in terms of form.

Consider the proposition:

The man Caius is mortal,

and think of the idea Caius as being arbitrarily changeable, so that in its place other ideas occur; for example, the ideas Sempronius, Titus, rose, triangle, etc. The resulting propositions are true as long as their subject ideas have objects. The proposition The man Titus is mortal is true, and so is the proposition The man Frege is mortal. On the other hand, The man rose is mortal is not even a proposition, since its subject idea, the man rose, has no object. And the same holds for expressions like The man plus is mortal’, ‘The man neither is mortal’, ‘The man Paris is mortal’, etc. In short, all the propositions derived from the given one which substitute for the idea of Caius a “proper” idea are true. Next, consider the proposition:

The man Caius is omniscient.

You can easily see that all of the propositions derived from this one by “proper” substitution are false. The proposition itself is false, and so are such propositions as The man Titus is omniscient and (even) The man Frege is omniscient. Finally, consider the proposition:

The being Caius is mortal.

If we substitute for the idea Caius the idea Titus, we get a true proposition, but if we substitute for it the idea God, we get a false proposition.

Here is another of Bolzano’s examples. He claims that the proposition:

This triangle has three sides

yields only true propositions as long as we treat only the idea this as a variable. What he has in mind, I think, is that a proposition like That triangle has three sides is true, while something like Caius triangle has three sides has a subject idea which lacks an object. I shall assume that this last string of words does not really represent a proposition at all. Given this assumption, we can describe Bolzano’s view as follows: If we think of an idea in a proposition as variable and replaced by other ideas, but in such a way that the result is a proposition and not nonsense, then there are three possible cases: (1) All the resulting propositions are true, (2) all of them are false, and (3) some are true and some are false. It is clear that there is no proposition such that we can change all of its ideas arbitrarily and get only true or only false propositions; for, as Bolzano remarks, if we can change all of the ideas, then we can turn any proposition into any other proposition and hence get some true propositions and some false propositions (Bolzano, 1981, par. 148).

We come now to Bolzano’s notion of analyticity:

But if there is even a single idea in a proposition which can be varied arbitrarily without destroying its truth or falsety . . ., then this characteristic of the proposition is peculiar enough to distinguish it from all of those [propositions] for which this is not the case. I allow myself, therefore, to call propositions of this sort, with an expression borrowed from Kant, analytic, all others, however, . . . synthetic.

(Bolzano, 1981, par. 148)

Bolzano then gives the following two examples of analytic propositions:

A morally evil person does not deserve respect,

and

A morally evil person may still enjoy continued happiness.

These two propositions are analytic, because they contain the idea person (Mensch) which, if varied, yields nothing but true propositions in the first instance and nothing but false propositions in the second. In effect, these two propositions are analytic because they are of the forms:

A morally evil X does not deserve respect,

and

A morally evil X may still enjoy continued happiness.

Bolzano later on gives the following examples of true logically analytic propositions:

(1) A is A,

(2) A, which is B, is A,

(3) A, which is B, is B,

(4) Every object is either B or not B.

And then he adds:

These examples of analytic propositions which I have just mentioned differ from those in number 1 [about the evil person] in that a judgment concerning their analytic nature necessitates no other but logical knowledge, since the concepts which form the invariable part of these propositions all belong to logic . . .

This difference, however, is not strict, since the domain of concepts which belong to logic is not so sharply delimited that one could never raise a controversy.

(Bolzano, 1981, par. 148)

Let us set aside this reservation and try to fully understand Bolzano’s notion of logical analyticity. As our example we shall take the proposition: Males, which are unmarried, are unmarried. This proposition is logically analytic, as I understand Bolzano, because it is of a certain form, namely, of the form: F’s, which are G’s, are G’s. This form is such that however we vary F and G, the result is always a true proposition. Furthermore, F and G are the only nonlogical ideas which can occur in propositions of this form. In a nutshell, our example is a logical analytic truth in virtue of its form.

In paragraph 186 of the Wissenschaftslehre, Bolzano discusses the quite different view that logic is a matter of form, not of content. He is especially critical of the view that all of the ideas which occur in a judgment belong to the content of the judgment. This view implies that “logical” words like ‘and’, ‘or’, ‘if’, ‘then’, etc. do not stand for ideas. Bolzano objects that one can only describe the form of a judgment by specifying that certain ideas are connected in this and no other way, precisely because a certain concept, the concept and, for example, connects them. So-called logical terms, he insists, represent concepts just as much as “ordinary” expressions. Logic, therefore, could not possibly be characterized by distinguishing between “descriptive” expressions which represent something and “logical” expressions which do not. Bolzano, I submit, has a decisive argument against such a conception of the nature of logic. If ‘and’ and ‘or’ have no “meaning” (if they do not represent relations, if they do not express ideas, or however you may wish to put it), then why can we not interchange these words willy-nilly in sentences? Surely, these words contribute just as much to the “meaning” of a sentence as other words do. It is therefore simply false to claim, as Quine does at one place, that logical truths are characterized by the fact that they are “invariant under lexical substitutions” (W. V. O. Quine, Philosophy of Logic, p. 102).

Bolzano concedes that there is a sense in which logic may be said to be a matter of form. In logic, he maintains, we are interested not in individual propositions, but in kinds of proposition. We divide propositions into groups, according to certain properties: “If one permits oneself to call such properties the common forms of these propositions, that is, their configuration (Gestalt), then one can rightly assert that all the divisions of propositions occurring in logic concern only their form, that is, concern only that which several, even infinitely many, propositions have in common” (Quine, 1970, p. 102). I do not think that propositions of the same form share a common property. In order to fully understand the fashionable view that logic is a matter of form, we must be clear about the ontological status of form. After we have cleared up this matter, we shall return to our main argument and explain in what sense logic is and in what sense it is not a matter of form.

d) The Ontological Status of Form

Let us return to our example: Males, which are unmarried, are unmarried. This proposition, I said, is of the form: F’s, which are G’s, are G’s. What, precisely, does it mean to say that a proposition is of a certain form? What is a form? What are those F’s and G’s in the form? In the world, there are no F’s and G’s; Bolzano would say that there are no ideas corresponding to ‘F’s’ and to ‘G’s’. There are only the expressions. Let us therefore shift from the form to the expression for the form: ‘F’s, which are G’s, are G’s’. What does this expression represent? The short answer is: Nothing! There is no entity, the form of a proposition, which is represented by this expression. It represents neither a property nor, in Bolzano’s terminology, an idea. However, there do exist a number of propositions (states of affairs, facts, circumstances) which are partially identical with the proposition of our example. There are propositions of this form. If we wish to talk about propositions which are partially identical with the given proposition, then we may use “schematic letters” like ‘F’ and ‘G’—or any equivalent device—in order to indicate what propositions we have in mind. For example, if we wish to talk about all of the propositions that start with a plural property idea, followed by the logical idea(s) which are, followed by another property idea, and so forth, we can do so more conveniently by mentioning the form: F’s, which are G’s, are G’s. Expressions for forms allow us to abbreviate talk about groups of complex things which are partially identical. It is important to notice that things of the same form do not, precisely speaking, share a common property, as white billiard balls share a common color. Rather, what they have in common is that they are partially identical. This partial identity can only be described in the cumbersome way of which we gave an example.

Expressions for forms contain “schematic letters.” Schematic letters are not “free variables.” They acquire their significance in the manner just explained: They serve to depict a partial identity among complex things. So-called free variables are either not variables at all, for example, when one mistakenly calls schematic letters “free variables,” or else they are bound variables in expressions that have been amputated. If we start out with the expression:

‘All f’s and all g’s and all x’s are such that: If x is f and x is g, then x is g’ and then drop the part before the ‘If’, we get an expression in which ‘x’, ‘f’, and ‘g’ occur “free,” that is, without a quantifier. But I insist that we can only make sense of such expressions by reminding ourselves of the sense which the original expressions have.

I said earlier that expressions for forms represent nothing. This sounds like a contradiction, but I think that you can now see what is meant. Such an expression does not represent an entity, but we can use it in order to make a long story short, namely, a story about complex things which are partially identical. In particular, we must realize that such expressions do not represent properties or functions (relations). To certain philosophers, the conviction comes naturally that an expression like ‘F’s, which are G’s, are G’s’ represents a function which yields true propositions as values for ideas (or concepts, or properties) as arguments. Russell even presents an argument to the effect that form expressions are needed in order to represent functions. Compare the expression just mentioned with the following expression: ‘A’s, which are B’s, are C’s’. If we hold that these two expressions do not represent functions, so Russell argues, but that the function is instead represented by something like ‘which are, are’, then we cannot distinguish between the first function and the quite different second one. Hence we are forced to hold that those form expressions represent functions (B. Russell, Principles of Mathematics, pp, 84-85, 508-509). Russell’s argument presupposes that we are confronted here with two different functions. But this is a mistake. The two expressions deal with the same function, namely, the function represented by ‘which are, are.’ We have here, not two different functions, but two different forms, forms which contain the same function. The fact that two propositions are not of the same form does not imply, as Russell assumes, that they must contain different functions. Another example may help to make this clear. Consider the two descriptions: The sum of 2 and 3 and the sum of 2 and 2. The first is of the form The sum of X and Y, while the second is of the form the sum of X and X. But though the forms are different, the function (relation) contained in the two descriptions is the same, namely, the sum function (relation).

Since we are talking about Russell, we may take this opportunity to say a few words about Russell’s version of the popular view that logic (and pure mathematics) is “purely formal” (see B. Russell, Introduction to Mathematical Philosophy, pp. 194-201). Russell asks: “What is this subject, which may be called indifferently either mathematics or logic?” The question, of course, makes an assumption which I consider to be most certainly false, namely, that logic and mathematics are the same. But never mind, what is, according to Russell, the nature of logic? He points out the familiar fact that the validity of arguments depends only on their form, and then he wrestles with the problem of the nature of form. He considers the form x R y which may be read: “x has the relation R to y.” Here I must interrupt our inspection of Russell’s view for a moment and amend something I said earlier. To talk about propositions of a certain form, I said then, is to talk about propositions which are partially identical. This was correct within the context, but must be changed for the general case, as Russell’s example shows. Propositions of Russell’s form may have no part in common. Rather, what they have in common is that they consist of two individuals and a relation in a certain order (or of ideas of a certain sort in a certain order). Still, my main contention stands: There is no such entity as the form of a proposition. In particular, the form of a proposition is not a property of the proposition.

Russell, in distinction from us, thinks of forms as things of a certain sort:

Given a proposition, such as “Socrates is before Aristotle,” we have certain constituents and also a certain form. But the form is not itself a new constituent; if it were, we should need a new form to embrace both it and the other constituents. We can, in fact, turn all the constituents of a proposition into variables, while keeping the form unchanged. This is what we do when we use such a schema as “x R y,” which stands for any one of a certain class of propositions, namely those asserting relations between two terms. We can proceed to general assertions, such as “x R y is sometimes true”—i.e. there are cases where dual relations hold. This assertion will belong to logic (or mathematics) in the sense in which we are using the word. But in this assertion we do not mention any particular things or particular relations; no particular things or relations can ever enter into a proposition of pure logic. We are left with pure forms as the only possible constituents of logical propositions.

(Russell, 1956a, pp. 198-99)

This paragraph clearly shows Russell’s confusion. From a proof that forms cannot be constituents of the propositions whose forms they are, he arrives at the conclusion that “forms are the only possible constituents of logical propositions.” The proof is relatively clear. If the form x R y were a constituent of the proposition Socrates is before Aristotle, then the proposition could not be of the form x R y, for this form does not show the form as a constituent of the proposition. Rather, the form would then have to be something like f-x R y, where the ‘f’ indicates the form. But now we can repeat the reasoning and argue that if this form is a constituent of the original proposition about Socrates and Aristotle, then the form of this proposition cannot be f-x R y either. Russell concludes, reasonably it seems to me, that the original proposition does not contain its form as a constituent. For us, the problem does not arise: since there are no such things as forms, they cannot be parts of propositions.

Next, Russell claims that we can turn all the parts of a proposition into variables: x R y. This shows, I take it, that the form was not part of the proposition to begin with; otherwise, there would occur a variable for it in the form (our ‘f’ above). But note that Russell speaks here of “variables.” This is a second mistake. The form is depicted, in our view, not by ‘x R y’ but by an expression with schematic letters, say, by ‘X R Y’. When we consider the expression with variables, we can only make sense of it, as I claimed earlier, as a part of a “closed” expression; for example as a part of the sentence: There is an individual thing x, and there is an individual thing y, and there is a relation R such that: x R y (that is, x stands in R to y)’ For a moment, Russell seems to take a step toward our view when he says, not, as we would expect, that ‘x R y’ stands for the form but that it stands “for any one of a certain class of propositions.” At this moment, he takes his talk about variables seriously. But the impulse is immediately repressed: the logical proposition that emerges is not our quantified proposition, but the rather strange “x R y is sometimes true.” And then Russell concludes from this example that the pure form x R y is a part of the logical proposition. Whatever happened, we may want to ask, to the claim that ‘x R y’ stands for any one of a class of propositions? It is clear that Russell is torn between treating the expression ‘x R y’ as a variable “ranging over certain propositions,” on the one hand, and as a name for a certain kind of entity, namely, a form, on the other.

How uncomfortable Russell feels can be seen from the next paragraph. He says, first, that he does “not wish to assert positively that pure forms—e. g. the form “x R y”—do actually enter into propositions of the kind we are considering.” But then he concludes two sentences later that “we may accept, as a first approximation, the view that forms are what enter into logical propositions as their constituents.” I have so far argued that we may not accept this view. Logic is not about a certain kind of entity called “form.”

e) Form and Logical Laws

Logic is not about forms. What is it about? Bolzano, as we see from his examples, thinks of identity statements of the form “A is A” as belonging to logic. I do not. Of course, it is in part “merely a terminological matter” whether we call the identity relation a “logical relation.” But our terminology should be guided by a desire to make philosophically illuminating distinctions. In this spirit, I think of identity statements as ontological statements rather than logical ones. I hold that there exists an ontological law of identity (or of self-identity): Every entity e is such that: e = e. This law holds for any entity whatsoever. It holds for properties as well as for individual things; it holds for numbers as well as relations; it holds for simple things and for complex things; it even holds for identity itself. In brief, it holds for all categories. But though it holds for everything, it does not hold for non-existents. Entities and only entities are self-identical. I do not believe, as some philosophers do, that Pegasus and Hamlet are self-identical. Hamlet can no more stand in the relation of identity than he can stand in the relation of being-a-son-of. What is true is, rather, that he is depicted as the son of the king of Denmark; and it goes without saying that he is thought of as being self-identical. This is a long story which I cannot here pursue (see my “Nonexistent Objects Versus Definite Descriptions,” pp. 363-77). What I wish to emphasize is that in my view the law of self-identity is an ontological law, and that an instance of this law is an ontological fact. The same is true, by the way, for the so-called axiom of infinity: it, too, if it is a fact, is an ontological rather than logical fact. That there are infinitely many things (as distinguished from that there are infinitely many numbers), if a fact, is a fact of ontology. The axiom of choice, on the other hand, seems to me to be a purely set-theoretical fact. That it is a fact, I have no doubt.

Since an instance of the law of self-identity is not an instance of a logical law, it is not, in our specific sense, an analytic truth. And neither are the axioms of infinity and of choice. However, this does not mean that these truths, if they are truths, are not necessary. Necessity, we must constantly emphasize, does not coincide with analyticity.

Returning to Bolzano, he also mentions as an example of a logically analytic statement: Every object is either B or not B. I take this to mean: Every entity either has the property F or does not have the property F. A fact of this form is an instance of the law: All properties f are such that: all entities e are such that: e is f or e is not f. (This is rather awkward and abominable English, but it has the advantage that I can fit different English sentences into the same mold.) This is a law of logic. I call it a “logical law,” because it says something about all properties. It tells us that all properties behave in a certain fashion. It is clearly different from the law of self-identity which tells us something not only about all properties, but about all entities whatsoever. An instance of this law of logic is thus a proposition of logic and therefore an analytic truth.

Ontological laws are about entities in general; (some) logical laws are about properties in general. According to this criterion, the second and third examples of Bolzano’s list of logically analytic truths belong to logic. And so do all of the laws of what is known as “the singulary functional calculus of higher order” or “the monadic predicate calculus of higher order” (see, for example, A. Church, Introduction to Mathematical Logic). However, logic as commonly understood is not restricted to a general theory of properties. There is also a general theory of relations. These two theories are usually combined into one. What we get then is “the higher functional calculus.” Finally, there exists also a quite different theory about states of affairs (propositions) in general. This theory is sometimes called “the propositional calculus” and counted as a part of logic. In short, what is commonly called “logic” consists in my opinion of three distinct theories, namely, (1) a theory about properties in general, (2) a theory about relations in general, and (3) a theory about states of affairs in general. The first theory describes under what general conditions properties are exemplified; the second states under what general conditions relations hold; and the third describes under what general conditions states of affairs obtain. I hope that it will become clear as our investigation proceeds how profoundly this conception of logic differs from almost all of the views about logic which have been proposed during the last hundred years.

Earlier, I explained Bolzano’s notion of logical analyticity in terms of the form of propositions. For example, I said that propositions of the form: F’s, which are G’s, are G’s are logical analytic truths. It must be emphasized, though, that I could have talked instead about instances of a certain law. From my point of view of logic, this would indeed have been the more perspicuous approach. Consider the logical law:

All properties f and g are such that: All entities e are such that: If e is f and e is g, then e is g.

Every instance of this law is a logical truth. It is also obvious that all of its instances share the same form and that all propositions of this form are instances of the law.

If one seeks an explication of logic which is centered around the notion of form rather than the notion of law, one may easily make the following mistake. It is obvious that there are many quite different forms which yield logical truths. F’s, which are G’s, are G’s is one such form. But so are P or not P, and also If P, then (if Q, then P). One is forced to ask what all of these diverse forms have in common that accounts for their logical nature. Well, it is obvious that what they have in common is a stock list of shared concepts, namely, the concepts or, and, if-then, all, some, not, etc. While all the other words are replaced by “variables” (actually schematic letters), words for these concepts remain “constant.” And this suggests to some philosophers that the forms under study are logical in nature because they contain these “logical concepts.” Thus logic is characterized by the fact that is is about (that it involves, that it is based on) a peculiar set of logical concepts (logical constants, logical terms).

But this conclusion is wrongheaded. What is essential to logic is not that it involves words like ‘not’, ‘all’, and ‘or’, but that it is about properties, relations, and states of affairs. The relation or is no more a logical relation than it is a biological relation; the quantifier all is no more a logical entity than it is a chemical thing. The relation or and the quantifier all are parts of all kinds of laws, not just of the laws of logic. They occur in laws of chemistry and geometry just as much as in the laws of logic. They are the stuff facts are made of. What describes a field of inquiry is its laws. And what distinguishes laws from each other is what they are about, that is, what the variables of the laws range over. In the so-called “propositional calculus,” the variables range over states of affairs (propositions). This is why it is a theory of states of affairs, not a theory of chemistry or biology. In the so-called functional calculus, the variables range over entities and properties and relations (let us agree to say that they range over entities and attributes). This is why it is a theory of attributes, not of geology or of geometry. Nor is it the same as set theory. In set theory, the same quantifiers and connectives occur as in “attribute theory,” but the variables of set theory range over sets, while the variables of attribute theory range over attributes. While set theory is about sets, logic (the “functional calculus”) is about attributes. From our point of view, it is absurd to hold that there is such a thing as propositional logic, but that there are no propositions (states of affairs, circumstances, etc.). From our point of view, it is absurd to teach that there is such a thing as the functional calculus, but that there are no attributes. This is just as absurd as to believe that there is chemistry without chemical elements, zoology without animals, and set theory without sets.

My main objection to the view that logic is a matter of form and that form is characterized by logical concepts (or logical terms) is not that it is impossible to draw a line between logical and nonlogical concepts. If challenged, we could simply enumerate the so-called logical concepts or terms. Rather our objection is that there are, precisely speaking, no logical concepts or terms. The concepts which are usually singled out as logical concepts are no more characteristic of logic than they are of chemistry. I share, therefore, Bolzano’s skepticism about how fruitful the notion of form is for an elucidation of the nature of logic. And in another respect, too, our views are related.

The view that logic is a matter of form, not of content, may give rise to the conviction that logic is not concerned with facts. This, in my opinion, is its most insidious corollary. The proposition:

Males, which are unmarried, are unmarried

is a logical truth. It is a logical truth because it is an instance of the logical law:

All properties f and g are such that: If an entity e is f and is g, then it is g, and this is a logical law, because it says something about properties in general. Two facts are involved: a fact about males and a fact about properties. The fact about males is true because it is an instance of the law, but the law itself is also a fact. The law, to emphasize, is just as much a fact as any law of biology, any law of set theory, any law of arithmetic. There are facts about properties in general just as there are facts about whales in general, or about certain sets in general, or about natural numbers in general. Just as the logical law is a law about properties in general, so is the law that all whales are mammals a law about whales in general. Assume that there exists a certain whale called “Moby.” It is a fact that Moby is a mammal. This is a fact because Moby is a whale and because of the law about whales. But we could also say that Moby is a mammal is true by virtue of its form, for any proposition of the form X is a mammal is true, assuming, of course, that X is a whale. However, would it not be absurd to conclude that it is not a fact that Moby is a mammal, or that zoology is not a matter of facts? Bolzano’s somewhat idiosyncratic conception of analyticity has the priceless virtue of showing that propositions which are obviously empirical can be said to be true in virtue of their form!

Of course, there is a difference between the law of logic and the law about whales. As a matter of fact, there are at least three important differences. Firstly, while the law of logic is about properties, the second law is about whales: there is a difference in subject matter. Secondly, while the law about logic may be said to be necessary in the sense that we cannot imagine exceptions to it, the law about whales is not necessary in this sense. Thirdly, and this is the difference I wish to dwell on, while the logical law is very general—it concerns all properties whatsoever—the second law is specifically about whales. There is thus a great difference in the degree of generality. This difference is partially hidden by the fact that we formulated the logical law, as is common, in terms of property variables. To emphasize its full generality, we should however give the following version of it:

All entities e₁, e₂, and e₃ are such that: If e₁ and e₂ are properties, and if e₃ exemplifies both e₁ and e₂, then e₃ exemplifies e₂.

It is this amazing generality of logical laws which accounts, at least in part, for our reluctance to give them up in the face of contrary evidence. Surely, to scrap a law about all properties will involve revisions a philosopher seldom dreams of.

f) Logic and Empiricism

If the logical law, in either form, is a fact, then we cannot avoid the Kantian question of how we know this fact. Before we turn to this important question, let us note another point about analyticity. Recall our paradigm of Bolzano’s logically analytic truth: A, which is B, is B, or: Males, which are unmarried, are unmarried. But what about the background law; is it an analytic truth? Bolzano’s answer can be glimpsed from the following, somewhat different, case:

The proposition: If all humans are mortal, and Caius is a human being, then Caius, too, is mortal, may perhaps (allenfalls) be called analytic in the wider meaning given in paragraph 148; however, the rule itself that from two propositions of the form A is B and B is C a third one follows of the form A is C, is a synthetic truth.

(Bolzano, 1981, par. 315, 2)

From this quotation it appears that Bolzano would say that our law of logic is a synthetic truth. Generally, the laws of the theory of attributes are synthetic. And so they are! With the Kantian notion of analyticity in mind, it is obvious that the concept of a property does not consist (in part) of “the concept that if something has two properties f and g, then it has the property g.” (I hardly know how to put this intelligibly.) This latter “concept” is no more contained in the concept of a property than it is contained in the concept of a whale that it is a mammal. Of course, it is true that properties behave in this way, just as it is true that whales are mammals. But the respective concepts of a property and of a whale, to speak Kant’s language, do not contain as parts those other concepts.

From our un-Kantian point of view, it is equally clear that the laws of logic are not analytic. This follows from our very explication of the term. A sentence is analytic, we said, if and only if, in its unabbreviated form, it represents an instance of a logical law. The laws of logic, though they follow from each other, are not instances of each other, except in that pickwickian sense in which every logical law is an instance of itself. And this shows once again how unimportant this notion of analyticity is for the philosophy of logic and mathematics.

How do we come to know logical laws? Bolzano, I think, gives the correct answer. He readily concedes that logic and mathematics are more reliable than other fields of inquiry, but he rejects Kant’s explanation of this fact:

And here, I believe, we have no reason to forsake an explanation which was given long before Kant. One has always said that these sciences [logic, arithmetic, geometry, and pure physics] enjoy such a high degree of certainty, only because they have the advantage that their most important theses can be tested easily and variously by experience, and have been so tested; and, furthermore, because those theses which cannot be tested directly, can be deduced by means of arguments which one has tested innumerable times and always found to be correct; and, finally, because the results, which one obtains from these sciences, do not affect the human passions, so that investigations can almost always be started and finished without bias, and with the proper leisure and calmness. The only reason why we are so certain that the rules Barbara, Celarent, etc. are correct is that they are confirmed by thousands of arguments to which we have applied them. This is also the true reason why we assert so confidently in mathematics that factors in a different order give the same product, or that the total angles in a triangle are equal to two right angles, or that the forces on a lever are in equilibrium when they stand in the inverse relation of their distances from the fulcrum, etc. But that the square root of 2 equals 1.414 . . . , that the content of a sphere is exactly two-thirds of the circumscribed cylinder, that in each body there are three free axes of revolution, etc. we assert with such confidence because they follow from propositions of the first kind by arguments which others, too, have used a hundred times and found to be correct; . . . .

(Bolzano, 1981, par. 315, 4; similar remarks, by the way, can be found in Russell)

The case for empiricism could hardly be stated any better. Bolzano clearly distinguishes between certain basic propositions of logic and arithmetic, on the one hand, and more complicated theorems, on the other. An example of the first kind from logic is the form Barbara; from arithmetic, the law that the order of the factors of a product does not matter. Why are we so certain that an argument of the form Barbara is valid? We are so certain, because we and many others have tested it innumerable times and found that it always leads from true premises to a true conclusion. Recall our earlier example of a logical law. We can easily verify—and it has been verified millions of times—that if a thing has the properties F and G, it has at least the property G. If you doubt this law, give me an example of a property, G, which does not behave in this fashion. Or show me that the negation of this law follows from some alleged truths. Or, finally, put forward an argument to the effect that the law in conjunction with some other true propositions yields a falsehood. No matter what you try, I am sure that you will fail. But though I am sure that you will not succeed, I am willing to consider this kind of refutation of the putative law. I admit that the putative law could be shown to be false in this manner.

It is at this point that empiricists like me part ways with many other philosophers. Ernest Nagel, for example, has argued that any suggestion that, say, modus ponens is not a valid rule of inference would be “dismissed as grotesque and as resting upon some misunderstanding” (Ernest Nagel, “Logic without Ontology”, p. 310). Of course, it would. Of course, we would suspect that one of the premises of the supposed exception to the rule must be false after all. Or else we may insist that the conclusion cannot but be true. But Nagel and I disagree about the reason for our stubbornness. I resist any suggestion that the rule is mistaken (that the underlying law is false) because of the fact that it has been confirmed a million times. Nagel, on the other hand, believes that “statements of the form: ‘If A and (if A then B) then B’ are necessarily true, since not to acknowledge them as such is to run counter to the established usage of the expressions ‘and’ and ‘if . . . then’ ” (Nagel, p. 310). According to him, “the laws which are regarded as necessary in a given language may be viewed as implicit definitions of the ways in which certain recurrent expressions are to be used or as consequence of other postulates for such usages” (Nagel, p. 319). We shall later discuss implicit definitions in some detail. For the moment, I wish to point out that what matters for the truth of statements of the form modus ponens is not just the “established usage of the expressions ‘and’, and ‘if . . . , then,’ ” but also the usage of the word ‘proposition’ (‘state of affairs’, ‘statement’). What is true is the statement: All propositions p and q are such that: If p and (if p, then q), then q. If it were not for the truth of this law, statements of form modus ponens would not be true. As soon as we realize that what we mean by ‘proposition’ (‘state of affairs’, ‘statement’, etc.) is as important for laws of (propositional) logic as what we mean by ‘and’, ‘or’, etc., we realize the futility of trying to base the certainty of logic on “established usage.” If we meant by ‘proposition’ something other than what we do mean by it, the logical law which justifies modus ponens might not be true. But by the same token, if we meant by ‘whale’ something different from what we do mean by it, the sentence ‘All whales are mammals’ might not be true.

Empiricists are sometimes challenged to name just one instance in which a logical law was abandoned for the reasons given above (see, for example, Nagel, p. 310). I shall take up the challenge. Consider the proposition that every property (which is exemplified) determines a set. Now, this is not strictly speaking a law of logic, although Frege thought of it as one, but I think that it is not any less evident to “informed common sense” than the laws of logic are. Surely, that a given property determines a certain set is confirmed millions of times: To the property of being a pen on my desk, there corresponds the set of all pens on my desk; to the property of being a citizen of the United States on a given date, there corresponds a set of persons; to the property of being a natural number, there corresponds the set of natural numbers; etc., etc., etc. Yet, the proposition is false: There are properties which do not determine sets! It took logicians and mathematicians many years to give up this law. They were so certain of it that they searched in a hundred different directions for a way of preserving it. How did we find out that the alleged law was false? In precisely the same way in which we on other occasions find out that an alleged law is not really a law: We found an exception! The property of being a set, it turns out, does not determine a set. Thus it is not true that all properties determine sets; only some properties do. Our strongest intuition, confirmed by millions of examples, turned out in the end to be mistaken.

The laws of logic and arithmetic, just like the laws of biology and chemistry, are “about the world.” They, too, have a specific “subject matter.” Part of logic is about propositions (states of affairs); another part is about attributes (properties and relations). Arithmetic, of course, is about numbers. The laws of logic and arithmetic, just like the laws of biology and chemistry, are established “empirically.” We are more or less certain that they hold, as long as we have found no exceptions. Does this mean that the vaunted necessity of logic and arithmetic is but a chimera? We shall have to discuss this question in the next chapter. One thing, though, is already clear. Whatever distinguishes between logic, arithmetic, and set theory, on the one hand, and the “empirical sciences,” on the other, cannot be a fundamental difference between their respective statements. On both sides, there are laws and instances of laws. In both cases, the laws necessitate their instances. In both fields, instances suggest (confirm) the laws. And on either side of the division, alleged laws are vulnerable to exceptions.

g) Propositions and the Murky Realm of Meaning

Bolzano breaks with the two-thousand-year-old tradition of Western philosophy and makes the category of proposition the center of his ontology. However, this revolution, like most revolutions, turns out to be a mixed blessing. On the positive side, there is the valuable insight that it is judgments and their contents which constitute knowledge. Bolzano holds that in addition to mental acts of judgment and linguistic expressions (sentences), there exists a third kind of thing which is neither mental nor linguistic. This kind is comprised of his so-called Saetze an sich, what I have in this chapter called “propositions.” Bolzano clearly sees that these propositions are “abstract” entities, that is, that they are not spatio-temporal. In addition to these major discoveries, Bolzano’s ontology contains numerous innovations that I cannot mention in this context.

On the negative side, Bolzano’s propositions turn out to be denizens of a murky realm of meanings, not the robust facts around us. The proposition Socrates is mortal, for example, is not the same as what I call the fact that Socrates is mortal. This fact contains Socrates as a constituent. It consists of Socrates, the property of being mortal, and the nexus of exemplification. Bolzano’s proposition, on the other hand, contains not Socrates, but the “objective idea” of Socrates. What, precisely, is this “objective idea”?

Bolzano distinguishes between subjective and objective ideas. A subjective idea, as one would expect, is a mental occurrence in an individual mind. When I think of Socrates, there occurs in my mind a certain subjective idea; and when you think of Socrates, there occurs in your mind a different subjective idea. If there were no minds, no such ideas would exist. But, according to Bolzano, there are also, corresponding to subjective ideas, certain objective ideas. These things are not mental and they are not confined to individual minds. Nor are they “concrete” like subjective ideas. They are abstract things, that is, they are not spatio-temporal. Propositions consist of these objective ideas. Bolzano holds that what the sentence ‘Socrates is mortal’ represents, the proposition, is of the form: Socrates has mortality. And this proposition consists of the three objective ideas: Socrates, the idea of having, and the idea mortality.

Every objective idea has an “extension,” that is, a certain set of objects (see, for example, Bolzano, 1981, par. 126, 127, and 131). This set may be empty, or it may consist of just one thing. The extension of the objective idea round square, for example, is empty. The objective idea Socrates, on the other hand, has just one object. The idea mortal, finally, has an extension with many objects. In order to get some understanding of what an objective idea is, we should ask: What, precisely, is the relationship between an objective idea and its extension? This seems to me to be the most important question about Bolzano’s ontology.

I think that there are two possible interpretations of Bolzano’s view. According to the first, the relation between an objective idea and its object is the intentional nexus. It is the relationship between an idea, in the ordinary sense of the word, and what the idea is an idea of, its object. The second possibility is that the relation is roughly the relationship between a property and the set it determines. I have the impression that Bolzano thinks of the connection between objective idea and extension in both ways simultaneously, and that this is one of the most important flaws of his theory. But be that as it may, the fact remains that Bolzano distinguishes between the objective idea Socrates, which is a constituent of the proposition, and the person Socrates. The objective idea (of) Socrates thus is placed somewhere in the middle between the subjective idea in the mind and the real person Socrates in the world. Some philosophers may be tempted to think of the objective idea Socrates as the “meaning” of the name ‘Socrates’, where the meaning must be distinguished from both the idea in the mind and the person in the world.

I do not think that there are propositions, as conceived of by Bolzano, because I do not think that there are “meanings” in this sense of the word. I do not deny that there is meaning. I believe that there is both meaning expressed and meaning represented by language. For example, the sentence ‘Socrates is mortal’ expresses, on a certain occasion, a judgment (or a belief, or an assertion, or a doubt, etc.), and it represents on that occasion the fact that Socrates is mortal. Thus we may say that it means the judgment and that it means the fact or, better, that its meaning is the judgment and that its meaning is the fact. (Of course, there are many other meanings of ‘meaning’, but they are not relevant to our discussion.) To say it again: I have no quarrel with meaning. But I do not believe that in addition to the judgment and the fact, there exists a third, Bolzano’s proposition.

Bolzano’s distinction between the subjective judgment, on the one hand, and its objective content, on the other, is not only correct, but essential for a rejection of Kant’s idealism. But Bolzano does not get the objective side right. What corresponds in the nonmental world to the subjective idea of Socrates is not an objective idea, but Socrates himself. What corresponds to the subjective idea mortality, is not an objective idea, but the property of being mortal. And what corresponds to the judgment that Socrates is mortal is not the proposition, but the fact. Bolzano overcomes Kant’s idealism, but the idealistic terminology lingers on, and with it, a tendency to subjectify the external world. To the realist, who has a firm grasp of the nature of the external world, this subjectified world must appear as a superfluous third, a “Mittelding,” neither fish nor fowl. Bolzano speaks of “Vorstellungen an sich” (what I have called “objective ideas”). These “Vorstellungen an sich” are divided into “Anschauungen” and “Begriffe.” These are all terms taken from the mental vocabulary. Yet they are supposed to represent nonmental things. What corresponds in the world to a mental “Anschauung” should not be called an “Anschauung an sich”; it is an individual thing. What corresponds in the world to a mental “Begriff’ should not be called a “Begriff an sich”; it is a property. And what corresponds in the world to a mental judgment should not be called a “Satz an sich”; it is a state of affairs (or a fact). Bolzano escapes from Kant’s idealism by projecting Kant’s mental categories into the outside world. In doing so, he creates a world which is neither mental nor the world of facts, things, and properties. He invents a world that is a mirror image of the mental world, a world of meanings.

The curse of the Kantian terminology haunts not only Bolzano’s philosophy, but Frege’s as well. Just think of Frege’s use of ‘concept’ (‘Begriff’). A concept turns out to be a property, something that has nothing at all to do with the mind (G. Frege, Translations from the Philosophical Writings of Gottlob Frege, p. 51). But the height of terminological confusion is reached with Frege’s use of ‘thought’ (‘Gedanke’). Surely, if anything is mental, it is a thought. Yet thoughts are supposed to be nonmental things. Roughly speaking, they correspond to Bolzano’s propositions (“Saetze an sich”). The sentence ‘Socrates is mortal’, according to Frege, expresses a thought and represents the truth-value true. The thought consists of the sense of ‘Socrates’ and of the “unsaturated” sense of ‘mortal’. What I wish to point out is that the person Socrates is not a part of the sense of the sentence. Senses, like propositions, are therefore not facts. Since truth-values are obviously not facts either, Frege’s ontology, like Bolzano’s, leaves out one of the most important categories, the category of fact.