1 Introduction

Towards the end of the 1960s,¹ the French philosopher Jules Vuillemin gave a presentation at the Maison française in Oxford entitled “Un Français peut-il encore comprendre les philosophes d’outre-Manche?”² Not long before, Vuillemin had published his 1968 book Leçons sur La première philosophie de Russell³ (henceforth: PPR)—based on his classes at the Collège de France⁴—in which he engages in a detailed discussion of Russell’s Principles of Mathematics (henceforth: PoM). Vuillemin opens his book with a polemical remark: “I would not have published these long studies on these texts, currently more than sixty years old, if it weren’t for the current state of French philosophy, its disdain for and ignorance of Anglo-Saxon thought, logic, and, more generally, any more or less exact style in the art of investigating the truth” (PPR, 5). It is clear, then, that Vuillemin wished to distance himself from a certain group of his French colleagues, without naming anyone in particular.⁵ In his Oxford presentation, he ascribes to those colleagues a conception of philosophical knowledge (savoir) on which it “needs no detour through analysis, and … is foreign to all exact sciences” (Mélès and Vuillemin 2015, 18). The object of such knowledge, Vuillemin says, is invariably the Absolute, whether it be called History, Logos, Structure, or God (Mélès and Vuillemin 2015, 18).⁶

In Russell, Vuillemin finds an alternative approach. For Vuillemin, the Principles of Mathematics is more than a mere work of philosophy. It is a manifesto, an announcement of a new age of philosophizing to come. Vuillemin writes that PoM “is for current philosophy [philosophie vivante] what Newton’s Principia and Hume’s Treatise were in the 18th century” (PPR, 6).⁷ He describes reading PoM as a revelation, which helped him realize that a different kind of philosophy is possible.⁸ Vuillemin explains that Russell taught him—through his engagement with Cantor’s set theory—how science can and should inform philosophy (Mélès and Vuillemin 2015, 21),⁹ how Russell taught him that it is not blasphemy to think that reality is not constructed by us (Mélès and Vuillemin 2015, 24), and, most importantly perhaps, how he finds in Russell a philosophy that replaces an analysis of consciousness with an analysis of reality (Mélès and Vuillemin 2015, 23). The enemy is the Absolute, the revolution is Russell’s pluralist and realist ontology of terms and sets, inspired by the scientific work of Cantor. In effect, Vuillemin seeks to repeat, in the French context, the revolt against idealism for which Russell and Moore became so famous. He writes that “the criticism that Russell made of Hegel appeared to me to apply verbatim to what is essential in current French philosophy” (Mélès and Vuillemin 2015, 21).¹⁰

My aim in this paper, briefly put, is to develop an account of the model for philosophy that Vuillemin extracts from Russell. To do this, I will proceed in two stages. The first stage is to look at PPR itself, from which I will emphasize two features. The first feature is Vuillemin’s attempt to codify the philosophy of PoM into a set of basic philosophical principles.¹¹ The second feature is Vuillemin’s prioritization of ontology, which I will illustrate with regards to two cases: Russell’s theory of denoting concepts, and Russell’s disagreement with Frege on the distinction between concept and object. Vuillemin rightly picks up on a certain primacy of ontology in Russell, but his own reading tends to magnify it, resulting in distorted readings of both Russell and Frege.

Through these (and other) features, PPR contains clear hints towards the model of philosophy that Vuillemin finds in Russell. That model is further developed and presented in his 1986 book What Are Philosophical Systems? (henceforth: WAPS), which is the subject of the second stage of my investigation. Although WAPS is not a historiographical work—and although Vuillemin does not explicitly refer to PPR in it—I believe that we can see WAPS as making explicit the model for philosophy that is already implicitly at work in Vuillemin’s earlier reading of Russell. This model revolves around two main pillars, corresponding to the two features of PPR from the first stage of my discussion: (1) transposing the axiomatic method to philosophy; (2) the primacy of ontology. The result is a conception of philosophy as what Vuillemin calls systematic ontology.¹² It is this conception that is meant to ground a principled rejection of the absolutist philosophies advanced by Vuillemin’s French colleagues. While WAPS is ostentatiously presented as a philosophically neutral taxonomy of philosophical systems, Vuillemin’s classification in fact incorporates a host of philosophical assumptions that lead to the elimination of precisely the sort of philosophical approaches that he finds objectionable.

2 Vuillemin’s Reading of Russell

My first task is to examine Vuillemin’s study of PoM. I begin by investigating in more detail the nature of Vuillemin’s architectonic historiography, which seeks to reduce the philosophical system of PoM to a set of basic philosophical principles.

2.1 The principles of Principles

Vuilemin presents the following basic philosophical principles as encapsulating the philosophical system of PoM (PPR 166):¹³

I cannot discuss in detail the content of these principles—or the way in which Vuillemin puts them to work—although we will encounter some examples below. As for the distinction between ontological and methodological principles, Vuillemin does not articulate its underlying principle—thinking, presumably, that the distinction is clear enough from the formulated principles themselves. For now, let us focus on the general features of Vuillemin’s architectonic approach. The aim of the historian, on this approach, is to systematize the concepts, ideas, and arguments of a philosophical work by tracing them to a set of basic philosophical principles—principles which the author of the work need not have made explicit themselves. Russell, for instance, mentions Occam’s razor only once in PoM and does not characterize it as an overarching principle that guides his thought.

Vuillemin did not, of course, come to PoM as a blank slate. His architectonic methodology should be traced to his own philosophical background, not in the least to the work of his mentor and predecessor at the Collège de France, Martial Gueroult.¹⁵ Gueroult is most famous, perhaps, for his book Descartes selon l’ordre des raisons (Gueroult [1953] 1968).¹⁶ As the title of the book makes clear, Gueroult’s approach to historiography may be characterized as rationalist. A philosophical work, according to Gueroult, is demonstrative in nature (Gueroult 1952, 31, 33). What the historian seeks to do, is to lay bare the logical structure (order of reasons) that governs this demonstrative organisation. Gueroult talks, in this vein, of a technology of systems, which seeks to understand philosophical doctrines by placing them against the background of the “architectonic structure” and “probationary techniques” from which they arise (Gueroult 1952, 34). Although I cannot discuss Gueroult’s views in detail, it is clear that Vuillemin’s systematization of PoM into a set of basic philosophical principles may be understood as one way of putting Gueroult’s rationalist, systems-oriented historiography into practice.¹⁷

A crucial feature of Vuillemin’s approach is that it is anti-holistic, in the following sense: he believes that a philosophical principle can be taken to have a self-standing content that is stable across its integration into distinct philosophical systems. That is, the meaning of a principle is prior to the philosophical system in which it occurs. That this is a commitment of Vuillemin, is made clear, for instance, in his Oxford presentation, where he writes:

I managed to isolate six principles—whose independence is not strict but appears each time it is possible to construct philosophies that maintain five of these principles but not the sixth, and, in some sense, one can regard the evolution of Russell as realising certain of these philosophies. (Mélès and Vuillemin 2015, 21)

One may compare this to axiomatic geometry, where one can “toggle” between different axioms so as to generate different geometrical systems. The most famous example, of course, is how one can “toggle” between elliptic, Euclidean, and hyperbolic geometry by keeping fixed the other axioms of Euclidean geometry, and postulating that, for any given line and a given point not on that line, there exist zero, one, or at least two lines through the point that are parallel to the line. Similarly, Vuillemin thinks, we can toggle between different basic philosophical principles so as to generate different philosophical systems. This also becomes clear when, in PPR itself, Vuillemin wonders whether Russell’s principle of realism is independent of his principle of external relations (PPR, 209). As a possible method of investigating this, he suggests that we could look for philosophical systems that accept the one but not the other—which replicates the method that Hilbert used to establish independence in geometry.¹⁸ He then presents the following table (PPR, 209):

To be sure, Vuillemin admits that this is not a proper method for ascertaining the independence of philosophical principles. His reasoning, however, is interesting:

A demonstration of independence presupposes that the theories in which the given axioms are not simultaneously affirmed is equally coherent as the theory in which they are. In philosophy, we have no proof of this. (PPR, 209)

The problem that Vuillemin identifies, is that the existence of philosophical systems in which one of Russell’s principles is ostensibly affirmed while the other is denied, cannot serve as a demonstration of their independence, because this does not establish the coherence of such a system. If, for instance, Kant’s system were to be shown to be incoherent, then we have not actually found a way of denying the principle of realism while affirming the principle of external relations, so that their independence remains in question. Where Vuillemin sees no problem, however, is in the supposition itself that the systems of Kant and Leibniz can be understood as each denying and preserving some of the very same principles that are affirmed by Russell—i.e., that the content of these principles is stable across the philosophical systems of Kant, Leibniz, and Russell.

In WAPS, we find the same idea. In talking about the task of the historian of philosophy, Vuillemin writes:

By reconstructing in their individuality some of the systems which constitute the object of his discipline, he meets open discord as well as seeming agreements with other systems, and is obliged to trace their cause to differences regarding a given premise or a given set of premises. (WAPS, 115)

Thus, the idea is that we should understand disagreement between philosophers in terms of disagreement about certain premises. For such a conception to work, those premises must have a content that is stable across different philosophical systems.¹⁹ Otherwise, there would be no determinate premises about which to disagree—unless we were to say that each accepts disjoint sets of premises, which is clearly not what Vuillemin has in mind.

This approach may be criticized. What understanding, for instance, do we have of the principle of logicism as a self-standing “premise” of distinct philosophical systems? Frege and Russell are commonly grouped together as the founders of logicism. Kremer, however, has argued convincingly that there is no logicist project that is shared by both—that what “logicism” amounts to in each case, is something fundamentally different. Such examples are familiar from the history of philosophy and could be multiplied at will.²⁰

There thus arises a dilemma. Either one sticks to the idea that basic philosophical principles—such as the principle of external relations and the principle of the legitimacy of analysis—have a self-standing content that is stable across different philosophical systems. In that case, however, that content must be so minimal as to become incapable of actually accounting for the details of those systems. If we say that the principle of logicism is accepted by Frege and Russell, then this principle can amount to little more than the blanket statement that mathematics is reducible to logic (or something along these lines). Once we start asking further questions—for instance, what the relevant notion of reduction is—the principle provides no answer. The alternative possibility is that one conceives of the basic principles as more substantive, in such a way that they genuinely account for the specifics of a philosophical system. In that case, however, the idea that distinct philosophical systems may share certain basic principles, must be abandoned for a more holistic conception.

In light of this, it should be emphasized that Gueroult himself does not characterize his historiographical method—the technology of systems—in terms of such an articulation of basic philosophical principles. This way of structuring the “order of reasons” is Vuillemin’s own, and, as we will see, will be further elaborated in What Are Philosophical Systems? In fact, it goes against one of Gueroult’s main ideas, which is that each philosophical system constitutes a “monument” that has a unique structure and coherence, which it is the task of the historian of philosophy to lay bare. In his inaugural lecture at the Collège de France, Gueroult criticizes his predecessor Etienne Gilson for believing that “beyond the doctrines that are actually presented throughout history, there are essential types of systems, which have to be unlocked in their abstract purity: realism, idealism, criticism, etc. … We discover these essences as they are imperfectly realized in historical doctrines; their purity finds itself altered by aberrant factors of individual or social origin” (Gueroult 1952, 24). This, Gueroult complains, is overly simplistic, and neglects the singularity of philosophical monuments. Against Gueroult, Vuillemin seems to be effecting a sort of return to Gilson. In WAPS, Vuillemin will try to uncover precisely such essential types of philosophical systems.²¹ He there echoes Gilson’s idea that the historian “ought to separate what genuinely follows from the highest principle of a system or of a class of systems from the consequences that are due to contingent or historical circumstances” (WAPS, 132). In other words: we need to separate the pure system from its merely contingent accretions.

This move away from Gueroult is pointed out by Vidal-Rosset (1995, 221), who defends Vuillemin’s approach as follows:

Because all philosophies are systems of demonstration [systèmes de preuves], each of them relies on principles to establish these demonstrations; since the principles recognized by human reason are finite, it is necessary that, if certain systems do not agree on the choice of principles, others do, and thereby constitute classes of philosophical systems. (Vidal-Rosset 1995, 219)

The argument is unclear to me. The amount of principles may be finite, but so is the amount of systems, so there is no need for any of them to overlap in their principles.²² Vidal-Rosset writes that “it is inevitable to take recourse to species and genus as soon as one begins the exposition of the history of systems and philosophical concepts” (Vidal-Rosset 1995, 219). I do not see why this is inevitable. Again, say that one is comparing the conceptions of “logicism” that one finds in Russell and Frege. Why should one be forced to group their views together under a purportedly common genus of “logicism”, rather than simply acknowledging the distinctiveness of their philosophical frameworks, along the lines of Gueroult’s conception of philosophical monuments? Of course, categorizing philosophers can be a helpful heuristic idealization for all sorts of reasons. The temptation, however, is to think that the heuristically fruitful grouping together of Russell and Frege as “logicists” must be indicative of a shared feature of their philosophical systems, which can then be articulated in terms of shared premises. In this way, thinking of philosophical systems in terms of species and genus leads to false analogies, resulting in distortions and undue generalizations. In taking the acceptance and rejection of certain independently given philosophical principles as the basic dimension along which philosophical systems are to be compared, Vuillemin heavily restricts the terms in which differences (and similarities) between philosophical systems are to be articulated. Below, I shall provide an illustration of this with regards to the disagreement between Russell and Frege on the concept/object distinction.

Of course, each is free to engage in the history of philosophy as one likes. Different approaches may yield different insights. The problem is that, in WAPS, Vuillemin will elevate his anti-holistic approach into a general principle for a purportedly neutral classification of philosophical systems as such, by conceiving of such systems as modelled on the axiomatic method of science. Before turning to that book, however, I wish to highlight a second feature of Vuillemin’s interpretation of Russell: its prioritization of ontology.

2.2 The primacy of ontology

Vuillemin writes:

We can date with precision the decadence of philosophy. It begins when philosophy, formerly the science of being [science de l’être], abandons its ancient pretensions and abases itself to the rank of science of representation. ‘Psychologism’, ‘Idealism’, ‘Kantianism’ are diverse forms of this decadence. (PPR, 326).

He continues:²³

With Frege, who remained unknown, Russell is the first to return to the Aristotelian conception of metaphysics as the science of being qua being. Russellian realism, and its distinction between being and existence, is the mark of this return. (PPR, 326)

Vuillemin, then, does not keep it a secret that he wishes to foreground the ontological dimension of Russell’s philosophy. I will discuss two instances of how this plays out: his treatment of Russell’s theory of denoting concepts, and his discussion of the disagreement between Russell and Frege on the distinction between concept and object.

2.2.1 Denoting concepts

In PoM Russell writes:

A concept denotes when, if it occurs in a proposition, the proposition is not about the concept, but about a term connected in a certain peculiar way with the concept. If I say “I met a man”, the proposition is not about a man: this is a concept which does not walk the streets, but lives in the shadowy limbo of the logic-books. What I met was a thing, not a concept, an actual man with a tailor and a bank-account or a public-house and a drunken wife. (PoM, §56)

The proposition “I met a man” contains the denoting concept a man as a constituent, but it is not about that concept. It is about the denotation of that denoting concept: a certain logical combination of all actual men—which Russell calls a variable disjunction (PoM, §59)—one of whom I met. This distinguishes denoting concepts from all other constituents of propositions. When we say something about Socrates—e.g., “Socrates is bald”—Socrates himself, the man of flesh and blood, is a constituent of the proposition.²⁴ This realist conception of the proposition—which Russell takes from Moore (1899)—also comes out nicely in his correspondence with Frege, where he criticizes Frege’s idea that it is the sense (Sinn) of “Mont Blanc” that is a constituent of assertions about Mont Blanc:

I believe that in spite of all its snowfields Mont Blanc itself is a component part of what is actually asserted in the proposition ‘Mont Blanc is more than 4000 metres high’. We do not assert the thought, for this is a private psychological matter: we assert the object of the thought, and this is, to my mind, a certain complex (an objective proposition, one might say) in which Mont Blanc itself is a component part. If we do not admit this, then we get the conclusion that we know nothing at all about Mont Blanc. (Frege 1980, 169)

In the regular case, then, constituency and aboutness come together. Not so, however, in the case of denoting concepts: the denoting concept is the constituent, while the proposition is about its denotation.²⁵

Vuillemin regards the introduction of denoting concepts as extravagant, and complains that “one does not understand why Occam’s razor does not apply itself in order to eliminate an entirely superfluous entity” (PPR, 77). According to Vuillemin, Russell is motivated by an undue attachment to the surface grammar of language:

The reason why the significations of denoting class-concepts are maintained is tied to a general principle, which contradicts the profound tendency of the theory of denotation, which is the principle of logico-grammatical parallelism: ‘Every word has a meaning’. This principle is, in turn, tied to a general metaphysical convention, in virtue of which, as soon as language employs a denoting word, a term must correspond to it. (PPR, 77)

It is because Russell is wrongly wedded to the idea that the words “a man” in the sentence “I met a man” must have a meaning—rather than merely denoting—that he is brought to introduce denoting concepts into his ontology. According to Vuillemin, logico-grammatical parallelism is given undue precedence over Ockham’s razor.

Vuillemin’s diagnosis is, to my mind, mistaken, and is revelatory of an important lacuna in his reconstruction of PoM. We can bring this out by asking why the proposition “I met a man” does not simply contain the variable disjunction of men itself as a constituent—which would preserve the unity of constituency and aboutness. The principle of logico-grammatical parallelism, after all, is neutral between taking a denoting concept or a logical combination of men as the meaning of “a man”.

Russell does not arrive at denoting concepts merely through an attachment to surface grammar. To fully understand his philosophical motivations, we must bring in at least two further principles. The first is what I will call the principle of logical combination. What I have in mind, is that Russell regards it as crucial that the proposition “I met a man” is indeed about a logical combination of actual men. For each of the denoting phrases “all men”, “every man”, “any man”, “some men”, “a man”, Russell introduces a distinct kind of logical combination of all actual men which is to serve as the denotation of those phrases (PoM, §§59–62). It is not merely that Russell does not yet have a full grasp of quantification, it is also that his underlying realist conception of the proposition requires the denotation of these phrases to consist of actual men.

Above, I said that the principle of logico-grammatical parallelism is neutral between taking a denoting concept or a logical combination of men as constituent of the proposition “I met a man”. We now see that, if we view the matter solely from the perspective of Russell’s realism, the preferred option would be to take the logical combination of men as constituent, since this would preserve a continuity with other propositions. As Mont Blanc itself is a component part of the proposition “Mont Blanc is more than 4000 metres high”, so the actual logical combination of men, one of whom I met, would be a component part of “I met a man”.

Ontologically, then, there is no reason to introduce denoting concepts. This, in effect, is what Vuillemin has correctly seen. The correct conclusion to draw, however, is not that denoting concepts are an extravagance, but that their introduction is motivated—not ontologically—but epistemologically. This involves what I will call Russell’s principle of acquaintance. It receives an explicit formulation in On Denoting: “In every proposition that we can apprehend …, all the constituents are really entities with which we have immediate acquaintance” (Russell 1905, 492).²⁶

Although Russell does not yet use acquaintance as a technical term in PoM, the principle is already at work in his theory of denoting, as can be seen from the following passage:

It may be said that the logical purpose which is served by the theory of denoting is, to enable propositions of finite complexity to deal with infinite classes of terms: this object is effected by all, any and every, and if it were not effected, every general proposition about an infinite class would have to be infinitely complex. Now, for my part, I see no possible way of deciding whether propositions of infinite complexity are possible or not; but this at least is clear, that all the propositions known to us (and, it would seem, all propositions that we can know) [my emphasis] are of finite complexity. It is only by obtaining such propositions about infinite classes that we are enabled to deal with infinity; and it is a remarkable and fortunate fact that this method is successful. (PoM, §141)

Take a proposition such as “All prime numbers greater than 2 are odd”. It is crucial for Russell that both of these are true: (1) Since it is known to us, this proposition is finitely complex; (2) This proposition is about all prime numbers greater than 2. The first requirement issues from the principle of acquaintance, the second from the principle of logical combination. The only way to satisfy both, is to introduce an entity that is at once logically simple—so that it may serve as a constituent of a finitely complex proposition—but which, as a constituent, has the effect of making the proposition be about a possibly infinite number of objects.²⁷ That entity is the denoting concept, and it is with this entity that we are acquainted in knowing the proposition “All prime numbers greater than 2 are odd”. We cannot be acquainted with infinitely many prime numbers, but we can be acquainted with the denoting concept all prime numbers greater than 2.²⁸ Similarly, we are not acquainted with every single man, but we are acquainted with the denoting concept a man.

For Russell, denoting concepts are not superfluous, but solve a real problem: how can we talk about (combinations of) entities with which we are not acquainted?²⁹ Russell would abandon his solution, but he would not abandon the problem. Denoting concepts only appear as extravagant to Vuillemin, because he does not acknowledge the epistemological problem that motivates their introduction. If we take Vuillemin’s architectonic methodology seriously, then the principle of acquaintance would have to be listed as one of the basic philosophical principles of PoM, since it accounts for fundamental aspects of the system of PoM, such as—as we have just seen—the theory of denoting, but also for the conception of propositions as complexes.³⁰ The reason that Vuillemin does not do so is internal to his own thought, which is here imposing itself on Russell.³¹ For Vuillemin, ontology is basic, as will further become clear when we discuss WAPS. It is because denoting concepts are not ontologically necessary—they are not necessary to account for the truth of “I met a man”—that Vuillemin regards them as superfluous.³²

2.2.2 Concept and object

I now proceed to my second illustration of Vuillemin’s prioritization of ontology: his discussion of the disagreement between Russell and Frege concerning the distinction between concept and object. As is well known, Frege believes that concepts and objects are categorically distinct. There are objects, which are the meaning (Bedeutung) of proper names such as “Seabiscuit”, and there are concepts, which are the meaning (Bedeutung) of concept names such as “ξ is a horse”. Whereas objects are complete, concepts are unsaturated and must be completed by an object as argument. Any judgment involves the completion of what is unsaturated by a suitable argument. For instance, the judgment “Seabiscuit is a horse” involves the completion of the concept ξ is a horse by the object Seabiscuit. Only objects can serve as arguments to complete concepts. There is no such thing as a (first-level) concept completing a (first-level) concept, or an object completing an object, etc. As Frege puts it:

Now it follows from the fundamental difference of objects from concepts that an object can never occur predicatively or unsaturatedly; and that logically, a concept can never stand in for an object. One could express it metaphorically like this: There are different logical places; in some only objects can stand and not concepts, in others only concepts and not objects. (Frege 1984, 281–82)

Frege takes the distinction between concept and object to be absolute. Objects are essentially complete, and concepts are essentially unsaturated.

Russell disagrees. In PoM there is one all-encompassing logical category of terms (PoM, §47). That category includes both what Russell calls things and concepts. Whereas things can only occur non-predicatively in propositions, concepts can occur as a verb, i.e., predicatively (PoM, §48). Concepts can occur predicatively, but they do not need to. In the proposition “Socrates is human”, for instance, the concept humanity occurs predicatively and therefore cannot be replaced by a thing. Because it occurs predicatively, humanity is not the logical subject of this proposition, the proposition is not about humanity. The proposition “Humanity belongs to Socrates”, on the other hand, is about humanity, which is the logical subject of this proposition. This is an illustration of Russell’s principle that “every constituent of every proposition must, on pain of self-contradiction, be capable of being made a logical subject” (PoM, §52).³³ This goes against Frege, who denies that a concept can be made into an object. For Frege, the concept ξ is human and the object humanity are categorically distinct entities.³⁴ The former is essentially unsaturated, whereas the latter is essentially complete. For Russell, however, their difference lies “solely in external relations, and not in the intrinsic nature of the terms” (PoM, §49).

Vuillemin characterizes Frege’s view as dividing reality into two fundamental categories: objects and non-objects (PPR, 203).³⁵ About these non-objects, Vuillemin says that “they do not receive the same degree of reality as the objects themselves” (PPR 203), so that non-objects only have “quasi-reality” (PPR, 204). On Russell’s view, on the other hand, concept and thing are viewed as “equivalent from the point of view of the real [egalité du point de vue du reel]” (PPR, 205). Thus, Vuillemin sees the disagreement between Frege and Russell as a disagreement whether concepts are to be regarded as fully real—whether, we could say, they are to be regarded as full-fledged members of the ontology of the system. This is not, however, how Frege understands the matter. To say that concepts are unsaturated, is not to say that they are any less real than objects. The logical stratification of the realm of Bedeutungen is not tied to a difference in degrees of reality, but to a difference in logical roles within the context of the proposition.

Vuillemin puts ontology before logic in a way that is foreign to Frege. Frege believes that both concepts and objects are the references (Bedeutung) of constituents of propositions—constituents which fulfil distinct logical roles in the proposition. For Frege, once these observations have been made, there are no further ontological questions to be asked about whether concepts are as “real” as objects. With Russell, however, Vuillemin thinks that ontological questions can (and should) be assigned a self-standing status. Russell believes that concepts have an intrinsic nature to which their appearance in the context of propositions is external. Thus, they have the capacity to maintain that nature, independently of the logical role they fulfil in a proposition. The same concept can occur predicatively in one proposition, non-predicatively in another. In this way, ontology is regarded as prior to and independent of the logical structure of propositions. Indeed, in the Principles, propositions are regarded as entities like any other, with no privileged logical status.³⁶

Frege rejects the idea that a concept (or an object!) has an intrinsic nature that is prior to and independent of the logical structure of propositions. For something to be an object, for something to be a concept, just is for it to be the reference of a name that fulfils the corresponding role. As Ricketts has put it: “Ontological categories are wholly supervenient on logical ones” (Ricketts 1986, 66). For Frege, the idea that there are self-standing ontological notions of “concept” and “object” that are prior to the logical analysis of the proposition, is an illusion.

From Vuillemin’s point of view, Frege’s notion of unsaturatedness is seen as tying the being of concepts to the context of the proposition—concepts inherit the unsaturatedness of concept names—so that they lose the sort of self-standing ontological status that Vuillemin takes objects to have. This is not, however, a faithful reconstruction of Frege’s view, but merely how Frege’s view must appear if one insists on seeing ontology as prior to and independent of the logical investigations that Frege takes to be fundamental. For Frege, we do not have a grasp on ontological categories independently of the logical unity of judgment, which always involves the completion of what is unsaturated.

What I hope my discussion brings out, is that we cannot characterize the disagreement between Russell and Frege as a first-order ontological disagreement about the constituents of reality—are both objects and concepts to be regarded as fully real, or only objects?—as Vuillemin does. Rather, it is a disagreement about the status of ontology—about the relation between ontology and the sort of logical investigation in which each takes themselves to be engaged. For Russell, to articulate the logical categories that govern the structure of propositions, and to articulate the ontological categories that govern reality, are related but ultimately distinct endeavours. For Frege, these are one and the same. Where Russell and Vuillemin see a self-standing task for ontology, Frege sees only the task of clarifying the logical structure of the proposition, which is one of the fundamental purposes of his Begriffsschrift notation (see Vanrie 2024, 23–29).

Here is another way of characterizing the difference between Frege and Russell. For Frege, the unity of the proposition is internal to its constituents. To say that concepts are unsaturated, and objects complete, just is to articulate the internality of the unity of the proposition—in which what is unsaturated is completed by an argument—to the ontological categories of “concept” and “object”. On Russell’s view, the unity of the proposition is external to its constituents. The relation in which terms stand in the context of a proposition, are external to their intrinsic ontological nature.

Vuillemin sees this as an application of Russell’s principle of external relations, and as a crucial dialectical move against Hegelian monism, which regards all propositions—and thereby also their constituents—as logically dependent on the overarching unity of the Absolute. Vuillemin writes:

The principle of external relations allows an affirmation of the principle of the invariance of the predicate despite its different grammatical uses, and of the principle of the reality of predicates—so that the consideration of psychological or linguistic significations remains foreign to logic. (PPR, 72)

For Vuillemin, this “invariance despite its different grammatical uses” is the mark of the real. Reality is tied to ontological independence. What cannot be seen as ontologically independent, can at most be regarded as “quasi-real”.

In PPR Vuillemin leaves no doubt that he considers Russell’s pluralistic, unabashedly realist, ontology of terms and sets to be revolutionary, and that he wishes to regard it as a paradigm of what philosophy should aspire to. It is now time to turn to WAPS to see how that paradigm evolves in Vuillemin’s hands.

3 What Are Philosophical Systems?

My guiding question is: what is the model of philosophy that Vuillemin extracts from Russell, and with which he wishes to replace the absolutist approaches of some of his French colleagues? In my discussion of PPR I highlighted two features of Vuillemin’s reading of Russell. First, Vuillemin seeks to effect an architectonic systematization of Russell’s thought into a set of basic philosophical principles. Second, Vuillemin takes ontology as primary. My next task it to show how Vuillemin—in WAPS—systematizes these features into an overarching model of philosophy.

Let me note straightaway that WAPS is an extremely ambitious and highly opaque book. In it, Vuillemin performs a deduction of possible categories of philosophical systems—a deduction that seems to be inspired by (inter alia) Carnap’s Aufbau, phenomenology, linguistics, and ordinary language philosophy. To adequately analyze that deduction itself would take us too far afield here. Although I will touch upon some of its features, my interest lies mainly in its results, and the surrounding conception of a philosophical system that Vuillemin advances.³⁷

Corresponding to two features of PPR that I highlighted above, there are two features of WAPS. The first is Vuillemin’s championing of the axiomatic method as the principle of unity of a philosophical system. The second is his idea that the highest principles of any philosophical system are ontological principles. Together, these result in a conception of philosophy as what Vuillemin calls systematic ontology, and it is this conception that Vuillemin wishes to advance as setting a standard for philosophy as such.

3.1 The axiomatic method

We have already seen that Vuillemin inherits Gueroult’s rationalist approach to the history of philosophy. The objects of the historian’s study are philosophical systems, and their task is to lay bare the internal unity and coherence of these systems. This, of course, raises the question: what is a philosophical system? What is the principle of its unity? In PPR we saw that Vuillemin seeks to systematize Russell’s PoM by identifying a set of basic philosophical principles from which Russell’s views follow. At that time, Vuillemin is not very explicit about the underlying rationale for this approach. As already mentioned, it does not come from Gueroult, nor does it come from PoM itself, which does not share the architectonic structure of Vuillemin’s study of it. To see that rationale, we must turn to WAPS, in which Vuillemin explicitly theorizes his conception of the unity of a philosophical system in terms of the axiomatic method of science.

Vuillemin regards the axiomatic method as the principal intellectual achievement of ancient Greece: “Greek science was a unique event in the history of mankind because of the axiomatic method it applied in arithmetic, geometry, logic, astronomy, harmonics and statics” (WAPS, 100).³⁸ This, Vuillemin says, is how the Greeks were able to abandon mythical modes of thinking: “The axiomatic method awakened the restless mind from its mythical slumber” (WAPS, 103). At first sight, this has little to do with philosophy, since it is science that is axiomatic, not philosophy. Not so for Vuillemin. Philosophy itself, he believes, must be modelled on the axiomatic method. Let me reconstruct his reasoning. Vuillemin remarks:

The axiomatic method precisely determines a field by listing a set of premises from which theorems flow with certainty. But it ignores the nature of its undefinable concepts and the justification of its indemonstrable principles. (WAPS, 104)

It is with regards to these undefinable concepts and indemonstrable principles that philosophy comes in: “It was left to Greek philosophy to inquire into the nature of the things presupposed by axiomatic systems but not included within them” (WAPS, 104). By itself, “axiomatics is … foreign to ontology” (WAPS, 104). The axioms, that is, “do not tell us what the elements are” (WAPS, 104).

Notice how Vuillemin straightaway frames the issue in ontological terms. What is “left out” of axiomatic science, and what thereby becomes philosophy’s task to investigate, is a specifically ontological issue: to inquire into the nature of the things presupposed by axiomatic systems. Vuillemin understands the very origin of philosophy in ontological terms, as an attempt to deal with an ontological question that was left hanging by axiomatic science: “The systematic form of philosophy was elaborated to answer the ontological question posed by the advent of axiomatics” (WAPS, vii). One could wonder, however, why this task—the task of the philosopher—is itself to be pursued according to the axiomatic method. This is because an ontological posit is equally “put to the test by uncoiling its consequences in a deductive chain” (WAPS, 105). Thus, “logic applies to ontology” (WAPS, 105), from which it follows that “philosophy is systematic in the same way that material axiomatic systems are” (WAPS, 105).³⁹ Let us look, then, how Vuillemin ties his axiomatic conception of the unity of a philosophical system to an ontological conception of its content.

3.2 Ontological posits

According to Vuillemin, the task of philosophy is to perform an ontological analysis of reality according to the axiomatic method.⁴⁰ Vuillemin summarizes his conception as follows:

Systematic ontology requires that (1) a minimal set of undefinable concepts and of undemonstrable principles should be given from which the whole furniture of the world may be derived, (2) this derivation should proceed according to legitimated rules, and (3) rival ontologies should be explained away as mere appearances. (WAPS, 116)

The second point is meant to acknowledge that different philosophical systems may proceed along different procedures of demonstration. Vuillemin’s preferred example is the contrast between intuitionism and realism, where only the latter accepts the law of excluded middle in its full generality. The third point pertains to the fact that a philosophical system cannot simply neglect the ontological categories of other systems: “The highest principle for an ontology must allow us to interpret in its own terms the objects that language has placed within the scope of other elementary sentences which do not belong to the basis of this principle” (WAPS, 118). For instance, “an ontology in which events constitute the highest category will have to construe substances and accidents as classes or as collections of events subject to particular conditions of connection, duration and place” (WAPS, 118).⁴¹

Vuillemin’s main focus, however, is on the first point. Given what has already been said, what do these undefinable concepts and undemonstrable principles look like? What is at issue here, are not the intricate details of specific philosophical systems, but the primordial dimension along which they may be grouped into different classes. Vuillemin divides the classes of philosophical systems according to what he calls their “highest principle”, which has the form “Everything which is is an X or a Y or …” (WAPS, 116). In other words: philosophical systems are classified according to what highest ontological category or categories they take as basic.⁴²

Vuillemin recognizes five such categories: ideas (conceived as objective and eternal entities), substance, events, constructions of the mind, and subjective representations (WAPS, 116). While I cannot discuss in detail how Vuillemin arrives at these categories, a couple of points may be noted. Vuillemin begins his deduction by investigating what he calls the perceptual organization of our experience. Crucially, Vuillemin takes that organization to be independent of language: “I deny … that language shapes perception, since perception precedes language” (WAPS, viii).⁴³ Language comes in when it becomes a matter of communicating that perceptual organization to others. Here, Vuillemin takes the singular elementary sentence—which is of the form “a is F”—as the basic unit for such communication.

Vuillemin arrives at his five ontological categories by analysing the possible ontological imports of the singular elementary sentence. The guiding question is: “What are and what may be genuine subjects or singular terms?” (WAPS, 43). Vuillemin’s procedure crucially depends on the idea that the ontological import of the singular elementary sentence is tied to the subject-term “a”. As Vuillemin puts it: “ontology is the business of the subject” (WAPS, 43).⁴⁴ This is a Russellian point, which Vuillemin inherits from his earlier discussion—in PPR—of the distinction between concept and object. Above, I explained how Frege does not regard concepts as any “less real” than objects, revealing how ontology is, for him, not the business of the subject at all. All logical constituents of the proposition have ontological import, irrespective of whether they occupy the subject position or not. This reveals how Vuillemin’s procedure depends on taking sides with Russell against Frege.⁴⁵ Universals, for instance, appear as a highest ontological category precisely because, according to Vuillemin, it is possible “to transform a universal into a singular term” (WAPS, 48), a possibility that Frege would deny.⁴⁶ In this way, Vuillemin is elevating Russell’s conception of ontology as prior to logic and his concomitant principle of nominalisation—that any term can be transformed into a subject without change of meaning—into a constitutive principle of his classification of philosophical systems.

On Vuillemin’s method, all philosophical systems are characterized in ontological terms. He ties his five fundamental ontological categories to five classes of philosophical systems: realism, conceptualism/nominalism of things,⁴⁷ nominalism of events, intuitionism, and scepticism. Thus, scepticism comes to be understood as consisting in the ontological posit that “Everything which is is subjective appearance”.⁴⁸ Similarly, Kant’s transcendental idealism, which Vuillemin understands as a form of intuitionism,⁴⁹ is understood as consisting in the ontological posit that “Everything which is is construction of the mind”. This brings home the far-reaching extent to which Vuillemin takes ontology as basic. By understanding such positions in ontological terms, their epistemological foundation—such as Kant’s conception of transcendental idealism as consisting in a critical investigation of the logical form of our rational capacities—is rendered invisible.

This puts under pressure, of course, the purported generality and applicability of Vuillemin’s classification to actual historical philosophical systems. To an extent, however, Vuillemin would regard this as a feature, not a bug. Although it is ostensibly presented in such terms, Vuillemin’s classification was never meant to be philosophically neutral. Properly understood, it does not offer us a descriptive framework that incorporates all that has, is, and will be done in philosophy, but rather a normative framework for how philosophy should be done. Vuillemin is using descriptive rhetoric in presenting us with a normative framework.

3.3 Pluralism, voluntarism, and neutrality

At this point, a comparison with Carnap is informative. Vuillemin’s classification contains clear echoes of Carnap’s metaphilosophy. As in Carnap, there arises a distinction between internal questions—which pertain to applying the chosen mode of demonstration in uncoiling the ontological posits—and external questions, which pertain to the choice between systems.⁵⁰ How does one select one’s ontological posits? Like Carnap, Vuillemin seems to regard this voluntaristically, as a matter of free choice that is not rationally grounded: “If a philosophical system were to select some principles and reject others, the choice would lack any justification” (WAPS, 113).⁵¹ As in Carnap, the result is a form of “methodological pluralism”, which Vuillemin takes to be “bound to the axiomatic method” (WAPS, 112).

At the same time, there are at least two crucial differences with Carnap, which we can once again trace back to his reading of Russell. The first is that the free choice between philosophical systems is explicitly framed in ontological terms: “At the basis of all philosophy, there is a judgment of taste pertaining to what it prefers to regard as real [ce qu’il convient de tenir pour réel]” (PPR, 333). Sticking to his Russellian inspiration, Vuillemin refuses to regard ontological questions as pseudo-questions. Second—and this is again reminiscent of Russell—Vuillemin does not believe that his methodological pluralism can eliminate polemics in philosophy: “as applied to ontology, axiomatics inevitably produces pluralism and disagreement. Indeed, philosophical reason is born and lives in contest” (WAPS, 113). We cannot simply rest content with the observation that we have chosen different systems.⁵²

Still, however, Vuillemin’s acceptance of philosophical disagreement is only partial, in a way that is reminiscent of Carnap. While there is a choice between systems, that choice is constrained by Vuillemin’s classification. Already in PPR Vuillemin claims that monism is incompatible with the axiomatic method:⁵³

A [monist] metaphysics entails the rejection of all science, since science cannot be conceived except as a sequence of determinate propositions, such that if the first are false, one can deduce anything whatsoever from them. The defence of science thus seems to demand that we reject monism and, consequentially, that we reject the priority of predication as well as the theory of significations. It was thus with good right that Moore tied his doctrine of judgment to a pluralist philosophy. (PPR, 72)

If we accept monism, then there are no determinate propositions, since propositions are taken to implicitly refer to the Absolute—however it is conceived—and are thereby regarded as possessing only partial truth at best. Thus, even though monism itself seems to be an ontological posit, we cannot regard it as offering a proper basis for a philosophical system. To avoid it, according to Vuillemin, we must accept both the principle of external relations and the principle of the reality of predicates.⁵⁴

In PPR it is the incompatibility of monism with the axiomatic method of science that Vuillemin emphasizes—guided, of course, by Russell’s own reflections in PoM. In WAPS, Vuillemin goes one step further. Since Vuillemin there advances the axiomatic method as proper to philosophy itself, the implication is that monism is seen as incompatible with philosophy as such. Vuillemin himself is explicit about this:

The classification entails no particular philosophy of the history of philosophy. Nevertheless, it does exclude a popular myth which identifies the True with the Totality, a presumably inconsistent concept. (WAPS, 130)

There is, of course, something self-undermining about this statement, since the admission that Vuillemin’s classification excludes Hegelian absolutism already means that it is not neutral with regards to the history of philosophy, nor with regards to philosophy itself. Vuillemin takes Russell’s pluralist, realist ontology—understood on a sufficiently abstract level—to be essential to the axiomatic method itself, and thereby to be essential to philosophy itself.

This makes clear how Vuillemin’s classification embodies several philosophical commitments that are far from philosophically neutral. They revolve around a specific understanding of the primacy of ontology, which equally comes with a neglect of epistemological issues. Consider the following passage:

Reality should be analysed into a few concepts and principles which will be taken as premises, and it is sufficient that from these premises one can draw results which, when compared with how reality looks, either are not distorted or obey a received rule of distortion. (WAPS, 105, my emphasis)

What is telling, is how Vuillemin seems to regard this dimension of comparison with reality as entirely unproblematic. We saw that Vuillemin is committed to there being a perceptual organization of experience that language seeks to communicate but which is itself independent of language. Although he is not entirely clear about this, the idea is presumably that we possess the capacity to ascertain to what extent a philosophical system—in uncoiling its basic ontological posits—accords with this pre-linguistic perceptual organization. Vuillemin explicitly states that the perceptual organization and its communication through the singular elementary sentence “a is F” does not, by itself, contain any ontological commitments: “We must, then, consider perceptual organization as it has been described as a subjective synthesis or as a phenomenology, not as an ontology” (WAPS, 32). This is crucial, since Vuillemin’s pluralism demands that the perceptual organization is neutral between the different ontological posits that can be chosen. What a philosophical system attempts to do, is precisely to approximate—in the uncoiling of its ontological posits—as closely as possible this perceptual organization, which thereby serves as an independent standard that seems to be simply given to us.

Is there nothing that may intervene in this relationship between philosophy and the pre-linguistic perceptual given? There is, but, as can be expected, it is not philosophy itself, but science. Vuillemin writes: “The present method … does not prevent science from contradicting not only the possibility of experience but also actual schemes of experience so well entrenched in our habits as to make their revision completely unexpected” (WAPS, 3). Thus, it may be the case that certain scientific developments enforce certain revisions in our philosophical systems. This does not put Vuillemin’s pluralism under pressure, because philosophy maintains a certain independence from science, so that “no scientific discovery is by itself able to force a philosophical decision” (WAPS, 132). Different ontological systems may account for scientific developments in different ways.

To what extent does Vuillemin regard his classification as final? It is only at the very end of the book that Vuillemin explicitly poses the following million dollar question:

Is [a philosophical choice] a priori confined within the bounds of the proposed classification, and, if it is, in what sense? In other words is, or in what sense is this classification complete? (WAPS, 133).

His answer, however, is exceedingly brief and limited. First, Vuillemin discusses but dismisses the idea that a form of Quinean pragmatism constitutes an objection to his classification. He then ends his book with the following observation:

There are, however, more serious objections coming from quantum physics which has shaken the whole conception we had of participation sentences.⁵⁵ … How must highest categories and principles be redefined in order to cope with the requirements of microphysics? Those are questions for the future, upon which scientists and philosophers are still working in the dark. There is but one certainty in all this. Whatever the issue of this further inquiry may be, future classifications will merge into the present one once they come back to describing our ordinary macroscopic experience. (WAPS, 134)

Thus, scientific developments may not only result in revisions that are internal to the ontological systems, they may even demand a revision of the classification of those systems themselves. At the same time, however, as long as we stick to macroscopic experience, Vuillemin regards his classification as final.

Read as a descriptive claim, this seems difficult to maintain. Vuillemin’s deduction relies on too many contentious points in order to be taken seriously as a neutral taxonomy in which “all philosophical systems [are] brought under a finite number of ultimate classes” (WAPS, viii). Like PPR before it, WAPS is not a descriptive work, but a prescriptive one. In PPR, Vuillemin is ostensibly presenting an interpretation of Russell’s philosophy. Indeed, he makes a rhetorical point of it:

Need I point out that I have merely tried to present this philosophy as it is and to add nothing of my own belief, in order not to weaken the exemplary virtue of these writings, which have imprinted a novel course on my thought and of which I have not stopped experiencing the persuasive and unsettling force? (PPR, 7–8)

If my arguments are correct, however, Vuillemin did let his own philosophical proclivities guide his reading of Russell. In itself, this is of course not a reproach: no historian of philosophy can fail to let their own philosophical perspective guide their work, nor should they. It is precisely where this point is denied, however, that one should become suspicious. Vuillemin is not merely reporting on the system of PoM—he is presenting it as a paradigm of how philosophy should be done, to the exclusion of philosophical approaches that he dislikes.

Similarly, we should not read WAPS as an attempt to present a philosophically neutral taxonomy of philosophical systems. Rather, we should read the book as articulating Vuillemin’s conception of how philosophy should be done, of what constitutes good philosophy. We should read it as further developing and making explicit the model of philosophy that is already implicitly at work in Vuillemin’s reading of Russell, and which is deeply inspired by Russell. Indeed, one of the upshots of WAPS is precisely that all proper philosophical systems proceed along the lines of the pluralist ontology laid out by Russell in PoM as understood by Vuillemin. As such, Vuillemin’s very notion of a philosophical system excludes the absolutist philosophies of his French colleagues that he sought to combat. Once Vuillemin’s classification is accepted, these are irrevocably out. Vuillemin is a pluralist, but only on his own conditions. In a Carnapian vein, he combines his pluralism and voluntarism with a rejection of the philosophical approaches that he finds objectionable, and which are thereby not available to be “freely chosen”.

Acknowledgements

The research for this paper was supported by an FWO Junior postdoctoral grant (project 3E008321). I am grateful to the audience at the TiLPS History of Analytical Philosophy workshop on The Global Reception of Russell’s Scientific Philosophy, where an earlier version of this paper was presented. I am grateful to the editors—Sébastien Gandon and Ties van Gemert—and to an anonymous referee for their helpful comments and suggestions, which allowed me to substantially improve the paper.