In the Tractatus Wittgenstein sketches a metaphysical position which displays nominalistic sympathies although it is certainly not nominalistic in either the terms of Ockham, or those of Quine and Goodman. In this note, I would like to explore a version of this position, and to say in what ways it is nominalistic and in what ways it is not. My aim is not, however, faithful exegesis of the Tractatus. The view which I call Tractarian Nominalism will have no part of Wittgenstein's interpretation of quantifiers, his fixeddomain treatment of possible worlds, or his quasi-intuitionist attitude towards infinity. Rather, I would like to give what I take to be the central idea of the Tractatus free play in less cramped quarters. The resulting metaphysics does, I believe, represent a viable ontological position. It has an advantage over traditional nominalism in the way in which it is connected to questions of epistemology. And it has an important consequence for the philosophy of logic. Tractarian Nominalism:

Wittgenstein's truly daring idea was that the ontology of the subject (nominalism) and the ontology of the predicate (platonism) were both equally wrong and one-sided; and that they should give way to the ontology of the assertion. We may conceive of the world not as a world of individuals or as a world of properties and relations, but as a world of facts – with individuals and relations being equally abstractions from the facts. John would be an abstraction from all facts-about-john; Red an abstraction from being-redfacts; etc.

WHAT IF THERE ARE NO ATOMS?

The Combinatorial scheme as so far developed postulates simples: simple individuals, simple properties and simple relations, all conceived of as abstractions from atomic states of affairs. But is the world made up of simples in this way?

May it not be that some, or all, individuals have proper parts which in turn have proper parts, ad infmitum? It is not certain that, just by itself, this supposition contradicts the idea that the individuals concerned are not made up of simples. For one might hold that such individuals are made up of an infinite number of simple individuals. For instance, there are those who hold that any extension is infinitely divisible into extensions, and yet that extensions are made up of continuum-many simple points. However, we can sophisticate our question to allow for this complication. Might it not be that some, or all, individuals have proper parts which in turn have proper parts, ad infmitum, without there being simple constituents even at infinity?

Similar questions can be raised about properties and relations.

It might be, for instance, that the property F is nothing but the conjunction of two wholly distinct properties G and H, that G and H, in their turn, are conjunctions of properties, and so ad infmitum. This progression, it may be, does not even end ‘at infinity’. There are no simple properties involved, even at the end of an infinite road.

MATHEMATICAL TRUTH KNOWN A PRIORI, ANALYTIC AND NECESSARY

In this chapter I sketch a view of mathematics which seems to go along harmoniously with a Combinatorialist Naturalism. We may distinguish, in routine manner, between mathematical entities and mathematical truths. The numbers 7, 5 and 12 are mathematical entities. That 7 + 5 = 12 is a mathematical truth. Let us begin with a discussion of the nature of mathematical truth.

The first point I want to make about the truths of mathematics is a traditional one: that mathematical results are arrived at a priori. This is not a very popular position at the present time. I believe that this is because the notion of the a priori carries a theoretical loading derived from past centuries, a loading that is objectionable. But this loading can be removed without great difficulty, leaving a workable concept of the a priori. It is plausible to think that the truths of mathematics are a priori in this purged sense.

The loadings that need to be removed from the notion of the a priori are the notions of certainty (a fortiori, the Cartesian notion of indubitable or incorrigible certainty) and knowledge.

One philosopher who has moved in this direction is Kripke, who said,

THE NOTION OF SUPERVENIENCE

In the preceding four chapters the theory may be said to have been on the defensive. I have been trying to show that the theory can meet, or has reasonable prospects of meeting, certain difficulties. At this point, however, we can relax and draw certain more or less interesting consequences from the theory.

We have a theory of possible worlds intended to be compatible with Naturalism. We can go on to use the possible worlds to define the notion of supervenience, and then use the latter notion to draw metaphysical conclusions of great importance.

I propose to work with the following simple definition of supervenience: If there exist possible worlds which contain an entity or entities R, and if in each such world there exists an entity or entities S, then and only then S supervenes on R. For instance, if there exist worlds in which two or more individuals have the property F, then, in each world containing such states of affairs, these same individuals stand in the relation of resemblance (at least in some degree). These resemblances are therefore supervenient on the individuals in question having property F.

This definition allows there to be cases where not only is S supervenient on R, but R is at the same time supervenient on S. This does not hold in the case just considered. Consider the individuals which are all F, and which therefore all resemble one another.

What is put forward in this essay is a new version of the metaphysic of Logical Atomism. It is a Logical Atomism completely purged of semantic and epistemic atomism. The idea that one can reach the atoms by analysing meanings is utterly rejected. In general, it is not for philosophers to say what the fundamental constituents of the world are. That question is to be settled a posteriori. It is a question for total science.

The version of Logical Atomism put forward here even abstracts from the question of whether there are any atoms at all at the bottom of the world. That too is a question to be decided a posteriori, if it can be decided at all. In Chapter 5 I argue that Logical Atomism can still be sustained even if we never get past merely relative atoms.

But if there may be no genuine atoms, why continue to speak of Logical Atomism? I do so because, with a little qualification, the scheme presented cleaves to the fundamental idea that the states of affairs into which the world divides (Wittgenstein's and Russell's atomic facts) are logically independent of each other. Each one is, as I will say, ‘Hume distinct’ from every other.

This becomes the basis of what I think is a simple (and naturalistic) Combinatorial theory of possibility. In his article ‘Tractarian Nominalism’ Brian Skyrms sketches a metaphysics of facts (states of affairs, as I put it), facts having as constituents individuals and universals (the latter divided into properties and relations).

Finally, what of the logical truths? They are necessary, it seems. But can we give an account of such things in terms of our possible worlds?

The problem we face here is part of a more general problem. I postulated a certain structure for the world which yields a certain theory of possibility, and I reasoned about this structure. Possibility was defined using that structure. But if so, what of the status of the postulation and of the theory and, again, the principles of reasoning used in their development? Are they to be taken as necessary truths? If they are necessary, how are they to be brought within the scope of my theory?

If my theory, including my theory of possible worlds, is true, then no exception will be found to the theory in any possible world, thus making the theory necessary. But this seems trivial. If that is all that there is to the necessity of the theory, it does not seem very necessary.

Consider a concrete instance: I postulated a world of states of affairs having individuals and universals as constituents. The constituents, I argued, cannot appear outside states of affairs. But what is the status of this prohibition? Why must it hold ‘in every possible world’?

Again, in defending the theory I had to reason about it. I pointed out what I take to be its consequences – for instance, that it forbids alien universals in other possible worlds.

In this chapter, we will present an alternative way of characterizing zeroorder logics. In some of its instances, this new way will be familiar to the reader with experience in elementary symbolic logic as “truth functional” logic. As we shall see, our characterization will be somewhat more general than the usual presentation. It is, however, in an important sense not as general as it might be. The overall methodology, as we shall see, consists of defining a class of mathematical structures which, when they are playing this role, are called model structures. One then defines a relation, analogous in some sense to “truth,” between wffs and model structures; this relation is usually called satisfaction. Substantial parts of the theory then are not specific to the particular version of satisfaction we will introduce in this chapter but are relative to whatever structures and “satisfaction” are used. We will indeed introduce a few of these alternatives in chapter 11. In this chapter (and in chapter 12), we will be addressing perhaps the mathematically simplest of these: those which have the variables ranging over arbitrary sets and which copy the functional structure of the logics over which they are defined. When necessary we will refer to this family of model theory as “extensional model theory.”

Our next task is to formulate exactly the notion of a derivation of a wff A from a set of wffs α (in a calculus). The basic notion is that of a string or sequence of wffs each of which is an element of α or follows from earlier wffs by application of a rule of the calculus and whose last element is A. For suitable calculi, a minor revision of this condition would suffice to define derivation. But because we also want to allow systems that use subordinate derivations (sometimes called “natural deduction” systems), we will have to take a more complicated path. Perhaps the paradigm of the kind of rule we have to accommodate is “conditionalizcuion, ” which asserts that if we can derive B from a set of premises a together with an additional premise A, we can derive “if A, then B” from the premise set a. Intuitively, we have a sequence of derivations in the first sense, except that the rules must allow reference to earlier “derivations” in the sequence. Here the matter is complicated by the fact that, normally at least, it is not the details of the subordinate derivation, but what conclusion follows from what premises, which is the relevant information. However, the fact that the principal derivation has certain premises and conclusions unfortunately does not specify what wffs are allowed to be premises and conclusions in the subordinate derivations.

Before starting our discussion of first-order systems, we want to point out that the expressive means provided in zero-order logic (variables and connectives), though simple, essentially exhaust what can be said on a single level (with a partial exception soon to be noted). Connectives provide in principle the means of expressing any transformation - provided the property in question holds of all wffs, a restriction implied by saying that we are talking about logic. From this point of view connectives can be considered the structural correlate of transformations from wffs to wffs that we have called wff functions.

The most basic feature of our generalization is that we provide two levels of well-formed expressions, called respectively terms and well - formed formulae (wffs). We will in addition want to extend the expressive means in two directions. The first of these generalizations consists of allowing operators whose character (“value”) and arguments are allowed to vary over the elements of either of the levels, though each argument position of these operators will be restricted to one level or the other. The second extension involves a kind of indexed operator of which the familiar universal and existential quantifiers are the best known examples.

Accordingly, our formal presentation is generalized so that our infinite family of sets {Sa} runs over an index set such that a decomposes into a natural number i and a sequence of i+1 0's and 1's.