N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).

4. Some Nominalistic Reductions

Some statements that seem to be about abstract entities can be rephrased in well-known ways as statements about concrete objects. Thus, where "A" and "B" are thought of as fixed terms and not as bindable variables, the statement:

Class A is included in Class B

may be rephrased as:

Everything that is an A is a B.

The phrases "is an A" and "is a B" here are predicates of concrete objects, and are regarded as naming nothing in themselves; that is to say, the positions that they occupy are treated as inaccessible to bound variables.

Certain statements that even involve explicit quantification over classes are replaceable by equivalent statements that conform to the tenets of nominalism. To take a simple example, the statement:

Class A is included in some class other than A

is equivalent to:

Something is not an A.

Statements purporting to specify sizes of finite classes of concrete objects are also easily accommodated. Thus the statement:

Class A has three members

may be rendered:

There are distinct objects x, y, and z such that anything is an A if and only if it is x or y or z;

i.e.:

(]x)(]y)(]z)(x =/= y & y =/= z & x =/= z & (w)(Aw <-->: w = x .V. w = y .V. w = z)).

Obviously any statement affirming or denying that there are just, or at least, or at most, a certain number of concrete individuals satisfying a given predicate can be readily translated in similar fashion, provided the translation is short enough to fit into the universe.⁸

The definition of ancestorhood in terms of parenthood according to Frege's method seems to involve a class-variable even more essentially. The definiens of "b is ancestor of c" would run thus:

b is distinct from c; and, for every class x, if c is a member of x and all parents of members of x are members of x then b is a member of x;

i.e.:

b =/= c & (x){c epsilon x & (y)(z)(z epsilon x & Parent yz .-->. y epsilon x) .-->. b epsilon x}.

But we can translate this sentence also with the help of the notation "Part st", meaning that the individual s is part (or all) of the individual t.⁹ We need only replace "class" by "individual", and "member" by "part", provided we also stipulate that b be a parent and c have a parent. This added stipulation insures that b and c be single whole organisms, rather than fragments or sums of organisms. In symbols, "b is ancestor of c" becomes:

b =/= c & (]u) Parent bu & (]w) Parent wc & (x){Part cx & (y)(z)(Part zx & Parent yz .--> Part yz) .--> Part bx}.

Clearly the above method of translation presupposes that an individual may be spatio-temporally scattered, or discontinuous. It presupposes that continuity is not necessary for concreteness. A broken dish is no less concrete than a whole one, but merely has more complicated boundaries; and any totality of individuals, however disperse in space and time, counts as an individual in turn. Individuals, thus liberally construed, serve some of the purposes of classes, as is evident from the above treatment of "ancestor". But it is by no means true that we can in general simply identify any class of individuals with a scattered single individual, and reconstrue "member" as "part". The individual composed of all persons, e.g., has many parts that are not persons; some of these parts are parts of persons, and some consist of many persons or of parts of many persons. In the above analysis of "ancestor", we were able to overcome this difficulty by inserting the clause "(]u) Parent bu & (]w) Parent wc". Commonly, however, this kind of difficulty admits of no such simple solution.

The two-place predicate "is ancestor of" is, to borrow terminology from the platonistic logic of relations, the (proper) ancestral of the two-place predicate "is parent of". We have seen, above, how it can be defined. But the scheme used there does not work for the ancestral of every two-place predicate of individuals. It works so long as every individual in the field of the predicate has some part that has no part in common with any other individual in that field. At the present writing we know of no way of defining the ancestral of every two-place predicate of individuals nominalistically.

A rather different problem is raised by such statements as:

There are more cats than dogs.

As pointed out earlier, we are already able to deal with such statements as "There is at least one cat and not at least one dog" and "There are at least two cats and not at least two dogs". An alternation of enough successive statements will be true if and only if there are more cats than dogs, and because it will contain at least one component statement that is true in view of the actual number of cats and of dogs. Use of this method requires, first, knowledge that in all space-time there are not more than so many (say fifty trillion) dogs, and second, a prodigious amount of writing or talking. Even though the requisite knowledge be available, the practical difficulties of actually writing or speaking the translation of the statement about cats and dogs would be prohibitive.

A better method of translation makes use of the predicate "is part of" and another simple auxiliary predicate: "is bigger than". The predicate "is a bit" is then so defined that it applies to every object that is just as big as the smallest animal among all cats and dogs. In other words, "x is a bit" is defined to mean that for every y, if y is a cat or a dog and is bigger than no other cat or dog, then neither is x bigger than y nor is y bigger than x. For brevity we shall call x a bit of z when x is a bit and is part of z. Now if and only if there are more cats than dogs will it be the case that every individual that contains at least one bit of each cat is bigger than some individual that contains at least one bit of each dog. (Such an individual will of course be spatio-temporally scattered.) Accordingly we may translate the sentence "There are more cats than dogs" as follows:

Every individual that contains a bit of each cat is bigger than some individual that contains a bit of each dog.

(Symbolic transcriptions are omitted here, as they will be given later for parallel cases: D9-10.)

This method of translation has the great advantage, over the first method suggested, that there is no practical difficulty about writing down an actual translation, regardless of the multiplicity of individuals concerned. But, like our method of defining the ancestral, it is not completely general. It will still work if, in place of "is a cat" and "is a dog", we choose any other two predicates each of which is such that the individuals fulfilling it are discrete from one another. Thus it holds good for such a case as:

There are more human cells than humans,

and indeed for most cases where such numerical comparisons are made in ordinary discourse. It has an important use in nominalistic syntax, as we shall see later. Moreover, by a relatively simple change it can be made general enough to work wherever each individual fulfilling either of the two predicates has a part that has no part in common with any other individual fulfilling that predicate. And in addition there are ways of modifying the method to take care of certain cases where even this latter condition is not satisfied But we have not found any general formulation that will cover all cases regardless of how the individuals concerned overlap one another.

The method will, however, help us in finding a nominalistic reduction for even so platonistic-sounding¹⁰ a statement as:

There are more age-classes than grade-classes in the White School.

We just replace this by:

There are more age-wholes than grade-wholes in the White School,

where an age-whole is the individual composed of all pupils in the school who were born during a single calendar year, and a grade-whole is an individual composed of all pupils who receive equally advanced instruction. The new sentence is then readily translated in the same way as the one about cats and dogs.

A combination of devices already described enables us to translate a statement like:

There are exactly one-third as many Canadians as Mexicans.

Letting "the Mexican whole" stand for the individual that is comprised of all Mexicans, the translation runs:

There are some mutually discrete wholes x, y, and z such that each is comprised of Mexicans and such that x + y + z = the Mexican whole; and there are exactly as many Canadians (in all) as there are Mexicans in x and as in y and as in z.

The last clause may then be further translated by a slight variation of the method used in the example of cats and dogs.

The foregoing samples will illustrate some of the means that remain in our hands for interpreting statements that prima facie have to do with abstract entities. Certainly we have not as yet reached our goal of knowing how to deal with every statement we are not ready to dispense with altogether. But there is as yet no convincing reason for supposing the goal unattainable. Some of the devices used above are rather powerful, and by no means all the possible methods have been explored.

Since, however, we have not as yet discovered how to translate all statements that we are unwilling to discard as meaningless, we describe in following sections a course that enables us -- strictly within the limitations of our language and without any retreat from our position -- to talk about certain statements without being able to translate them.

---

Notes

⁸ The nominalist need not necessarily regard such a sentence as "There are 10¹⁰⁰⁰ objects in the universe" as meaningless, even though there be no translation along these lines. For, this sentence can be translated as "The universe (as an individual) has 10¹⁰⁰⁰ objects as parts" where "has 10¹⁰⁰⁰ objects as parts" is taken as a primitive predicate of individuals. But while this translation satisfies purely nominalistic demands, there may be extranominalistic reasons of economy or clarity for wanting a translation that contains no such predicate. And wherever and for whatever reasons a translation of an expression is wanted in terms of certain predicates or a certain kind of predicates, the search for such a translation is a problem for the nominalist -- though of course neither he nor anyone else claims that every predicate can be defined in terms of every possible set of others.

⁹ A systematic treatment of "part" and kindred terms will be found in "The Calculus of Individuals and its Uses" by Henry S. Leonard and Nelson Goodman in Journal of Symbolic Logic, Vol. 5 (1940), pp. 45-55. Earlier versions were published by Tarski and Lesniewski. Although all of these would have to undergo revision to meet the demands of nominalism, such revision is for the most part easily accomplished and does not affect any of the uses to which the terms in question are put here. [See SA II].

¹⁰ We use "platonistic" as the antithesis of "nominalistic". Thus any language or theory that involves commitment to abstract entity is platonistic.

---

Contents -- Go to §5