N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).
3. The Nominalist's Problems
By renouncing abstract entities, we of course exclude all predicates that are not predicates of concrete individuals or explained in terms of predicates of concrete individuals. Moreover, we reject any statement or definition -- even one that explains some predicates of concrete individuals in terms of others -- if it commits us to abstract entities. For example, until we find some way of construing "is an ancestor of" in terms of "is a parent of" other than the way the ancestral of a relation is usually defined in systems of logic,6 the relationship between these predicates remains for us unexplained. We shall, then, face problems of reducing predicates of abstract entities to predicates of concrete individuals, and also problems of constructing certain predicates of concrete individuals in terms either of certain others or of any others that satisfy some more or less well-defined criteria. Apart from those predicates of concrete objects which are permitted by the terms of the given problem to appear in the definiens, nothing may be used but individual-variables, quantification with respect to such variables, and truth functions. Devices like recursive definition and the notion of ancestral must be excluded until they themselves have been satisfactorily explained. We are not as nominalists concerned with the motives behind the demand that a given predicate of concrete individuals be defined in terms of certain other such predicates. Naturally the demand may often arise from a feeling that the latter predicates are in some sense the clearer, and we may as persons often share this feeling; but purely as nominalists we know no differences of clarity among predicates of concrete individuals.7 Our problem is solely to provide, where definitions are called for, definitions that are free of any terms or devices that are tainted by belief in the abstract We shall naturally first try to find definitions where, for varied reasons, we feel they are most urgently needed; and we shall not waste time looking for definitions in terms of predicates that we suppose to be ambiguous or self-contradictory. But, as has perhaps been illustrated by the case of "ancestor" and "parent", it cannot be said that the explanation of one predicate in terms of another is of interest only if the latter is regarded as clearer. Indeed, if we have only a pseudo-explanation (involving abstract entities) relating predicates of individuals, the problem of replacing it by a genuine construction has as immediate interest as the problem of defining a given predicate in terms of others that come up to a certain standard of clarity, or the problem of explaining a predicate of abstract entities. Notes 6 The usual definition, which was first set forth by Frege in 1879 (Begriffschrift, p. 60), has become well known through Whitehead and Russell and other writers. It is presented once more in the next section. 7 It might be supposed that the nominalist must regard as unclear any predicate of individuals for which there is no explanation that does not involve commitment to abstract entities. But unless 'explanation' as here intended depends upon standards of clarity, which do not concern the nominalist as nominalist, a suitable explanation can always be supplied trivially by equating the predicate in question with any arbitrarily concocted single word. Contents -- Go to §4