We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether.

We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write ‘x is a dog,’ or ‘x is between y and z’; for here ‘is a dog’ and ‘is between … and’ can be construed as syncate-gorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values. In ‘x is a dog,’ only concrete objects are appropriate values of the variable. In contrast, the variable in ‘x is a zoölogical species’ calls for abstract objects as values (unless, of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.

We shall be concerned in this paper with properties of relations. For simplicity we shall consider only two relations, a binary relation R and a ternary relation S, but the generalization to the case of any number of any relations presents no difficulty. We shall denote by A0 the field of the relations R and S and assume that A0 is infinite (not inductive).

The properties to be dealt with are expressible in a symbolic language L whose principal features will be sketched below.

Alonzo Church suggested to the writer that a certain problem of Thue [6] might be proved unsolvable by the methods of [5]. We proceed to prove the problem recursively unsolvable, that is, unsolvable in the sense of Church [1], but by a method meeting the special needs of the problem.

Thue's (general) problem is the following. Given a finite set of symbols α1, α2, …, αμ, we consider arbitrary strings (Zeichenreihen) on those symbols, that is, rows of symbols each of which is in the given set. Null strings are included.

In the first paper of the above main title, one of us has proved that any formula of the first order predicate calculus is equivalent (as to being satisfiable or not) to some binary first order formula having a prefix of the form (Ex1)(x2)(Ex3) … (xn) and containing a single predicate variable. This result is an improvement of a theorem of Ackermann stating that any first order formula is equivalent to another with a prefix of the above form but saying nothing about the number of predicate variables appearing therein. Hence the question arises if other theorems reducing the decision problem to the satisfiability question of the first order formulas with a prefix of a special form can be improved in like manner. In the present paper we shall answer this question concerning Gödel's reduction theorem stating that any first order formula is equivalent to another the prefix of which has the form

The possibility of constructing a numerical equivalent of a system of trans-finite ordinals, in recursive number theory, was briefly indicated in a previous paper, where consideration was confined to ordinals less than ε (the first to satisfy ε = ω). In the present paper we construct a representation, by functions of number-theoretic variables, for ordinals of any type.

In addition to definite numerals, and numeral variables, we introduce majorant variables σ, ω, ωr , r ≧ 1. A relation containing a single majorant variable σ is required to hold, not necessarily for all non-negative integral values of σ, but for all values greater than some assigned constant.

In previous papers we developed two functional calculi of first order based on strict implication which we called S2 and S4. In the present paper, these systems will be extended to include a functional calculus of second order with the purpose of introducing the relation of identity of individuals.

Primitive symbols. ̂ {the abstraction operator, the blank space to be replaced by an appropriate variable}.

The philosophical dispute over whether there are universale, or abstract entities, is not a dispute over the admissibility of general terms. Both sides will agree that general terms, e.g. ‘man,’ and perhaps even abstract singular terms such as ‘mankind’ and ‘7’, are meaningful, in the sense at least of participating in statements which as wholes are true or false. Where the platonists (as I shall call those who accept universale) differ from their opponents, the nominalists, is in positing a realm of entities, universals, corresponding to such general or abstract words.

The platonist is likely to regard the word ‘man’ as naming a universal, the class of men or the property of being a man, much as the word ‘Caesar’ names the concrete object Caesar. Actually, however, this is an extraneous detail. The platonist could equally well concur with the nominalist in the view that general terms, though meaningful in context, are not names at all.

There are logicians, myself among them, to whom the ideas of modal logic (e. g. Lewis's) are not intuitively clear until explained in non-modal terms. But so long as modal logic stops short of quantification theory, it is possible (as I shall indicate in §2) to provide somewhat the type of explanation desired. When modal logic is extended (as by Miss Barcan1) to include quantification theory, on the other hand, serious obstarles to interpretation are encountered—particularly if one cares to avoid a curiously idealistic ontology which repudiates material objects. Such are the matters which it is the purpose of the present paper to set forth.

In his Mathematical logic Quine characterizes the theorems of quantification theory by six principles: