It is a scientific platitude that there can be neither precise control nor prediction of phenomena without measurement. Disciplines are diverse as cosmology and social psychology provide evidence that it is nearly useless to have an exactly formulated quantitative theory if empirically feasible methods of measurement cannot be developed for a substantial portion of the quantitative concepts of the theory. Given a physical concept like that of mass or a psychological concept like that of habit strength, the point of a theory of measurement is to lay bare the structure of a collection of empirical relations which may be used to measure the characteristic of empirical phenomena corresponding to the concept. Why a collection of relations? From an abstract standpoint a set of empirical data consists of a collection of relations between specified objects. For example, data on the relative weights of a set of physical objects are easily represented by an ordering relation on the set; additional data, and a fortiori an additional relation, are needed to yield a satisfactory quantitative measurement of the masses of the objects.

The major source of difficulty in providing an adequate theory of measurement is to construct relations which have an exact and reasonable numerical interpretation and yet also have a technically practical empirical interpretation. The classical analyses of the measurement of mass, for instance, have the embarrassing consequence that the basic set of objects measured must be infinite. Here the relations postulated have acceptable numerical interpretations, but are utterly unsuitable empirically. Conversely, as we shall see in the last section of this paper, the structure of relations which have a sound empirical meaning often cannot be succinctly characterized so as to guarantee a desired numerical interpretation.

Plutarch berichtet, Chrysipp habe gesagt, “daß die Konjunktionen aus 10 Aussagen zahlenmäßig 1.000.000 übertreffen, ohne daß er das zuvor selbst genau untersucht hat oder von Fachleuten die Wahrheit erkundet hat …. Den Chrysipp widerlegen alle Arithmetiker, unter ihnen Hipparch, der beweist, daß ihm (Chrysipp) ein sehr großer Rechenfehler unterlaufen ist insofern, als nämlich das Bejahende 103.049 Konjunktionen ausmache, das Verneinende 310.952.”

Ohne Zweifel liegt hier eine der frühesten uns bekannten Aufgaben aus dem Gebiete der Kombinatorik, − neben der Berechnung der Zahl der Silben durch Xenokrates, die sich aus der Zusammenstellung von Buchstaben ergeben −, vor. Außerdem handelt es sich um die erste Anwendung der Kombinatorik in der Logik, bei der Werte auftreten, die nicht durch einfaches Abzählen gewonnen sein können.

Es ist bisher weder eine befriedigende Deutung der von Plutarch gegebenen Zahlenwerte noch, von der Interpretation ausgehend, eine Wiederherstellung der von den Alten benutzten Formeln gelungen.

Adam of Balsham (Parvipontanus) writing his Ars Disserendi in 1132 A.D. reaches the notion of a set A equinumerous with a set B containing A as a proper sub-set. References by page and line are to [Minio-Paluello].

It can be asked, says Adam (92,13) whether this one is something, and whether that one is something, and these are two askables (interrogabilia), and again whether a third is something, and similarly of anything whatever. He thinks, then, of the totality of ‘objects’ (92,12) as numbered. But this question can be put not only about those which are not askables; of any askable too it can be asked whether it is something. Whether the questions be actually put or not, makes no difference; even when they are not put, the points remain askable. Thus the totality of askables is seen to be equinumerous with the totality of objects, which last comprises both the objects which are not askables and those which are. And so the same set is more numerous than itself.

Conscious that this paradoxical situation may be ascribed to his novel terminology and methods, Adam points out that a similar puzzle arises over things enunciated. Even if you do not want to term them ‘enunciables,’ still they are generally said to be true or false. Take the true ones, and compare them as to number with the totality of objects of which enunciations can be made. There he leaves this second version, but it is easy to see how the same result as before can be reached.

The argument lightly paraphrased in the last paragraph but one concludes with the words ‘which is impossible’ (92,21). Instead, however, of rejecting some of his premisses, Adam lets everything stand, and breaks off the discussion on the ground that his purpose is to expound a finite process of achievement (artis viam finitam) rather than what infinitely exceeds our capability to construct − or perhaps, the infinitely extending realm of possibility (possibilitatis excessum infinitum, 93,3–5).

In 1885 Peirce axiomatised the propositional calculus on the basis of five ‘icons’ or axioms, which with Cpq for ‘If p then q’ and o for a false proposition, and with the source in Peirce's [2] beside each, may be represented as follows: 1. Cpp (3.376); 2. CCpCqrCqCpr (3.377); 3. CCpqCCqrCpr (3.379); 4. Cop (3.381); 5. CCCpqpp (3.384). From these Peirce proves, among other things, 6. CpCCpqq (3.377), 7. CpCqp (3.378), and with Np (Not p) for Cpo, 8. CNNpp (3.384).

By a well-known result of Wajsberg's II. 93, the entire two-valued propositional calculus is derivable by substitution and detachment from 3, 4, 5, and 7 (with Df. N). In Berry [1] this is taken to show the completeness of Peirce's basis; but this proof will not do as it stands. For Peirce's proof of 7 consists in first passing from 1 to CqCpp by the argument that ‘to say that (x ⥽ x) is generally true is to say that it is so in every state of things, say in that in which y is true’; and then from CqCpp to 7 by 2. This procedure clearly involves, formally, the use not only of substitution and detachment but of a rule to infer ⊦ Cqα. from ⊦α. Peirce's basis is sufficient with substitution and detachment nevertheless; for by a less well-known result of Wajsberg's II. 93, 3, 4, 5, and 6 are sufficient in this sense, and Peirce does prove 6 by substitution and detachment from 1 and 2.

In attempting to reconstruct Rose's proof of Lemma 3.2 of [1], the present authors found what is apparently a different and simpler method, which moreover leads to a far stronger conclusion.

We are operating in the Heyting prepositional calculus as formulated on p. 3 of [1] or on pp. 82 and 101 of [2], and shall make use of relevant theorems on pp. 90, 113–119 of [2]. We shall use a, b, c, w, x, y, z as propositional variables.

We say that a conjunction is simple if each factor has one of the forms: (i) a, (ii) ¬a, (iii) a⊃b, (iv) a⊃(b∨c), (v) (a&b)⊃c, (vi) (a⊃b)⊃c.

The basic aim of the present paper is to simplify the axioms for many-valued quantification theory which were developed by Rosser and Turquette in Many-valued logics ([10], pp. 33–34 and pp. 63–64). The simplification which is achieved results not only from a reduction in the number of axiom schemes and rules of inference, but also by obtaining greater formal similarity to some elegant axioms for standard 2-valued quantification theory. This result will not be accomplished by establishing the formal interdeducibility between our simplified axioms and the Rosser-Turquette set. On the contrary, it will be shown that these two sets of axioms are not formally interdeducible.

What will be shown is that whenever the Rosser-Turquette axioms are used to define a plausible set of statements, this set of statements can be defined using our simplified axioms. The exact meaning of “plausible” will become clear from what follows, so for the moment it will be sufficient to remark that a set of statements is plausible if and only if each statement of the set is analytic in the sense that the statements take designated truth-values exclusively. In general, a Rosser-Turquette set of axioms would be used to define a set of plausible statements, and in such a case our simplified axioms could be used to achieve the same result.

In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question.

We shall discuss relational systems of the form

where A and R are non-empty sets; each element of R is an ordered triple 〈a, b, c〉, with a, b, c ∈ A.1 If the triple 〈a, b, c〉 belongs to the relation R, we write R(a, b, c); if 〈a, b, c〉 ∉ R, we write (a, b, c). If x0, x1 and x2 are variables, then R(x0, x1, x2) and x0 = x1 are predicates. The expressions (x0, x1, x2) and x0 ≠ x1 will be referred to as negations of predicates.

We speak of α1, …, αn as terms of the disjunction α1 ∨ … ∨ αn and as factors of the conjunction α1 ∧ … ∧ αn. A sentence (open, closed or neither) of the form

where each Qi (if there be any) is either the universal or the existential quantifier and each αi, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form.

The principal aim of the present paper is to sketch some mathematical applications of the work of the Hilbert school in the foundations of mathematics. Most of them depend on the ε-theorems of Hilbert-Bernays II [6], or of Ackermann [1], but instead the work of Gentzen [5] or Schütte [25], which does not use the ε-symbol, could have been applied.

The avowed purpose of all this work is described in the introduction to volume I of Hilbert-Bernays: the consistency of the usual principles of proof is to be established by means of finitist methods, or at least, by means of methods which are more “evident” or more “constructive” than the principles under discussion. This formulation seems to have several defects:

• (1) Since the notion of constructive proof is vague, the whole formulation of the program is vague; and though an exact formulation constitutes, of course, an interesting problem for the logician, because of this vagueness the mathematician does not find the program attractive.

• (2) The formulation does not cover too well the actual substance of the material contained in [6]; e.g. the ε-theorems for the predicate calculus go far beyond establishing mere consistency of the predicate calculus. It is significant that it is precisely these theorems which lead to the more interesting applications, e.g. the use of the ε-theorems with equality in the solution of Hilbert's 17th problem given below.

Hilbert's tenth problem is to find an algorithm for determining whether or not a diophantine equation possesses solutions. A diophantine predicate (of positive integers) is defined to be one of the form

where P is a polynomial with integral coefficients (positive, negative, or zero). Previous work has considered the variables as ranging over nonnegative integers; but we shall find it more useful here to restrict the range to positive integers, no essential change being thereby introduced. It is clear that the recursive unsolvability of Hilbert's tenth problem would follow if one could show that some non-recursive predicate were diophantine. In particular, it would suffice to show that every recursively enumerable predicate is diophantine. Actually, it would suffice to prove far less.

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.

The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.

The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:

(1) 0, 1, and the number variables are nfs.