It has been pointed out by Carnap that ‘probability’ is an equivocal term, which is used currently in two senses: (i) the degree to which it is rational to believe a hypothesis h on specified evidence e, and (ii) the relative frequency (in an indefinitely long run) of one property of events or things with respect to another. This paper is concerned only with the first of these two senses, which will be referred to as ‘the concept of confirmation,’ in order to avoid equivocation.

We may distinguish a quantitative and a comparative concept of confirmation. The general form of statements involving the former is

The degree of confirmation of the proposition h, given the proposition

e as evidence, is r (where r is a real number between 0 and 1).

In this paper the notation for a statement of this form is

The general form of statements involving the comparative concept is The proposition h is equally or less confirmed on e than is the proposition h′ on e′,

which may be symbolized as

The following abbreviations will also be used in discussing comparative confirmation:

The former of these two abbreviations means intuitively

The proposition h is less confirmed on evidence e than h′ is on e′. The latter means

The proposition h is confirmed on evidence e to the same degree that h′ is on e′.

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.

In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.

The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

In this paper I am going to set forth a formal system based on five inter-class relations. These relations exist respectively between a class of a's and a class of b's.

(i) if and only if every a is a b and every b is an a,

(ii) if and only if every a is a b and not every b is an a,

(iii) if and only if it is not the case that either every a is a b or every b is an a or no a is a b,

(iv) if and only if every b is an a and not every a is a b,

(v) if and only if no a is a b.

These relations between classes, which correspond, as will be seen, to the five relations between two circles a and b shown in the well-known Eulerian diagrams,

are of course connected in an intimate way with the four forms of proposition, A, E, I, O, of the traditional syllogistic logic. The French mathematician, J. D. Gergonne, seems to have been the first to recognize these relations explicitly and to understand their importance in syllogistic theory. It is therefore appropriate that they should be called by his name.

Gergonne first of all showed with reference to these relations what are the sufficient and necessary conditions of the truth of propositions of each of the four traditional forms: for example, an A proposition, ‘All a is b’ is true if and only if either the first or the second relation exists between the class of a's and the class of b's.

In certain applications of truth-functional logic it is of interest to determine classes of formulas equivalent to a given formula Φ under the hypothesis that certain conjunctions of letters of Φ are always false. Of especial interest is the case where the class to be determined is that of simplest normal truth functions. The problem of giving a calculation procedure for this question is evidently a more general form of the simplification problem as considered in [1] and [2]. The purpose of this note is to indicate how the procedure of [1] applies.

Let Π, Π′, Π″, … be fundamental formulas, in the sense of [2], such that every literal of Π, etc., is a literal of Φ and such that Π, Π′, Π″, … are all false for all values of the constituent literals. Then we say that Ψ is a weak simplest disjunctive normal equivalent of Φ if and only if Ψ ≡ Φ, under the hypothesis, and there is no Ψ′ such that Ψ′ is simpler than Ψ and Ψ′ ≡ Φ, under the hypothesis. Similarly, under exactly the same hypothesis, we say that X is a weak simplest conjunctive normal equivalent of Φ. When the class of always false fundamental formulas is empty, as in [1] and [2], we may speak of strong simplest forms and hence of the more general problem of strong simplification.

In [1] Quine has presented a method for finding the simplest disjunctive normal forms of truth functions. Like the tabular methods of [2] and [3], Quine's method requires expansion of a formula into developed normal form as a preliminary step. This aspect of his method to a certain extent defeats one of the purposes of a mechanical method, which is to secure simplest forms in complicated cases (perhaps by using a digital computer) [4]. In the present paper we develop a method for both disjunctive and conjunctive normal truth functions which is in some respects similar to Quine's but which does not involve prior expansion of a formula into developed normal form. Familiarity with [1] is presupposed.

We use the notations and conventions of [1] with the following exceptions and additions. ‘Φ’ names any formula, ‘Ψ’ any conjunction of literals, and ‘χ’ any disjunction of literals. Any disjunction of conjunctions of literals is a disjunctive normal formula and is designated by ‘ψ’; any conjunction of disjunctions of literals is a conjunctive normal formula and is designated by ‘X’. Note that we do not make use of Quine's notion of fundamental formulas. A formula Ψ occurring in a disjunctive normal formula ψ, provided it is a disjunct of ψ, is a clause; similarly for χ. We use ‘≠” for logical equivalence of formulas and ‘=’ for identity of formulas to within the order of literals in clauses and the order of clauses in normal formulas.

We prove the following theorem: If S is a consistent set of axioms for set theory, among whose consequences are the axioms of pairing and extensionality, and if S′ is the extension of S to all classes of sets of S, then S′ is consistent and remains so upon addition of an axiom expressing that the universal class has a well-ordering wherein every non-initial element has an immediate predecessor.

The basic idea will be use of the denumerable model of S guaranteed by the Skolem-Löwenheim theorem to obtain (i) a model of S′ in the domain of classes of natural numbers, and (ii) a correspondence between representatives (in the model of S′) of sets of S′ and natural numbers. One can then select a class of natural numbers in the model of S′ which will well-order (in the sense of the model of S′) the representatives of sets of S′ according to the numbers corresponding to them.

Specifically, let the formulas of S involve only one kind of variable and the sole predicate ‘ϵ’. The formulas of S′ can be thought of as identical with those of S; but the axioms of S′ will consist of (i) those of S with quantifiers restricted to sets by clauses ‘(∃z)[y ϵ z]’, (ii) those provided by the schema ‘(∃x) ▪ y ϵ x ≡y [(∃z)[y ϵ z] ▪ A]’, where ‘A’ is any formula of S′ in which ‘x is not free, and (iii) the axiom of extensionality.

In applying the method of arithmetization to a proof of the completeness of the predicate calculus, Bernays has obtained a result which, when applied to set theories formulated in the predicate calculus, may be stated thus.

1.1. By adding an arithmetic sentence Con(S) (expressing the consistency of a set theory S) as a new axiom to the elementary number theory Zμ (HB II, p. 293), we can prove in the resulting system arithmetic translations of all theorems of S.

It then follows that things definable or expressible in S have images in a simple extension of Zμ, if S is consistent. Since S can be a “strong” system, this fact has interesting consequences. Some of these are discussed by me and some are discussed by Kreisel. Kreisel finds an undecidable sentence of set theory by combining 1.1 and the Cantor diagonal argument. I shall prove below, using similar methods, a few further results, concerned with the notions of truth and designation. The method of numbering sets which I use (see 3.1 below) is different from Kreisel's. While the method used here is formally more elegant, Kreisel's method is much more efficient if we wish actually to calculate the numerical values.

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].

One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula (Ex)(x, a), with Gödel-number a, is provable or not. Here (x, y) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y.

In this note we present a solution of the previous problem with respect to the system Zμ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function (k, l) used below is definable.

The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Zμ containing no free variables, whose Gödel number is a, then ({}) stands for (Ex)(x, a) (read: the formula with Gödel number a is provable in Zμ); if is a formula of Zμ containing a free variable, y say, ({}) stands for (Ex)(x, g(y)}, where g(y) is a recursive function such that for an arbitrary numeral the value of g() is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing (), where is an arbitrary numeral, for (Ex){x, ).

In the following pages we wish to clarify some questions concerning definitions of relations in terms of other relations. A study is made of how a definition of a relation R1 in terms of another relation R2 works in a given specified universe U. The definition is regarded as a correlation rule between the “possible extensions” of R1 and R2. There are no difficulties in formulating necessary, and in a certain sense also sufficient, formal conditions for the existence of a definition relation between R1 and R2.

In particular, we are concerned with questions such as: “Is every relation of the form s1 definable in terms of some relation of the form s2?” (§ IV) A related question concerns the criterion used by Goodman1 for degrees of complexity, namely, whether for every relation R2 of the form s1 there is some relation R2 of the form s2 such that R1 and R2 are mutually definable (§ V). Some concrete results concerning this condition are stated in the theorems 22–24.

The “complexity function” which is defined (definition 17) and which very naturally suggests itself from the discussion of the definability conditions, seems to furnish an adequate measure of the sort of complexity discussed by Goodman.

Consider the statement:

(1) What I am saying cannot be proved.

Let us suppose that this statement can be proved. Then it must be true, i.e., in its own words, it cannot be proved, which contradicts our supposition.

Therefore the statement cannot be proved, since our supposition has led to a contradiction. In other words, the statement is true. In this way we have proved the statement.

Therefore the statement both can and cannot be proved.

In the second part of the argument, if we speak of proving (1) in a given formal system S, then we cannot draw the conclusion that we have proved the statement in S. For it is possible that the argument cannot be formalized in S. Indeed, as we know, Gödel shows in his famous article of 1931 that for suitable systems S, it is possible to construct in S a statement saying in effect that the statement itself is not provable in S. The conclusion Gödel draws, we may recall, is that the statement thus constructed is true but not provable in S. Thus no contradictions arise in this way if we are confined to provability in a given system.

A dual of (1) is the following:

(2) What I am saying can be refuted.

Let us suppose that this statement is true, or, in its own words, that it can be refuted. Then it must be false, which contradicts our supposition.

Therefore the statement is false, since our supposition has led to a contradiction. In this way, we have refuted the statement. We see that the statement can be refuted; in other words, it is true.

The purpose of this paper is to analyze rational betting. In particular, we concentrate on one necessary feature of rational betting, the avoidance of certainty of losing to a clever opponent. If a bettor is quite foolish in his choice of the rates at which he will bet, ah opponent can win money from him no matter what happens.

This phenomenon is well known to professional bettors—especially bookmakers, who must as a matter of practical necessity avoid its occurrence. Such a losing book is called by them a “dutch book.” Our investigations are thus concerned with necessary and sufficient conditions that a book not be “dutch.”

De Finetti [3] has started with the same idea and used it as a foundation for the theory of probability. It is our aim to consider the same subject more precisely and attempt to answer some questions about desirable features of a confirmation function. We wish to connect the ideas of De Finetti with those of Carnap [1] and Hossiasson-Lindenbaum [6]. The results expressed by Theorem 1 are essentially contained in De Finetti's work. Theorems 3 and 4 seem to be new.

The confirmation functions which we consider will be functions with two sentences as arguments taking real numbers as values. Intuitively, C(h, e) will represent the rate at which a bettor would be willing to bet on the hypothesis h if he knew the information expressed by the sentence e, the evidence.

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.

On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.

The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

Carnap (in [1], p. 566) has remarked that his measure function m* is fitting for finite languages using a fixed number of one-place predicates, i.e., for any sentence i, m*(i) is the same in all such restricted finite languages in which i occurs. The main purpose of this brief note is to show by means of a counter-example that m* does not have the intuitively desirable property of fittingness when we consider languages using two-place predicates.

We first state for finite languages a general theorem (related to the results in [2]) which guides the construction of large numbers of counter-examples.

Theorem. Let be the symmetric group of n letters. In a language which consists of a finite number of predicates and n individual names, the number of state descriptions in a structure description is equal to the index of some subgroup G in .

Proof. Let Σ be a structure description (of a language satisfying our hypothesis) and let S be an arbitrary state description in Σ. Let π be a permutation in , and let π*S be the state description which results from S by applying the permutation π to the individual names of . It is easily verified that for every π in π*S ∊ Σ. Also, for π, ψ in ,

We define the set G of permutations as follows :

.

A system SF, closely related to NF, is outlined here. SF has several novel points of simplicity and interest, (a) It uses only one basic notion, from which all the other concepts of logic and mathematics may be built definitionally. Three-notion systems are common, but Quine's two-notion IA has for some time represented the extreme in conceptual economy, (b) The theorems of SF are generated under just three rules of analysis, which unify into a single postulational principle, (c) SF is built solely in terms of what is commonly, known as the “natural deduction” method, under which each theorem is attacked primarily as it stands, by means of a very small body of rules, rather than less directly, through a very large, potentially infinite backlog of theorems. Although natural deduction is by no means new as a method, its exclusive applications have previously been relatively limited, not even reaching principles of identity, much less set theory, relations, or mathematics proper, (d) SF is at least as strong as NF, yielding all of its theorems, which are expressed here in forms analogous to those of the metatheorems in ML. If NF is consistent, so is SF. The main points in the relative consistency proof are set forth below in section seven.