In Part I of this paper no official significance was given to free occurrences of restricted-variables. Indeed doubt has been expressed as to the desirability or feasibility of such usage. Thus, to quote from Rosser [4], p. 146:

“We now raise the question of the significance of F (α), in which the occurrences of α are free [α a variable subject to the restriction K(α)]. In deciding to take (x).K(x) ⊃ F(x) and (Ex).K(x).F(x) as the meanings of (α)F(α) and (Eα)F(α), we were guided by the intuitive meanings. In the case of F(α), the intuitive meaning does not furnish a satisfactory guide. In everyday mathematics, if it has been agreed that α stands for a quantity satisfying the restriction K(α), it is commonly the case that, if one is assuming F (α), then K(α)&F(α) is understood, but if one is trying to prove F (α), then K(α) ⊃ F (α) is understood. It seems that in symbolic logic perhaps it is best not to give any especial significance to α in F(α) when it occurs free.”

Despite this anomalous behaviour we shall, in this Part, show how one can have unhampered use of free restricted-variables in appropriate contexts.

Recent attempts to analyse or explicate assertion-statements and other kinds of indirect discourse in formal terms seem to have overlooked an even greater difficulty than the familiar problems arising about equivalence, synonymy, translatability, etc. This difficulty is created by the fact that in judicial, journalistic, or historical fact-finding, as well as in everyday conversation, we frequently cite statements about a witness's truthfulness alongside the report of his testimony as the premisses from which we argue about the facts. It is then an essential part of our argument that it should appear to jump from one level of statement to another, and perhaps back again. So that the difficulty emerges: how can the formalisation of such an argument adopt any of the usual hierarchy-principles as a guarantee against semantical antinomies?

Consider, for example,

Alonzo Church has proposed a system of “restricted recursive arithmetic” for the analysis of computer circuits which include time delays and feedback. Restricted recursive arithmetic, which will be referred to as rra, was introduced by Church in this Journal, vol. 20 (1955), pp. 286–287, in a review of an article by Edmund C. Berkeley. It is the purpose of this paper to present some results which have been found for the system rra.

The concepts of ω-consistency and ω-completeness are closely related. The former concept has been generalized to notions of Γ-consistency and strong Γ-consistency, which are applicable not only to formal systems of number theory, but to all functional calculi containing individual constants; and in this general setting the semantical significance of these concepts has been studied. In the present work we carry out an analogous generalization for the concept of ω-completeness.

Suppose, then, that F is an applied functional calculus, and that Γ is a non-empty set of individual constants of F. We say that F is Γ-complete if, whenever B(x) is a formula (containing the single free individual variable x) such that ⊦ B(α) for every α in Γ, then also ⊦ (x)B(x). In the paper “Γ-con” a sequence of increasingly strong concepts, Γ-consistency, n = 1,2, 3,…, was introduced; and it is possible in a formal way to define corresponding concepts of Γn-completeness, as follows. We say that F is Γn-complete if, whenever B(x1,…, xn) is a formula (containing exactly n distinct free variables, namely x1…, xn) such that ⊦ B(α1,…,αn) for all α1,…,αn in Γ, then also ⊦ (X1)…(xn)B(x1,…,xn). However, unlike the situation encountered in the paper “Γ-con”, these definitions are not of interest – for the simple reason that F is Γn-complete if and only if it is Γ-complete, as one easily sees.

The inferences countenanced by the traditional rules of modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism, and the complex types of dilemma may be regarded as single applications of one rule of inference, “the rule of complementary elimination”. In the present paper, we shall discuss this rule informally and illustrate it in application to expressions written in the language of Principia Mathematica. Our illustrations will contain no connectives except for those for conjunction, disjunction, and negation; we use parentheses in place of dots; and we allow disjunction and conjunction to have any number of operands more than two.

In applying complementary elimination to a set of premises, we take the following three steps, (i) We form, merely by disjoining the premises, an expression which we shall call a premise disjunction, (ii) If we have n premises, we eliminate n minus one (or fewer) pairs of the following sort from our premise disjunction: each pair is such that one of its members is the negation of the other and both members are specific occurrences of disjuncts of our premise disjunction. We shall call such pairs complementary pairs, (iii) The formula obtained by means of our second step is one that may be made well-formed merely by eliminating parentheses or connectives other than negation; we make such elimination, and any formula we thus obtain is a consequence of our premises.

Faris [1] presents a system for non-void classes employing four primitive functors with another defined, and based on ten asserted and eleven rejected axioms, with a special rule of rejection analogous to Słupecki's for the syllogistic. Primitive functors are here reduced to three, axioms are simplified, and it is shown that the system is decidable by the easier syllogistic means, e.g. the tabular method of Thomas [3].

From the theses (original numbering to the right with axioms starred)

we can deduce

In Chapter X of his book on The structure of appearance, Goodman [1] studies the concept of linear order in finite linear arrays in terms of a two-place symmetrical predicate M, M(x, y) being read as x matches y. Although arising in the development of a nominalistic philosophical system, many problems relating to the properties of M are essentially of a mathematical and not of a philosophical nature. Some were considered by Fine in [2]. In the terminology of [1] and [2], the main result of this paper is that there is an effective method for embedding a weakly mapped array (consistently) in a uniform array. This carries with it the result that it is possible to obtain effectively a minimal uniform extension of the original array, that is, a uniform extension with minimum span for such an extension and with the minimum number of elements possible for a uniform extension with this span.

Greater economy can be effected in the primitive rules for the system K of basic logic by defining the existence operator ‘E’ in terms of two-place abstraction and the disjunction operator ‘V’. This amounts to defining ‘E’ in terms of ‘ε’, ‘έ’, ‘o, ‘ό’, ‘W’ and ‘V’, since the first five of these six operators are used for defining two-place abstraction.

We assume that the class Y of atomic U-expressions has only a single member ‘σ’. Similar methods can be used if Y had some other finite number of members, or even an infinite number of members provided that they are ordered into a sequence by a recursive relation represented in K. In order to define ‘E’ we begin by defining an operator ‘D’ such that

Here ‘a’ may be thought of as an existence operator that provides existence quantification over some finite class of entities denoted by a class A of U-expressions. In other words, suppose that ‘a’ is such that ‘ab’ is in K if and only if, for some ‘e’ in A, ‘be’ is in K. Then ‘Dab’ is in K if and only if, for some ‘e and ‘f’ in A, ‘be’ or ‘b(ef)’ is in K; and ‘a’, ‘Da’, ‘D(Da)’, and so on, can be regarded as existence operators that provide for existence quantification over successively wider and wider finite classes. In particular, if ‘a’ is ‘εσ’, then A would be the class Y having ‘σ’ as its only member, and we can define the unrestricted existence operator ‘E’ in such a way that ‘Eb’ is in K if and only if some one of ‘εσb’, ‘D(εσ)b’, ‘D(D(εσ))b’, and so on, is in K.

Hasenjaeger [2] has answered a question raised by Hilbert and Bernays [4] p. 279, by showing that the axiom

which, within lower predicate calculus, is independent of the remaining axioms of the system (B) p. 273, is no longer independent if the axiom 0 ≡ 0 of (B) is replaced by a ≡ a. The axioms from which (Cov) can be proved form the following system (B1) (obtained from (B), of course, by dropping (Cov) and replacing 0 ≡ 0 by a ≡ a):

In what follows we consider (B1) and also the systems (B′) and (B′1) which are obtained from (B) and (B1) respectively by dropping the induction axiom and which clarify the role of the induction axiom.

1. (B1) is equivalent to (B). For (B) can be obtained from (B1) by Hasenjaeger's result. On the other hand, by pp. 263 and 261, the axioms (<1) and (J1) of (B1) can be proved from the axioms (<3) and

which belong to the system (A) p. 263. Therefore (B1) can be obtained from (A) and the induction axiom, and hence from (B).

La théorie de la quaternalité, telle que Piaget et Gottschalk l'ont appliquée aux connectifs binaires du calcul bivalent, appelle quelques précisions et compléments.

Les seize connectifs ne comportent que deux quaternes complets: celui des jonctions et celui des implications. Leurs similitudes formelles ne doivent pas dissimuler une différence dans leur mode de construction. Elle apparaît sur leurs diagrammes (inspirés du “carré logique” traditionnel) par la place de la cellule initiale et par celles des signes barrés du trait vertical de la négation:

En effet, la dualité qui se combine avec celle de la négation contradictoire (DN, flèche diagonale) est, pour les jonctions, la dualité “morganienne” (DM, flèche verticale) et pour les implications, la dualité des réciproques (DR, flèche horizontale). Car dans les jonctions, liaisons symétriques, il n'y a point de vraies réciproques; inversement, les implications ne comportent pas (pourvu, bien sûr, qu'on ne les traduise pas en termes de jonctions) de dualité morganienne. Le quatrième terme, obtenu par la combinaison des deux relations génératrices, sera donc, pour les jonctions, une contreduale (DM × DN) et, pour les implications, une contreréciproque (DR × DN).

Logical falsehoods are usually barred in confirmation theory from serving as evidence statements. I wish to study here some three major ways in which this restriction can be lifted.

1. Consider the following axioms for the two-argument confirmation function C:

A1. 0 ≤ C(h, e) ≤ 1;

A2. If ⊦h ≡ h′ and ⊦e ≡ e′, then C(h, e) = C(h′, e′);

A3. If ⊦e ⊃ h, then C(h, e) = 1;

A4. If ⊦e ⊃ ˜(h.h′), then C(h⋁h′, e) = C(h, e)+C(h′, e);

A5. C(h.h′, e) = C(h,e)×C(h′, h.e);

where h, h′, e, and e′ are any four statements from a consistent language L and where ‘⊦’ is short for ‘is logically true in L.’ A1—A5 are consistent, as shown by the following construction:

C1. Let all four of h, h′, e, and e′ be some logical truth lt of L, and let C(h, e) equal 1 for any two h and e;

under which they become true. But they yield as consequences:

and hence

No statement e of L is logically false.

Since all the languages for which confirmation theories have been proposed contain logical falsehoods, (1) is undesirable. Restrictions must accordingly be placed upon some of A1—A5.

There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists of

Another one, closely connected with A1–A3, is comprised of

B1–B3 jointly assert the inferential equivalence of a sentence of the form (p ⊃ q) ⊃ r and two sentences of the forms q ⊃ r and ∼p ⊃ r respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.

The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by ‘⊦’. ‘H’ will stand here for inferential equivalence. It may be looked upon as a combination of signs ‘⊦’ and ‘⊣’. ‘H’ is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role ‘H’ plays is determined by **1 –*5 below. Besides that ‘H’ remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.

The notion of a variable restricted in range to some set of values, or to values having some property, is widespread in logic and mathematics. To avoid continual explicit mention of such a restriction the common device is to resort to a typographically different set of letters for such variables. But a notation of this kind is clearly inadequate when a general discussion, including more than one kind of restriction, is under consideration. For example, two restrictions may have some inner connection, as in the case of mutually exclusive restrictions, and this the usual symbolism doesn't depict; or, one may wish to study the dependence of such restrictions upon one or more parameters as in the case of bounded quantifiers of recursive function theory; or, one can even conceive of restrictions which may in turn depend upon restricted variables. In each of these cases the currently used notation is deficient.

If x is an individual-variable and Q a formula of the predicate calculus, then for the idea ‘x such that Q’ the notation ‘vxQ’ is proposed as one having the requisite degree of notational multiplicity – the ‘vx’ acting as a binding operator for free occurrences of x in Q. The expression ‘vxQ’ is referred to as a restricted-variable and it may occupy positions in formulas appropriate to individual-variables, in argument places as well as in quantifiers. In § 2 a first-level predicate calculus, called ‘‘, is presented in which the restricted-variable is an integral part.

In this paper we show how a modification of results due to Simons ([6]) yields a set of independent axiom schemata for von Wright's M ([8], p. 85), with a single primitive rule of inference. We first describe a system M*, then show its equivalence with M, and finally show that our schemata are independent.

1. Axiomatization ofM*. We adopt the notational conventions of McKinsey and Tarski ([4], p. 2), as amended by Simons ([6], p. 309), except that we take “(α ⊰ β)” as an abbreviation for “˜◇˜(α 0→ β)”, rather than for “˜◇(α ∧ ˜β)”. Our only rule of inference is:

Rule. If ⊦ α and ⊦ (˜◇˜α → β). then ⊦ β.

We have six axiom schemata:

We require a number of theorems for the proof of equivalence of M* with M.

Theorem 1. If ⊦ α and ⊦ (α → β), then ⊦ β.

Theorem 2. If ⊦ α and ⊦ (α ⊰ β), then ⊦ β.

Proof by hypothesis, A5, and Theorem 2 (twice).

Theorem 3. If ⊦ (α ⊰ β), then (˜◇β → ˜ ◇α).

Proof by hypothesis, A6, and Theorem 2.

Theorem 4. If ⊦(α ⊰ β), then ⊦[˜˜(γ ∧ α) ⊰ ˜˜(β ∧ γ)].

I once asked myself the question: How were the famous axiom systems, such as Euclid's for geometry, Zermelo's for set theory, Peano's for arithmetic, originally obtained? This was to me more than merely a historical question, as I wished to know how the basic concepts and axioms were to be singled out, and, once they were singled out, how one could establish their adequacy. One possible approach which suggests itself is to take typical theorems, proofs, definitions, and examine case by case what assumptions and concepts are involved. The obstacle in such an empirical study is, apart from the obvious demand of excessive time and energy, the lack of conclusiveness in both result and justification.

The attempt to find an answer to this question led me to some interesting fragments of history. For example, in 1899 Cantor distinguished consistent collections (the “sets”) from inconsistent collections ([1], p. 443), anticipating partly the distinction between the two kinds of classes stressed by von Neumann and Quine. Cantor had already proposed a form of the axiom of substitution ([1], p. 444, line 3), although Fraenkel and Skolem, more than twenty years later, had to adjoin it to Zermelo's list of axioms as a supplement. In another direction, the history of the development of axioms of geometry makes clear how natural it was for Hilbert to raise in 1900 the consistency question of analysis ([2], p. 299) quite independently of the emphasis on set-theoretical paradoxes.

By far the best piece of good fortune I had in these historical researches was, however, my findings with regard to Peano's axioms for arithmetic. It is rather well-known, through Peano's own acknowledgement ([3], p. 273), that Peano borrowed his axioms from Dedekind and made extensive use of Grassmann's work in his development of the axioms.

This paper contains examples T1 and T2 of theories which answer the following questions:

(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?

(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)

The theory T1 represents a modification of a theory described by Myhill [7]. The common feature of theories T1 and T2 is that in both of them pleonasms are essential in the construction of the axioms.

Let T1 be a theory with identity = which contains one binary predicate R(x, y) and is based on the axioms A1, A2, A3, B1, B2, B3, B4, Cnm which follow.

A1: x = x. A2: x = y ⊃ y = x. A3: x = y ∧ y = z ⊃ x = z.

(Axioms of identity.)

B1: R(x, x). B2: R(x, y) ⊃ R(y, x). B3: R (x, y) ∧ R(y, z) ⊃ R(x, z).

(Axioms of equivalence.)

B4: x = y ⊃ [R(z, x) ≡ R(z,y)].

Let φn be the formula

which express that there is an abstraction class of the relation R which has exactly n elements.

Let f(n) and g(n) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets X1 and X2.

We now specify the axioms Cmm.

It is obvious that the set composed of the formulas A1−A3, B1−B4, Cnm (n,m = 1,2, …) is recursive.

The theory T1 is essentially undecidable; for if there were a complete and decidable extension T′1 (of it, then the recursive sets Z = {n: φn is provable in T′1} and Z′ = {n: ∼φn is provable in T′1} would separate the sets X1 and X2.

Kleene and Post [2] have shown that between each degree a and its completion a′ there are an infinity of mutually incomparable degrees of unsolvability, and that between a and its nth. completion a(n) there are an infinity of mutually incomparable degrees each incomparable with a′, …, a(n−1). They have left open the following questions: whether there exist complete degrees in similar profusion; whether indeed there exist incomparable complete degrees at all; and whether certain degrees, such as a(ω) ([2] p. 401), are complete. All these questions are answered affirmatively by the corollary of the present paper, which states that every degree greater than or equal to 0′ is complete. Since every complete degree is already known to be greater than or equal to 0′, it follows from the present paper that being greater than or equal to 0′ is both a necessary and a sufficient condition for completeness.

This note is a sequel to the previous paper of mine which was entitled Identity, variables, and impredicative definitions and published in this Journal, vol. 21 (1956), pp. 225–245. That early paper served to call attention to the dependency of the set-theoretic paradoxes on the interpretation of the variables that may occur in the critical abstraction principle:

where K is a well-formed formula containing free occurrences of y but not those of x.

A partial reinterpretation of the variables occurring in the above formalization of the abstraction principle yields Frege's unsuccessful way out of Russell's antinomy:

with the same kind of K as above. Another gives us the alternative schema

where K+ is obtained from a K of the kind described above by replacing every well-formed part of the form (∃z)L by (∃z)(z≠x & L) and every well-formed part of the form (z)L by (z)(z≠x ⊃ L). Still a third choice of an interpretation yields the schema

where K+ is subject to the same conditions as in (1) and z1, z2, …, zk are all the free variables of (2). These three interpretations are only some of the most conspicuous ones among a multitude of possible ways of interpreting the variables that may occur in the abstraction principle.