The attempt to find usable diagrams for n terms of the sort devised by John Venn seems to have originated with Venn himself, who published diagrams for up to five classes (the fifth class, however, was shaped like a doughnut, and contained an area outside itself — like the hole in the doughnut). Venn then suggested that “if we wanted to use a diagram for six terms (x, y, z, w, v, u) the best plan would probably be to take two five-term figures, one for the u part and one for the non-u part of all the other combinations …” Such a method would, as Venn admits, be somewhat confusing to the eye, and thus fail to fulfil one of the major purposes of the diagram; nevertheless, it is a method which could be extended to cover n classes.

Venn also suggests (but without illustration) another method for drawing diagrams for n terms: “… the rule of formation would be very simple. We should merely have to begin by drawing any closed figure, and then proceed to draw others subject to the one condition that each intersect once, and once only, all the existing subdivisions produced by those which had gone before.” Then, in a rather surprising footnote, he adds: “It will be found that when we adhere to continuous figures, instead of the discontinuous five-term figure given above, there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five. If we begin by circles or other rounded figures the teeth are curved, if by parallelograms then they are straight.

1. Introduction. Since this paper is a written version of one presented as a survey lecture at a meeting of the Association for Symbolic Logic, its form and content have been essentially determined by the form and content of that lecture, and these latter were themselves considerably influenced by a deliberate attempt to avoid the lecture consisting essentially either of a catalogue of results in some fairly wide field or of the presentation of detailed proofs of a few particular results in a very limited field, with, in either case, a consequential serious limitation of the time available for discussion of the motivation for the results and of the connections between them. There is thus no attempt to cover, even in outline, the full range of topics which could reasonably fall within the scope of its title.

The paper consists mainly of a general discussion of some results concerned with finite models of axiomatically defined calculi of propositional form, with particular reference to the occurrence of such models in proofs that certain propositional calculi are decidable. In addition, there is an account of some results concerned with the existence of undecidable propositional calculi and of others connected with the decidability or undecidability of certain structure problems which relate either to sets of propositional calculi or to different presentations of the same calculus.

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.

The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.

Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

In the design of multiple input-output logical systems, the authors have previously formulated and proved a computer program for obtaining optimal system representations satisfying three optimality criteria [1]. In that program, the need for obtaining prime implicants occurred. Two algorithms for this purpose are available in the literature; one due to Quine [2] and one due to Nelson [3], [4]. Nelson's, briefly, is to express the function (whose prime implicants are desired) as a product of sums of literals, multiply out, and discard repeated literals save one and proper multiples. (To obtain the function in the initial form, one can start with the complement of the function expressed as a sum of products and complement it using De Morgan's theorem.) In using these algorithms, it became apparent to us that in certain cases one of these two algorithms required less computer time than the other one. In particular, if the given system has a large number of don't cares then the Nelson algorithm seemed to be preferred [5]. Also in the case of a single function, if there are few zeroes in the truth table representation for the function, then Nelson's algorithm is preferable. It also became apparent to us that in certain situations one needs a generalized version of the Nelson algorithm.

This note shows that the system of natural deduction proposed by Copi in this Journal (1956), made by varying one restriction on Universal Generalization (UG) of the system of his Symbolic logic (1954), is incorrect. The original Symbolic logic system, also incorrect, was corrected in the third printing (1958) by modification of another restriction on UG; but combining this modification with that of the Journal article does not give a correct system.

1. In Symbolic logic — cited as SL (or as SL54 to indicate the first printing) — Copi precedes his formal statement of the quantification rules by a set of conventions applying to all the rules.

The expression ‘Φμ’ will denote any propositional function in which there is at least one free occurrence of the variable denoted by ‘μ’. The expression ‘Φν’ will denote the result of replacing all free occurrences of μ in Φμ by ν, with the added provision that [a] when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.

In this paper we prove:

Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.

Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.

Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.

The proof of Theorem 2 will follow from the lemmas which we establish below.

We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.

Clearly, in every theory of sets we have:

(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),

(2) (x = y) ↔ (P(x) = P(y)),

(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),

(4) ⋃P(x) = x.

In view of (2), (3) and the definition of equality, we have:

Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.

We have also, in view of (4):

Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.

In this paper we show that for n-th order languages L′ with p nonlogical constants, 2 ≦ n < ω, 0 ≦ p < ω, a notion of satisfaction can be defined in an n-th order language containing one additional nonlogical constant, say S. By the usual methods we also show that this notion cannot be defined in L′. Hence, in its present formulation, Beth's Theorem in [1] for first order languages has no analogue for L′.

Our defining expression is such that, given any values of the other nonlogical constants and any appropriate m-tuple, it allows us to determine whether or not the m-tuple belongs to the value S of S without considering the totality of objects which are of the same type as S. Whether every definition in an n-th order language is equivalent to one thus “predicative”, and hence whether there is a formulation of Beth's Theorem which generalizes to higher orders, we do not know.

The falsehood for L′ of the analogue of Beth's Theorem implies the falsehood of an analogue of an interpolation theorem for first order languages. The above definability of satisfaction for L′ implies a result on finite axiomatizability in slightly richer languages. Details are given in § 4.

We consider some natural deduction systems for quantification theory whose only quantificational rules involve elimination of quantifiers. By imposing certain restrictions on the rules, we obtain a system which we term Analytic Natural Deduction; it has the property that the only formulas used in the proof of a given formula X are either subformulas of X, or negations of subformulas of X. By imposing further restrictions, we obtain a canonical procedure which is bound to terminate, if the formula being tested is valid. The procedure (ultimately in the spirit of Herbrand [1]) can be thought of as a partial linearization of the method of semantical tableaux [2], [3]. A further linearization leads to a variant of Gentzen's system which we shall study in a sequel.

The completeness theorem for semantical tableaux rests essentially on König's lemma on infinite graphs [4]. Our completeness theorem for natural deduction uses as a counterpart to König's lemma, a lemma on infinite “nest structures”, as they are to be defined. These structures can be looked at as the underlying combinatorial basis of a wide variety of natural deduction systems.

In § 1 we study these nest structures in complete abstraction from quantification theory; the results of this section are of a purely combinatorial nature. The applications to quantification theory are given in § 2.

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.

In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.

Let Pp, Pd, and N be the intuitionistic systems of prepositional calculus, predicate calculus, and elementary arithmetic, respectively, described in [3].

Kleene [4] introduces a metamathematical predicate Γ ∣ A for each of the systems Pp, Pd, and N, where Γ ranges over finite sequences of wffs, and A ranges over wffs, of that system. In the case of N, if Γ is consistent, then ‘ Γ ∣ A’ is essentially the result of deleting all references to recursive functions from the metamathematical predicate ‘A is realizable-(Γ ⊦)’ described in [3], pp. 502–503.

Through use of this predicate, Kleene [4] obtains elegant constructive proofs of the following results for N:

Metatheorem 0.1. If B ∨ C is a closed theorem of N, then ⊦ B or ⊦ C.

Metatheorem 0.2. If (∃a)D(a) is a closed theorem of N, then there is a numeral n such that ⊦D(n).

Metatheorem 0.3. If A is a closed wff of N, then A ∣ A is a necessary and sufficient condition that, for all closed B, C, (∃a)D(a) inN:

(0.3.1) ⊦ A ⊃ B ∨ C implies ⊦ A ⊃ B or ⊦A ⊃ C

and

(0.3.2) ⊦A ⊃ (∃)D(a) implies there is a numeral n such that ⊦ A ⊃ D(n).

The Stoic “indemonstrables” were inference rules; a rule about rules was the synthetic theorem: if from certain premisses a conclusion follows and from that conclusion and certain further premisses a second conclusion follows, then the second conclusion follows from all the premisses together. Similar things occur as medieval “rules of consequence”, although not usually on a metametalevel; and (with the same proviso) the following might be deemed a contemporary avatar of that Stoic theorem.

If every formula which occurs once or more often in the list A1, A2, …, An, B1, B2, …, Bm occurs also at least once in the list C1, C2, …, Cr then:

This rule [Church: Introduction to Mathematical Logic, 1956, pp. 94, 165], which may be called the rule of modus ponens under hypotheses (MPH), is worthy of attention for the following reasons:

A. MPH and the axioms A ⊃ A yield precisely the positive implicative calculus (and very easily, too).

B. MPH and the axioms A ⊃ f ⊃ f ⊃ A yield a new formulation of the full classical propositional calculus (in terms of f and ⊃).

C. MPH and the axioms ∼A ⊃ A ⊃ A and A ⊃. ∼A ⊃ B yield the classical calculus in terms of ∼ and ⊃.

Our ultimate purpose is to give an axiomatic treatment of recursion theory sufficient to develop the priority method. The direct or abstract approach is to keep in mind as clearly as possible the methods actually used in recursion theory, and then to formulate them explicitly. The indirect or experimental approach is to look first for other mathematical theories which seem similar to recursion theory, to formulate the analogies precisely, and then to search for an axiomatic treatment which covers not only recursion theory but also the analogous theories as particular cases.

The first approach is more general because it does not depend on the existence of (familiar) analogues. A concrete mathematical theory, it seems, need have no such analogues and still be important, as e.g. classical number theory. In such a case, an axiomatic treatment may still be useful for exhibiting the mathematical structure of the theory considered and the assumptions on which it rests. However, it will lack one of the most heavily advertised advantages of the axiomatic method, namely, the “economy of thought” which results from an uniform theory for several different and interesting cases: we cannot hope for this if, by hypothesis, we know of only one particular case. In contrast, the second approach, if successful at all, is bound to achieve such “economy” because we start out with several interesting particular cases. Another possible virtue of the second approach is that of field work over insight: the abstract pattern that we are looking for and hoping to formalize in axioms, may not be evident in any one mathematical theory, but may spring to the eye if one happens to look simultaneously at several theories which happen to realize the pattern.

In the fourth edition of Hilbert-Ackermann's Grundzüge der Theoretischen Logik (Springer-Verlag: Berlin-Göttingen-Heidelberg 1959) the following assertion is “proved” (p. 131):

Ein Ausdruck, der keine freien Individuenvariablen enthält, ist dann und nur dann allgemeingültig, wenn allgemeingültig ist. Die Ausdrücke, die die Identität enthalten, sind dabei eingeschlossen.

As a counter example one can cite: ∃x(¬∃xFx ∨ Fx), which is universally valid yet the generalized form: ∀x(¬∃xFx ∨ Fx) is not (e.g. let the domain of individuals be the natural numbers and let ‘Fx’ be ‘x is factorable’).

It may be of interest to see where the alleged proof went astray. It is argued (correctly) that any expression can only be derived by the given rules from a formula , or from this formula with any other variable in place of y. Moreover (still correctly) any upper formula of such a formula is of the form (“Upper formula” in the transitive sense of an upper formula of an upper formula being an upper formula). But then it is stated (incorrectly) that if is an upper formula of , then is an upper formula of . To show that this is incorrect we here append a proof of the universally valid formula mentioned above:

Introduction. This paper is a sequel to our paper [3]. In that paper we introduced the notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2]. The results of the paper [3] show relationships between the models of an infinitely long sentence and the models of its finite approximations. In the present paper we shall apply the main result of [3] to prove a number of theorems about ordinary finitely long sentences; these theorems are of a type which one might call “preservation theorems”. The following two known results are typical preservation theorems: A sentence φ is preserved under substructures if and only if φ is (logically) equivalent to some universal sentence (Łoś [7] and Tarski [11]); φ is preserved under homomorphic images if and only if φ is equivalent to some positive sentence (Lyndon [8]). An expository account of preservation theorems may be found in Lyndon [9].

We shall use freely all of the notation introduced in [3], and shall not attempt to make this paper selfcontained. However, the reader should have no difficulty following this paper after he has read [3]. In § 1 we cover some preliminary topics. In § 2 and § 3 we shall illustrate our method in familiar situations by proving anew the two preservation theorems mentioned above. We shall then obtain four new preservation theorems in §§ 4, 5, and 6, involving respectively direct powers and roots, strong homomorphisms and retracts, and direct factors. The result in § 6 was stated in the abstract [4], while the results in § 5 were stated in the abstract [5].

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:

Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.

This paper deals with Hilbert's substitution method for eliminating bound variables from first order proofs. With a first order system S framed in the ε-calculus [2] the problem is to associate a system S' without bound variables and an effective procedure for transforming derivations in S into derivations in S′. The transform of a formula A derived in S is to be an “ε-substitution instance” of A, i.e. it is obtained by replacing terms εxB(x) in A by terms of S′. In general the choice of these terms will depend on the particular derivation of A, and not on A alone. Cf. [4]. The present formulation sharpens Hilbert's original statement of the problem, i.e. that the transform of A should be finitistically verifiable, by making explicit the methods of verification used, namely those formalized in S′; on the other hand, it generalizes Hilbert's formulation since S′ need not be restricted to finitist systems.

The bound variable elimination procedure can always be taken to be primitive recursive in (the Gödel number of) the derivation of A. Constructions which transcend primitive recursion can simply be built into S′.

In this paper we show that if S′ is taken to be a second order system with constants for functionals, then the existence of suitable ε-substitution instances can be expressed by the solvability of certain functional equations in S′. We deal with two cases here. If S is number theory without induction, i.e. essentially predicate calculus with identity, then we can solve the equations in question by taking for S′ the free variable part S* of S with an added rule of definition of functionals by cases (recursive definition on finite ordinals), which is a conservative extension of S*.