1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.

It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form

(1) (Ex1)(x2)(Ex3)(x4)…(xm).

On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove the

Theorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.

2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,

(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

Professor C. I. Lewis, in Lewis and Langford's Symbolic logic, designates the system (S2) determined by the postulates used in Chapter VI—namely, 11.1–7 (B1–7) and

as the system of strict implication. For certain reasons, he prefers it to either the earlier system (S3) determined by the stronger set of postulates of his Survey of symbolic logic, as emended, namely, A1–7 and

or the system (S1) determined by the weaker set of postulates B1–7 or A1–7.

But Lewis and others, following O. Becker, have also given consideration to systems which contain some additional principle effecting the reduction of complex modalities to simpler ones. Notable are the system (S4) determined by B1–7 plus

which includes (is stronger than) S3; and the system (S5) determined by B1–7 plus

which includes S4.

It seems worth while to investigate further the system of the Survey (emended), S3, which is intermediate between S2 and S4. This is the purpose of the present paper. We first prove some additional theorems in S2 and S3. These enable us to reduce all the complex modalities in S3 to a finite number, viz. 42; and it is shown that no further reduction is possible. Finally, several systems which include S3 are considered.

In this address, no attempt to deal with any problem of symbolic technique will be made, for even if a problem of this kind should be found that was suitable to the present occasion, I should probably lack sufficient competence in the handling of symbols to deal with it satisfactorily. It occurred to me, however, that it might be interesting to consider the nature of symbols in general, to point out certain characteristics peculiar to the symbols used by logicians and mathematicians, and to say something concerning the relations of these symbols to the knowledge of Nature. Although developments in the science of logic are not dependent on such an inquiry, it yet provides us with a perspective on the nature and importance of that science which we cannot gain so long as we attend only to the problems arising inside its field.

The symbolic relation. Our attempt to gain this perspective may well begin with the trite remark that nothing is intrinsically a symbol, but that anything is a symbol if and only if it symbolizes. Moreover, the relation called symbolizing is not a dyadic but rather a tetradic relation. That is, in order for something A to be a symbol of something B, there must be in addition C, a mind trained in a special way, and D, a certain manner in which that mind is occupied at the time. For although we do say, for instance, that a mark consisting of a little cross is the symbol of addition, the fact is of course that at times when that mark is not present to a mind, it does not symbolize addition or anything else. Moreover, even when it is present to a mind, it does not symbolize addition unless that mind has been trained in a certain manner; for obviously such a mark does not symbolize addition to the mind of a Hottentot or other wholly illiterate person.

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).

I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).

At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.

1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.

In my American Academy paper of 1937 I gave a mathematical clarification and extension of C. I. Lewis's theory of strict implication using as the base (K, T, ×, ′, ≣). In that paper the element a⊰b (read “a hook b”) was defined by (a⊰b) = (ab′=cc′). The main purpose of the present paper is to show that this definition may be replaced by the simpler definition (a⊰b) = (ab≣a), provided Postulate 15 is replaced by two simpler postulates, 15a and 15b (see below). The paper may also serve to throw further light on the essential difference between Lewis's theory of “strict” implication and the more familiar theory of “material” implication. This paper and the earlier one, when read together, are entirely self-contained, all proofs (including proofs of independence) being given in full and no reference to earlier literature being required.

We begin by reviewing briefly the rôle which logical deduction plays in the writing of any scientific book.

Suppose you sit down to write. What do you actually do? In the last analysis, your activity as an author is a process of voluntary selection, restricted by certain inescapable rules. From an indefinite number of possible propositions which may occur to your imagination, you voluntarily select certain propositions which you accept as being worthy of being written down, one after another, in your book. Every proposition which occurs to you is either accepted, or rejected, or laid aside for further consideration.

In dem vorstehenden Aufsatz hat A. Tarski den Begriff der inhaltlichen Widerspruchsfreiheit (content-consistency) einer Aussagenmenge eingeführt. Der Verfasser hebt hervor, daß die inhaltliche Widerspruchsfreiheit keine innere Eigenschaft der Aussagenmenge selbst ist, sondern vielmehr auf die jeweilige Definition dieser Menge bezogen werden soll. Es ist also a priori möglich, daß wir, von zwei verschiedenen Definitionen einer und derselben Aussagenmenge E ausgehend, das eine Mal feststellen, daß E inhaltlich widerspruchsfrei ist, und das andere Mal, daß E nicht inhaltlich widerspruchsfrei ist. Daß dies tatsächlich in vielen Fällen vorkommt, zeigt folgender einfacher Satz:

I. Eine definierbare Satzmenge E, die alle gültigen Formeln des Aussagenkalküls enthält und in Bezug auf die Abtrennungsregel abgeschlossen ist, ist dann und nur dann in Bezug auf jede ihrer Definitionen inhaltlich widerspruchsfrei, wenn sie ausschließlich aus wahren Aussagen besteht.

Daß diese Bedingung hinreichend ist, hat Tarski (a. a. O. Satz 7.1) bewiesen. Nehmen wir nun an, E enthalte einen falschen Satz f (ohne freie Variable), und es sei ϕ irgendeine Definition von E. Die Aussagefunktion ψ = f¯→ϕ ist offenbar ebenfalls eine Definition von E. Wird ψ als Definition von E angenommen, so hat x(E) die Form einer Implikation mit dem Vorderglied f¯, welche auch die Aussage x sein könnte. x(E) gehört daher immer der Menge E an, woraus wir mit Rücksicht auf Bedingung 3.4 aus dem vorstehenden Artikel schließen, daß E in Bezug auf die Definition ψ nicht inhaltlich widerspruchsfrei ist, w. z. b. w.

In this paper I shall show that no one of the four primitive symbols of Heyting's calculus of propositions is definable in terms of the other three. So as to make the paper self-contained, I begin by stating the rules and primitive sentences given by Heyting.

The primitive symbols of the calculus are “⅂”, “∨”, “∧”, and “⊃”, which may be read, respectively, as “not,” “either…or,” “and,” and “if…then.” The symbol “⊃⊂”, which may be read “if and only if,” is defined in terms of these as follows:

The rule of substitution is assumed, and the rule that S2 follows from S1 and S1⊃S2; in addition it is assumed that S1∧S2 follows from S1 and S2. The primitive sentences are as follows:

This paper is an attempt to explain as non-technically as possible the principles and devices used in the various proofs of Gödel's Theorems and Church's Theorem.

Roman numerals in references shall refer to the papers in the bibliography. In the statements of Gödel's Theorems and Church's Theorem, we will employ the phrase “for suitable L.” The hidden assumptions which we denote by this phrase have never been put down explicitly in a form intelligible to the average reader. The necessity for thus formulating them has commonly been avoided by proving the theorems for special logics and then remarking that the proofs can be extended to other logics. Hence the conditions necessary for the proofs of Gödel's Theorems and Church's Theorem are at present very indefinite as far as the average reader is concerned. To partly clarify this situation, we will now mention the more prominent of these assumptions.

I. In any proof of Gödel's Theorems or Church's Theorem, two logics are concerned. One serves as the “logic of ordinary discourse” in which the proof is carried out, and the other is a formal logic, L, about which the theorem is proved. The first logic may or may not be formal. However L must be formal. Among other things, this implies that the propositions of L are formulas built according to certain rules of structure.

In the present paper, the various means used by the author in attempting to find a contradiction in Quine's New foundations are sketched, and the reason why each method failed is indicated. It was not Quine's system itself which was tested, but a stronger one obtained by adding a rule of Kleene's type to Quine's system. Reasons are presented why a contradiction in the stronger system should be as damning to Quine's system as a contradiction in Quine's system itself. Two other features of the system are worthy of note. One is the fact that only two symbols are used in building up the formulas of the system. Other systems have been built using only two symbols, but the particular method used in this paper is very flexible and simple, and is peculiarly adapted to the use of the Gödel technique. It was suggested by the consideration that the formulas of any system can be written by the use of only two symbols by first assigning Gödel numbers to the formulas and then writing those numbers in the binary scale of notation. The second feature is that ι (to be used in ιxp, meaning “the x such that p”) is an explicit and integral part of the system, and the axioms and rules governing its use are such as to make it very simple to handle. By a process similar to “Die Eliminierbarkeit der ι-Symbole” of Hilbert-Bernays, it is shown that the system involving the ι can be reduced to one not involving it. The points of difference with the Hilbert-Bernays technique make possible an especially unhampered use of ι.

Let “the hypothesis of similarity” be the hypothesis that all infinite classes are similar (have the same power or cardinal number). It will be shown that the system of logic of Whitehead and Russell's Principia mathematica (without the axiom of reducibility) is consistent when to its axioms are added all the following:

(1). The hypothesis of similarity.

(2). The axiom of infinity.

(3). The axiom of choice.

(4). The contradictory of the axiom of reducibility.

If (2) – (4) but not (1) are employed in Principia, it must therefore be the case that Cantor's theorem is not deducible, since it contradicts (1). If (2) and (3) but not (1) and (4) are employed, then a proof of Cantor's theorem must require the use of some sort of reducibility principle.

The first step in the required consistency proof is to modify in certain respects the system S of a previous paper. Immediately after definition 3.5 the following definition is to be added:

3.5.1. If a is of the form (b,c), where x is an S-variable and where b and c are S-constants neither of which is of higher order than that of x, then a is an S-proposition.

In a previous paper a method was given for representing modal functions in two-valued logic. The purpose of the present note is to indicate how the results of that paper can be improved in certain respects and to point out a simple Boolean representation of Lewis's system of strict implication. The definitions of the previous paper will be presupposed.

I. The present paper contains a formal proof of the following theorem of the elementary system ⊨ [ 1 0 ] c:

This theorem means that if there were a theorem of ⊢0 ( 3 2 ) 0 stating that there is no contradiction in ⊢2 ( 5 4 ) 2, we would have a contradiction in ⊢2 ( 5 4 ) 2. (This is in accordance with the second theorem of Gödel.)

Our proof is based on the calculus of the system ⊨ [ 1 0 ] c and of the metasystem ⊢0 ( 3 2 ) 0.

Note that we could take

instead of Ax E .1 ( L ) (.1 ( L ) .0 ( L ) L) Z. We would then have to do with the simple systems and metasystems of Hetper, which do not contain propositional variables or the logico-semantical axiom. Our method applies equally to the systems and metasystems of New Foundation and to the simple systems and metasystems of Hetper.

Our proof can be considerably simplified by using a recent result of Hetper concerning ancestral functions. Hetper introduces the following abbreviations:

If A ( Z1c), B ( x1cy1cZ1c) are propositions, the expression will be called an ancestral function (we prove without difficulty that this expression is a proposition). The expressions will be called respectively the principal term and the term of derivation of this function.

The question of the independence of the axioms of the theory of sets has been dealt with in a number of works, although not in a final manner. The writer will be concerned solely with the axiomatic system of Zermelo and Fraenkel, and only with that feature of the system whereby all the objects of the underlying domain are sets (so that there is no difference between objects in general and sets in particular).

A special place among the axioms is occupied by a purely relational one, an axiom of definiteness, which establishes the character of equality within the system.

Zermelo introduces equality intensionally—if two symbols x and y represent the same object, we write = (x, y). According to this there is no necessity for axioms to assure the interchangeability of equal objects as arguments of the primitive relation ϵ( , ). For under the intensional interpretation it is clear that:

Similarly it is clear that there is no necessity for a separate axiom that will assure to equality the properties of an equivalence relation. Therefore Zermelo introduces only the following axiom of extensionality:

(if one set is a subset of another set, and the second set is also a subset of the first, then the two sets are equal).