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Inner models for set theory—Part II
Authors: Shepherdson J. C. 
Published: May 2006

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I (page 186). The condition (c) of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition (c) and the completeness conditions 1.41, 1.42 in terms of Vm, we may state the definition in the form:3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM(U)’ (‘Udetermines a super-complete model’) thus:whereψ(U) is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ(U) contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form (after having replaced all defined symbols which occur by the corresponding formulae in primitive notation) and then replacing ‘(Am)ϕ(Am) bywhich is in turn replaced by, and similarly replacing ‘(xm)ϕ(xm)’ by ‘(xm)ϕ(xm)’ by ‘(X)(X ϵ U ▪ ⊃ ▪ ϕ(X)), andThusψ(U) is obtained in primitive notation.
A note on the theory of quantification
Authors: Shaw-Kwei Moh 
Published: May 2006

In the paper On the logic of quantification Prof. W. V. Quine showed that for the theory of quantification we have a mechanical process to determine whether or not a monadic expression is a valid logical formula, and that to deduce the polyadic theory from the monadic theory we need only the generalized modus ponens, which reads:If a conditional is valid, and its antecedent consists of zero or more quantifiers followed by a valid schema, then its consequent is valid, i.e.:That we have a mechanical process to determine the validity of a monadic expression was pointed out by Löwenheim long ago; Quine's method, however, has the merit that it is very simple and hence practical. The present paper is to show that we may deduce the polyadic theory by means of only a mechanical process and the ordinary modus ponens alone, i.e., the schema,or even by means of only the weakened modus ponens, i.e., the schema,For, if an expression becomes monadic when we omit the initial all-quantifiers (i.e., those preceding every existence-quantifier) and regard the corresponding variables as constants, then evidently we may determine by a mechanical process whether or not it is a valid formula by examining the resulting monadic expression. For example, the expressionis of the same validity as the expressionand hence as the expression
A demonstrably consistent mathematics—Part II
Authors: Fitch Frederic B. 
Published: May 2006

13.1. If ‘[p f a]’ is in K′, we will say that ‘p’ is a value of ‘f’ at ‘a’.13.2. If ‘a’ and ‘b’ are rational U-reals and if ‘b’ is not less than ‘a’, then ‘{a, b}’ will be called a rational U-interval.13.3. If ‘f’ has a value at ‘d’, if ‘d’ is within a U-interval ‘{c, e}’, and if all values of ‘f’ at U-reals with ‘{c, e}’ are within the U-interval ‘{r, t}’, then we will say that ‘{c, e}’ restricts ‘f’ to ‘{r, t}’ at ‘d’, and that ‘{c, e}’ is an ‘f’-restrictive correlate of ‘d’ with respect to ‘{r, t}’.13.4. If ‘d’ is within ‘{a, b}’, we will say that ‘{a, b}’ “contains” ‘d’.13.5. If ‘f’ has a value at ‘d’ and if corresponding to every U-interval ‘{r, t}’ that contains a value of ‘f’ at ‘d’ there is an ‘f’-restrictive correlate of ‘d’ with respect to ‘{r, t}’, then ‘f’ will be said to be “continuous” at ‘d’. Without loss of generality the ‘f’-restrictive correlate can be required to be a rational U-interval.13.6. If ‘f’ is continuous at every U-real in ‘{a, b}’, we will say that ‘f’ is continuous in ‘{a, b}’.13.7. Theorem. If ‘k’ is a positive U-real and ‘f’ is continuous in a U-interval ‘{a, b}’, then ‘{a, b}’ is covered by the class S of rational U-intervals that are ‘f’-restrictive correlates of rational U-reals in ‘{a, b}’ with respect to those rational U-intervals ‘{r, t)’ that contain values of ‘f’ at rational U-reals in ‘{a, b}’ and that are such that’∣t − r∣‘ is equal to ‘k’ (in the sense of 10.10).
Correction to a paper on modal logic
Authors: Anderson Alan Ross 
Published: May 2006

In a recent paper (here referred to as “IDP”) the writer outlined a decision procedure for Lewis's system S4 of modal logic. One of the clauses in definition 3.1 of IDP requires correction. Clause II of 3.1 (2) should read as follows.II. Some constituent of the form ◊β, of degree n1 ≤ n, has the value T in Row (i), and some constituents of the forms ◊δ1, … ◊δh, and ◊η1, …, ◊ηm, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ1 ∨ … ∨ δh ∨ ◊η1 ∨ … ∨ ◊ηm) is an (n1 − 1)-tautology of S4.This change is required in order to carry out the proof of Metathcorcm 3.19. In particular, the change guarantees the following. If the expression η of the second paragraph of 3.20 is of degree n − 1, then the antecedent λ of formula ζ on page 210 of IDP is also of degree n − 1; and consequently the formula ζ of 3.19 is of degree n (since ◊λ is a constituent of ζ). (If we fail to make the correction, then it might be the case that both ζ and (3) are of degree n − 1, in which case Row (i) would not satisfy clause II as originally stated, contrary to the claim at the end of 3.20.) The proofs for the remaining cases of 3.20 can then be carried out, using the revised clause II, in the way originally indicated.The proof of Metatheorem 3.2 requires only trivial corrections for case 2.
On the syntactical construction of systems of modal logic
Authors: Mckinsey J. C. C. 
Published: May 2006

When C. I. Lewis developed his theory of strict implication, he left open the question which of his various systems should be regarded as being closest to our intuitions—though he was inclined to favor the system S2. There are to be found in the literature numerous discussions of this question; most of these have condemned S2 as being too strong, and have proposed ways of weakening it.In the present paper I shall attempt to throw some light on this question by setting up a syntactical definition of “possibility.” I shall show that every system of modal logic constructed on the basis of this definition is at least as strong as the Lewis system S4.As the intuitive basis for the syntactical definition of possibility, I take the position that to say a sentence is possible means that there exists a true sentence of the same form. Thus, for example, it would be said that the sentence, “Lions are indigenous to Alaska,” is possible, because of the fact that the sentence, “Lions are indigenous to Africa,” has the same form and is true.
On absolute properties of relations
Authors: Mostowski Andrzej 
Published: May 2006

We shall be concerned in this paper with properties of relations. For simplicity we shall consider only two relations, a binary relation R and a ternary relation S, but the generalization to the case of any number of any relations presents no difficulty. We shall denote by A0 the field of the relations R and S and assume that A0 is infinite (not inductive).The properties to be dealt with are expressible in a symbolic language L whose principal features will be sketched below.
New foundation of formal metamathematics
Authors: Chwistek L. , Hetper W. 
Published: May 2006

Our method is based on the same principles which we have assumed in a paper about the foundations of metamathematics which we published in collaboration with J. Herzberg some years ago. Nevertheless we have introduced some important modifications which enable us to use a very restricted preliminary language and to avoid any artificial construction.The problem of a restricted language in mathematics appears to be a very important one, as our ordinary language is really ambiguous and obscure. On the other hand there is the problem of restricting the intuitive logic to be used in our construction. We shall see that for this purpose we need only substitutions and modus ponens.If we avoided any abbreviation, we could reduce our preliminary language to propositions of the following forms:X is a theorem.If P, then Q.But we have to do with very long symbolic expressions and as there is not yet any machine to build up such expressions we must use abbreviations. For this purpose we shall use propositions of the following forms in the preliminary language:X is an expression.X is an abbreviation of Y.Any proposition which does not belong to one of our four forms will be excluded from our construction.
On reduction to a symmetric relation
Authors: Craig William , Quine W. V. 
Published: May 2006

In a current paper, Church and Quine show how any dyadic relation of natural numbers can be defined in terms of a symmetric dyadic relation of natural numbers together with the truth functions and quantification over natural numbers. The purpose of the present note is to extend their result to the following: Where R1, R2, …, ad infinitum are any classes or relations (of any degree) of natural numbers, they can all be defined in terms of a single symmetric dyadic relation of natural numbers together with the truth functions and quantification over natural numbers.Let pi, for each i, be the tth prime > 1. Let the degree of Ri, for each i, be di. Let R be the dyadic relation which each number x bears to each number y such that either x is a factor of y or else x is 0 and y is for some such that . Since R is dyadic, we know from Church and Quine's result that R is definable in terms of a symmetric dyadic relation. Moreover, as seen in the incidental constructions in §1 of Church and Quine, ‘y = x + 1’ is expressible on the same basis.
A correction to Lewis and Langford's Symbolic logic
Authors: McKinsey J. C. C. 
Published: May 2006

The purpose of this note is to call attention to a minor error in Lewis and Langford's Symbolic logic. On page 221, in discussing the Tarski-Łukasiewicz three-valued logic, the authors make the following assertion: “Let T(p) be any proposition, involving only one element, whose analogue holds in the two-valued system; if T(p) does not hold in the Three-valued Calculus, then pC.T(p) and Np.C.T(p) both hold.”I shall show, by means of a counter-example, that this assertion is not true. Let T(p) be the sentence:It is then easily verified that T(0) = T(1) = 1, and that T(½) = 0. Thus T(p) holds in the two-valued calculus, but not in the three-valued calculus. On the other hand, pC.T(p) does not hold, since ½.CT(½) = ½C0 = ½; similarly, Np.C.T(p) does not hold, since N½.C.T(½) = ½C0 = ½.
The deduction theorem in a functional calculus of first order based on strict implication
Authors: Barcan Ruth C. 
Published: May 2006

In a previous paper, a functional calculus based on strict implication was developed. That system will be referred to as S2. The system resulting from the addition of Becker's axiom will be referred to as S4. In the present paper we will shw that a restricted deduction theorem is provable in S4 or more precisely in a system equivalent to S4. We will also show that such a deduction theorem is not provable in S2.The following theorems not derived in Symbolic logic will be required for the fundamental theorems XXVIII* and XXIX* of this paper. We will state most of them without proofs.
A formal system of logic
Authors: Wang Hao 
Published: May 2006

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.
Modality and quantification in S5
Authors: Prior A. N. 
Published: May 2006

In the first of her papers on functional calculi based on strict implication, Ruth Barcan Marcus takes as her starting point the Lewis systems S2 and S4, supplemented by one of the normal bases for quantification theory, and by one special axiom for the mixture, asserting that if possibly something φ's then something possibly φ's. In the symbolism of Łukasiewicz, which will be used here, this axiom is expressible as CMΣxφxΣxMφx. In the present note I propose to show that if S5 had been taken as a startingpoint rather than S2 or S4, this formula need not have been laid down as an axiom but could have been deduced as a theorem.It has been shown by Gödel that a system equivalent to S5 may be obtained if we add to any complete basis for the classical propositional calculus a pair of symbols for ‘Necessarily’ and ‘Possibly,’ which here will be ‘L’ and ‘M’; the axiomsthe ruleRL: If α is a thesis, so is Lα;and the definitionDf. M: M = NLN.
Reduction to a dyadic predicate
Authors: Quine W. V. 
Published: May 2006

Consider any interpreted theory Θ, formulated in the notation of quantification theory (or lower predicate calculus) with interpreted predicate letters. It will be proved that Θ is translatable into a theory, likewise formulated in the notation of quantification theory, in which there is only one predicate letter, and it a dyadic one.Let us assume a fragment of set theory, adequate to assure the existence, for all x and y without regard to logical type, of the set {x, y) whose members are x and y, and to assure the distinctness of x from {x, y} and {{x}}. ({x} is explained as {x, x}.) Let us construe the ordered pair x; y in Kuratowski's fashion, viz. as {{x}, {x, y}}, and then construe x;y;z as x;(y;z), and x;y;z;w as x;(y;z;w), and so on. Let us refer to w, w;w, w;w;w, etc. as 1w, 2w, 3w, etc.Suppose the predicates of Θ are ‘F1’, ‘F2’, …, finite or infinite in number, and respectively d1-adic, d2-adic, …. Now let Θ′ be a theory whose notation consists of that of quantification theory with just the single dyadic predicate ‘F’, interpreted thus:The universe of Θ′ is to comprise all objects of the universe of Θ and, in addition, {x, y) for every x and y in the universe of Θ′. (Of course the universe of Θ may happen already to comprise all this.)Now I shall show how the familiar notations ‘x = y’, ‘x = {y, z}’, etc., and ultimately the desired ‘’, ‘’, etc. themselves can all be defined within Θ′.
Begründung einer strengen Implikation
Authors: Ackermann Wilhelm 
Published: May 2006

Die Gründe, die C. I. Lewis [5], [6] bewogen haben, neben der gewöhnlichen Implikation eine strikte Implikation einzuführen, sind bekannt. In der vorliegenden Arbeit wird aus ähnlichen Gründen eine strenge Implikation eingeführt, die jedoch einen engeren Begriff darstellt als die strikte Implikation. Mit einer Arbeit von Arnold Schmidt [7] hat meine nur geringe Berührungspunkte, da der Verfasser sich mit der strikten Implikation beschäftigt. Für diese wird ein relativ einfaches Axiomensystem angegeben und gezeigt, wie man durch geeignete Definitionen von Notwendigkeit und Möglichkeit bei eventueller Hinzufügung von Zusatzaxiomen die bekannten Systeme des Modalitätenkalküls erhält. Den Hauptteil meiner Arbeit bildet die Einführung meines Implikationsbegriffs; im zweiten Teil der Arbeit komme ich dann auf dieser Basis auch zu einer Einführung der Modalitäten.Um die Art der Einführung des Kalküls zu motivieren, wollen wir kurz auf das eingehen, das uns mehr oder weniger deutlich vorschwebt, wenn wir von der strengen Implikation sprechen. Die strenge Implikation, die wir durchA → Bwiedergeben, soll ausdrücken, daß zwischenAundBein logischer Zusammenhang besteht, daß der Inhalt vonBein Teil des Inhaltes vonAist, oder wie wir es sonst ausdrücken wollen. Das hat mit der Richtigkeit oder Falschheit vonAundBnichts zu tun. So würde man die Allgemeingültigkeit einer FormelA→ (B → A) ablehnen, da sie den Schluß vonAaufB → Aeinschließt und da die Richtigkeit vonAnichts damit zu tun hat, ob zwischenBundAein logischer Zusammenhang besteht.
Inner models for set theory – Part III
Authors: Shepherdson J. C. 
Published: May 2006

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, 1.51, viz.:This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in (1) this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system.
An informal exposition of proofs of Gödel's theorems and Church's theorem
Authors: Rosser Barkley 
Published: May 2006

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.
A system of strict implication
Authors: Vredenduin P. G. J. 
Published: May 2006

Lewis has made an attempt to construct a logistic system containing an implication-relation ⊰ in such a way that p⊰q shall have the meaning: q is deducible from p. Lewis and Langford admit that “the one serious doubt which can arise concerning the equivalence of p⊰q to the relation of deducibility … arises from the fact that strict implication has its corresponding paradoxes:19.74 ˜⋄ p ⊰ · p ⊰ q,‘If p is impossible, then p strictly implies any proposition q’; and19.75 ˜⋄˜p ⊰ · q ⊰ p,‘If p is necessary, then any proposition q strictly implies p.’”Indeed, it is not obvious that 19.74 and 19.75 should hold. It is true that in many cases 19.75 (as well as 19.74) is valid; for in the system of Lewis and Langfordp · ˜p · ⊰ qis provable, and the conformity of this law with real deduction is shown by them. In all cases in which a proposition of the form p·∼p can be derived from an impossible proposition, 19.75 holds. But it is not obvious that there are no other kinds of impossible propositions.Fortunately it is easy to see what is the origin of the paradoxes. They are introduced into the system by the definition11.02 p ⊰ q · = ˜ ⋄(p · ˜q).Assume this definition. Then if q is necessary, p·∼q is impossible, and so, in accordance with 11.02, p⊰q is valid.
On simplifying truth-functional formulas
Authors: Bing Kurt 
Published: May 2006

The following are some consequences of a recent result concerning the prime implicants of disjunctive normal formulas.I. Introduce the void fundamental formula λ, and define the consensus of αϕ and as in [5], but allowing ϕ or ψ or both to be λ. Then a normal formula Φ is valid if and only if the operations of dropping subsuming clauses and adjoining consensus formulas end by transforming Φ into the single formula λ. (For under the notions of fundamental formula and consensus as now extended, λ is a (unique) prime implicant of all valid formulas and of no others; and the argument of [5] now shows that the operations mentioned transform every normal formula, whether valid or not, into the disjunction of its prime implicants.)II. A normal formula ψ implies a normal formula Φ if and only if every clause of ψ subsumes one of the prime implicants of Φ, which are found as in I above. (The condition is clearly sufficient. It is necessary, for if ψ implies Φ, then every clause of ψ implies Φ and subsumes a shortest fundamental formula implying Φ.) In this condition, the prime implicants of Φ may be replaced by the clauses of the result of merely adjoining, as long as possible, consensus formulas to Φ.
A rudimentary definition of addition
Authors: Ritchie R. W. 
Published: May 2006

In [S, pp. 77–88], Smullyan introduced the class of rudimentary relations, and showed that they form a basis for the recursively enumerable sets. He also asked [S, p. 81] if the addition and multiplication relations were rudimentary. In this note we answer one of these questions by showing that the addition relation is rudimentary. This result was communicated to Smullyan orally in 1960 and is announced in [S, p. 81, footnote 1]. However, the proof has not yet appeared in print. (Shortly after the publication of [S], James H. Bennett, using much more subtle methods than those of this note, showed that the multiplication relation is also rudimentary. That result appears in his doctoral dissertation [B], and is being prepared for publication.)Let us begin by reviewing Smullyan's definition [S, p. 10] of dyadic notation for the positive integers. Each positive integerais identified with the unique stringanan−1…a1a0of 1's and 2's such thata= Σin=0ai2iBecause of this identification, we are able to speak of typographical properties of numbers.