bluuebunny/crossref_metadata_2025_split huggingface dataset:
On truth and multiple denotation
Authors: Martin R. M. 
Published: May 2006

If we think of denotation in a broad sense, we may say not only that, e.g., ‘Marni’ denotes the actual dog Marni but also that ‘dog’ denotes severally the dogs Marni, Fido, etc. What are ordinarily called class names thus come to denote severally the members of the class but not the class itself. This kind of a multiple denotation relation is a very simple as well as a very powerful semantical relation. In fact it can be taken as the fundamental relation of formal (extensional) semantics.Conceptually this denotation relation is akin to the relation of satisfaction studied by Tarski, although the two must be carefully distinguished. Satisfaction (in its simplest form) is a relation between objects and sentential functions of one variable. Denotation is taken as a relation holding, on the one hand, between what is ordinarily called a proper name and the object which it names, and, on the other, between what may be called a virtual class denotator and the objects which are members of the virtual class in question. (This relation will be characterized more precisely below.) Clearly the two relations, satisfaction in its simplest form and multiple denotation, are interdefinable in an appropriate formalism.
On natural numbers, integers, and rationals
Authors: Fitch Frederic B. 
Published: May 2006

A theory of natural numbers will be outlined in what follows. This theory will also be extended to give an account of positive and negative integers and positive and negative rational numbers. The system of logic used will be that of Whitehead and Russell's Principia mathematica with the simple theory of types. It will be assumed that the reader is familiar with the more elementary properties of relations and with such notions as the relative product of two relations, the square of a relation, the cube of a relation, and the various other whole-number powers of relations.The guiding principle of this theory is that the natural number zero is to be regarded as the relation of the zevoth power of a relation А to А itself, and the natural number 1 is to be regarded as the relation of the first power of a relation А to А itself, and the natural number 2 is to be regarded as the relation of the square of a relation А to А itself, and so on.
A simplification of basic logic
Authors: Fitch Frederic B. 
Published: May 2006

The program of “basic logic” can be summarized as follows:I. To treat every syntactical system as a subclass of a certain fixed infinite classUof “U-expressions.” This can always be done by modifying in trivial ways the notation of each syntactical system which is not already such a subclass. As a result all syntactical systems become comparable with each other in the sense that they are merely different subclasses of a single class of expressions. The class U can be chosen in such a way as to be inductively definable thus in terms of a fixed symbol ‘σ’: (1) The symbol ‘σ’ is a U-expression. (2) The result of placing two U-expressions (or two occurrences of the same U-expression) next to each other, and enclosing this total expression within a pair of parentheses, is a U-expression.II. To formulate a particular syntactical systemKwithin which every syntactical system (and indeedKitself) is “represented.” Such a system is here said to be a “basic system,” and an appropriate interpretation of it is said to be a “basic logic.” Within such a logic every finitary logic is definable, as well as the basic logic itself. Such a logic should be of fundamental importance, especially if it is so constructed as to be the weakest such logic and so contain no theorems that are not essential to its being basic.With the above considerations in view, the system K has been defined in such a way that it is a subclass of U and is a basic syntactical system. A simpler definition of K will be given than heretofore in previous papers, and the minimum character of K will be made more clear.
Towards a consistent set-theory
Authors: Myhill John 
Published: May 2006

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.
A Boolean derivation of the Moore-Osgood theorem
Authors: Blake Archie 
Published: May 2006

A fundamental problem of symbolic logic is to define logical calculi sufficient to comprise important parts of mathematics, and to develop systematic methods of calculation therein.The possibility of progress in this direction has been severely limited by Gödel's proof that a consistent system sufficient to comprise arithmetic must contain propositions whose truth-value cannot be decided within the system, and by Church's extension of Gödel's method to the result that even in the first order logical function calculus the general decision problem cannot be solved.
Note on truth-tables
Authors: Kalicki Jan 
Published: May 2006

In this paper a method will be described which is intended to exhibit some relationships between sets of tautologies determined by truth-tables. This method is an attempt to form an algebra of truth-tables. The results sketched below are restricted to sets of tautologies determined by truth-tables with a finite number of elements and involving a single binary connective Δ. However, most of the results can be easily extended to the case of Tarski's logical matrix and even to a more general case.We denote by S() the set of all tautologies (-tautologies) according to a given truth-table . Let describe a binary connective Δ. Then Δ()(x, y) stands for the truth-value of ΔPQ, when P has the truth-value x and Q has the truth-value y. If no ambiguity may arise we write Δ(x, y) or Δ() for Δ()(x, y).
Note on a problem of Paul Bernays
Authors: Sobociński Bolesław 
Published: May 2006

In this Journal, vol. 18 (1953), p. 350 (Problem 7), Prof. P. Bernays proposed the following problem on propositional calculus: What is the smallest number n such that the propositional calculus, formulated with substitution and modus ponens as the only rules of inference, can be based on a set of initial formulas each of which contains at most n propositional letters (counted with multiplicity) ?
In this note I give a solution to this problem, viz., that this number n = 5. For a system of propositional calculus in which the primitive functors are “C” (implication) and “N” (negation) and in which there are only two rules of inference, i.e. the rules of substitution and detachment (modus ponens), the following can be proved. (1) A set of propositional theses each of which contains at most 4 propositional letters is inadequate to give the complete bi-valued calculus of propositions. (2) There are axiom systems for this calculus in which each axiom contains at most 5 propositional letters.
§ 1. Consider the following normal metrix, in which the designated value is I:

This satisfies the two rules of inference, and the following.
(a) The law of commutation, i.e. the thesis

CCpCqrCqCpr.
(b) The following theses:

Furthermore, in this matrix “N” is defined in such a way that:
(c) For any well-formed formula α and any value m of this matrix, α = m if and only if NNα = m.
Note on an idea of Fitch
Authors: Myhill John R. 
Published: May 2006

The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:Every recursive predicate is recursively definite. If R(x1, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x1, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.
On Quine's axioms of quantification
Authors: Berry George D. W. 
Published: May 2006

In his Mathematical logic, Quine adopts as axioms of quantification all statements specified by the first five of the six metatheorems listed below. The sixth serves as a rule of inference, and is the only primitive rule of inference used. The metatheorems in question are as follows:The theme of the present paper is that *101 may be eliminated by means of an otherwise trivial shift in the meanings of *100, *102, *103, and *104.In the statement of the above metatheorems and throughout the paper as a whole, we deviate from Quine's notational conventions only in using double quotes rather than corners and in using English letters rather than Greek ones as syntactical variables. Accordingly, ‘a’, ‘b’, and ‘d’ ambiguously denote variables ‘x’, ‘y’, ‘z’, and so on. The letters ‘p’ and ‘q’ ambiguously denote formulae, i.e. statements, and expressions which become statements when their contained free variables are supplanted by constants. Since we replace Quine's corners with double quotes, “(a)p” (for instance) is not the result of prefixing a parenthesized occurrence of the first letter of the alphabet to an occurrence of the sixteenth: “(a)p” is rather the result of prefixing any parenthesized variable a to any formula p.Following Quine, we define the closure of p as p itself when p is a statement and as “(a1)(a2)…(an)p” where a1, a2, …, an are the sole variables free in p and where each ai is alphabetically prior to the corresponding ai+1.
Leibniz's interpretation of his logical calculi
Authors: Rescher Nicholas 
Published: May 2006

The historical researches of Louis Couturat saved the logical work of Leibniz from the oblivion of neglect and forgetfulness. They revealed that Leibniz developed in succession several versions of a “logical calculus” (calculus ratiocinatororcalculus universalis). In consequence of Couturat's investigations it has become well known that Leibniz's development of these logical calculi adumbrated the notion of a logistic system; and for these foreshadowings of the logistic treatment of formal logic Leibniz is rightly regarded as the father of symbolic logic.It is clear from what has been said that it is scarcely possible to overestimate the debt which the contemporary student of Leibniz's logic owes to Couturat. This gratitude must, however, be accompanied by the realization that Couturat's own theory of logic is gravely defective. Couturat was persuaded that the extensional point of view in logic is the only one which is correct, an opinion now quite antiquated, and shared by no one. This prejudice of Couturat's marred his exposition of Leibniz's logic. It led him to battle with windmills: he viewed the logic of Leibniz as rife with shortcomings stemming from an intensional approach.The task of this paper is a re-examination of Leibniz's logic. It will consider without prejudgment how Leibniz conceived of the major formal systems he developed aslogicalcalculi – that is, these systems will be studied with a view to the interpretation or interpretations which Leibniz himself intends for them. The aim is to undo some of the damage which Couturat's preconception has done to the just understanding of Leibniz's logic and to the proper evaluation of his contribution.
A definition of negation in extended basic logic
Authors: Fitch Frederic B. 
Published: May 2006

In a previous paper it was shown that the system K of basic logic could be formulated in a simpler way owing to the fact that the proper ancestral could be defined in terms of the other concepts of that system. In the present paper analogous but more far-reaching results will be obtained for the system K′ of extended basic logic. In particular we will show that negation and the dual of the proper ancestral, as well as the proper ancestral itself, are definable in terms of the other concepts of K′. Hence, in order to define K′, we need to add only a single non-finitary rule to the rules used to define K. This rule was already among the rules originally used to define K′. It asserts that ‘Aa’ is in K′ if (and only if) every ‘b’ is such that ‘ab’ is in K′.We will also show that a large class of non-finitary classes and relations are represented in K′, among which is K′ itself, just as all finitary syntactical classes and all finitary two- and three-place syntactical relations are represented in K, one of which is the finitary syntactical class K itself. The point is that K′ is adequate to handle the sort of transfinite induction that is essential in formulating K′, just as K is adequate to handle the ordinary finitary mathematical induction required in defining K′.
The recursive irrationality of π
Authors: Goodstein R. L. 
Published: May 2006

A primitive-recursive sequence of rational numbers sn is said to be primitive-recursively irrational, if there are primitive recursive functions n(k), i(p, q) > 0 and N(p, q) such that:1. (k)(n ≥ n(k) → ∣sn – sn(k)∣ < 2−k).2. (p)(q)(q > 0 & n ≥ N(p, q) → ∣sn ± p/q∣ > 1/i(p, q)).The object of the present note is to establish the primitive-recursive irrationality of a sequence which converges to π. In a previous paper we proved the primitive-recursive irrationality of the exponential series Σxn/n!, for all rational values of x, and showed that a primitive-(general-) recursively irrational sequence sn is strongly primitive-(general-)recursive convergent in any scale, where a recursive sequence sn is said to be strongly primitive-(general-)recursive convergent in the scale r (r ≥ 2), if there is a non-decreasing primitive-(general-) recursive function r(k) such that,where [x] is the greatest integer contained in x, i.e. [x] = i if i ≤ x < i + 1, [x] = —i if i ≤ —x < i+1, where i is a non-negative integer.A rational recursive sequence sn is said to be recursive convergent, if there is a recursive function n(k) such that.If a sequence sn is strongly recursive convergent in a scale r, then it is recursive convergent and its limit is the recursive real number where, for any k ≥ 0,.
On the theory of types
Authors: Quine W. V. 
Published: May 2006

In this paper the theory of logical types will be examined, and certain departures from it will be suggested. Though the purpose of the paper is not primarily expository, an approach has been possible which presupposes no familiarity with special literature. Matters at variance with such an approach have been confined to appendices and footnotes.In the early pages the logical paradoxes will be considered—an infinite series of them, of which Russell's paradox is the first. Then Russell's simple theory of types will be formulated, in adaptation to a minimal set of logical primitives: inclusion and abstraction. Two aspects of the theory will be distinguished: an ontological doctrine and a formal restriction. It will be found that by repudiating the former we can avoid certain unnatural effects of the type theory—notably the reduplication of logical constants from type to type, and the apparent dependence of finite arithmetic upon an axiom of infinity. But the formal restriction itself has unnatural effects, which survive, even in an aggravated form, after the type ontology has been dropped. A liberalization of the formal restriction will be proposed which removes the more irksome of these anomalies.
On the consistency of an axiom of enumerability
Authors: Beneš Václav Edvard 
Published: May 2006

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.
Modality and description
Authors: Smullyan Arthur Francis 
Published: May 2006

There are logicians who maintain that modal logic violates Leibniz's principle that if x and y are identical, then y has every property of x. The alleged difficulty is illustrated in the following example due to Quine.1A. It is logically necessary that 9 is less than 10.B. 9 = the number of the planets.C. Therefore, it is logically necessary that the number of the planets is less than 10.
Set-theoretical basis for real numbers
Authors: Wang Hao 
Published: May 2006

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.
Solution of a problem of Tarski
Authors: Myhill John 
Published: May 2006

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.
On predicates in algebraically closed fields
Authors: Robinson Abraham 
Published: May 2006

Many properties of curves, surfaces, or other varieties in Algebraic Geometry can be formulated in the lower functional calculus as predicates of the coefficients of the polynomial or polynomials which define the variety (curve, surface) in question. For example, the property of a plane curve of ordernto possess exactlymdouble points, or the property to be of genusp— wheremandpare specified integers — can be formulated in this way. Similarly many statements on the relation between two or more varieties, e.g., concerning the number and type of their intersection points, can be expressed in the lower functional calculus. It is usual to study the properties of a variety in an algebraically closed field. Accordingly, it is of considerable interest to investigate the general structure of the class of predicates mentioned above in relation to algebraically closed fields. The following result will be proved in the present paper.Main Theorem.LetFbe a commutative algebraic field of arbitrary characteristic, and letF′ = F[x1, …,xn]be the ring of polynomials of n variables with coefficients inF.With every predicate Q{x1, …,xn)which is formulated in the lower functional calculus in terms of the relations of equality, addition, and multiplication and (possibly) in terms of some of the elements ofF,there can be associated an ascending chain of ideals inF′,such thatfor every extensionF*ofFwhich is algebraically closed. In this formula, V0, …, V2k+ 1are the varieties of the ideals J1, …, J2k+iin the coordinate spaceSn: (x1, …xn) over F*, and VQis the set of points ofSnwhich satisfyQ.