bluuebunny/crossref_metadata_2025_split huggingface dataset:
Politics and Pseudopolitics: A Critical Evaluation of Some Behavioral Literature
Authors: Bay Christian 
Published: May 2006

A curious state of affairs has developed within the academic discipline that bravely calls itself Political Science—the discipline that in a much-quoted phrase has been called “a device, invented by university teachers, for avoiding that dangerous subject politics, without achieving science.” A growing and now indeed a predominant proportion of leading American political scientists, the behavioralists, have become determined to achieve science. Yet in the process many of them remain open to the charge of strenuously avoiding that dangerous subject, politics.Consider a recent essay on the behavioral persuasion in politics. The conclusion stresses the purpose of political inquiry: “The Goal is Man.” There is to be a commitment to some humane purpose after all. But what kind of man? A democratic kind of man, a just man, or perhaps a power-seeking man? The answer follows: “These are philosophical questions better left to the philosophers.”
Contested Elections and Voter Turnout in a Local Community: A Problem in Spurious Correlation
Authors: Beyle Thad L. 
Published: May 2006

This paper presents an attempt, in a case study, to infer causal relationships between contests for local elective office and voter turnout—an attempt which required, in turn, the use of a recently developed mathematical technique for identifying and eliminating spurious correlations. The method no doubt has wider applications.The case study involves a small New England town and its annual February town elections over the period 1947–1963. The town was chosen as one of the six affected by the newly established Cape Cod National Seashore. The time period was chosen to cover the interval during which the town was faced with the rapid increase in tourism which transformed it from a declining agricultural town to a service-based town. The policies and methods by which the town dealt with this basic change in its economic structure were of immediate interest, especially in the latter years when the National Park Service became involved in the problems the Cape has faced in this postwar period.
On the semantic non-completeness of certain Lewis calculi
Authors: Halldén Sören 
Published: May 2006

I intend to show that for S1, S2, and S3 the class of true formulas cannot coincide with the class of theorems. This is to hold irrespective of the meaning assigned to “◊”. when only “∼”, “▪”, and the variables are interpreted in the customary manner.The idea of the proof is extremely simple and can be illustrated by the following argument concerning S3. The formula ~(◊(p▪~p)⥽▪p▪~p) ∨ (◊(q▪~q)⥽▪q▪~q) is a theorem of S3. Suppose now that all S3-theorems are true. Then either ~(◊(p▪~p)⥽▪p▪~p) is true (for every p) or (◊(q▪~q)⥽▪q▪~q) is true (for every q). But none of these formulas is an S3-theorem. Then some true S3-formula is not an S3-theorem. Hence, if all S3-theorems are true, some true S3-formula is not an S3-theorem. Then the class of S3-theorems cannot be identical with the class of true S3-formulas.
Postulates for the calculus of binary relations
Authors: McKinsey J. C. C. 
Published: May 2006

The theory of relations was discussed, along with the theory of classes, by Peirce and Schröder. While the calculus of classes has subsequently been presented (by Couturat, and by Huntington) as an abstract mathematical system, no similar formulation has been published of the calculus of relations. Professor Alfred Tarski has proposed to me, the problem of developing a similar formulation for the theory of relations. In this paper I present a set of postulates for the calculus of relations.The postulates have been chosen so as to enable us to prove Theorem A, below, which asserts a kind of completeness of the system. A set of postulates is called semi-categorical, if every two realizations which involve the same number of elements are isomorphic. From Theorem A one can easily prove that my postulates are semi-categorical, if one assumes the following theorem from Mengenlehre: if α and β are cardinal numbers such that , then α=β. This theorem has not been proved, however, except on the assumption of the axiom of choice and the generalized continuum hypothesis, and is not assumed in the present paper.It should be pointed out, that the sort of completeness I have chosen is not the only sort, or even necessarily the most interesting sort, of completeness which a set of postulates for the calculus of relations could possess. For example, the question arises, whether it would be possible to find a set of postulates from which all true equations (involving free variables and the operations used by Schröder) could be derived, and which was the weakest possible such set. The present work should therefore be regarded only as a first step in the investigation of the axiomatization of the calculus of relations.
Proofs of non-deducibility in intuitionistic functional calculus
Authors: Mostowski Andkzej 
Published: May 2006

It has been proved by S. C. Kleene and David Nelson that the formulais intuitionistically non-deducible, i.e., non-deducible within the intuitionistic functional calculus.The aim of this note is to outline a general method which permits us to establish the intuitionistic non-deducibility of many formulas and in particular of the formula (1).
The identity of individuals in a strict functional calculus of second order
Authors: Barcan Ruth C. 
Published: May 2006

In previous papers we developed two functional calculi of first order based on strict implication which we called S2 and S4. In the present paper, these systems will be extended to include a functional calculus of second order with the purpose of introducing the relation of identity of individuals.Primitive symbols. ̂ {the abstraction operator, the blank space to be replaced by an appropriate variable}.
A system of axiomatic set theory—Part II
Authors: Bernays Paul 
Published: May 2006

For the formulation of the remaining axioms we need the notions of a function and of a one-to-one correspondence.We define a function to be a class of pairs in which different elements always have different first members; or, in other words, a class F of pairs such that, to every element a of its domain there is a unique element b of its converse domain determined by the condition 〈a, b〉ηF. We shall call the set b so determined the value of F for a, and denote it (following the mathematical usage) by F(a).A set which represents a function—i.e., a set of pairs in which different elements always have different first members—will be called a functional set.If b is the value of the function F for a, we shall say that F assigns the set b to the set a; and if a functional set f represents F, we shall say also that f assigns the set b to the set a.A class of pairs will be called a one-to-one correspondence if both it and its converse class are functions. We shall say that there exists a one-to-one correspondence between the classes A and B (or of A to B) if A and B are domain and converse domain of a one-to-one correspondence. Likewise we shall say that there exists a one-to-one correspondence between the sets a and b (or of a to b) if a and b respectively represent the domain and the converse domain of a one-to-one correspondence. In the same fashion we speak of a one-to-one correspondence between a class and a set, or a set and a class.
A system of axiomatic set theory—Part VI
Authors: Bernays Paul 
Published: May 2006

Comparability of classes. Till now we tried to get along without the axioms Vc and Vd. We found that this is possible in number theory and analysis as well as in general set theory, even keeping in the main to the usual way of procedure.For the considerations of the present section application of the axioms Vc, Vd is essential. Our axiomatic basis here consists of the axioms I—III, V*, Vc, and Vd. From V*, as we know, Va and Vb are derivable. We here take axiom V* in order to separate the arguments requiring the axiom of choice from the others. Instead of the two axioms V* and Vc, as was observed in Part II, V** may be taken as well.
On direct products of theories
Authors: Mostowski Andrzej 
Published: May 2006

This paper deals with the notion of direct product in the theory of decision problems.Elementary mathematical theories are always concerned with certain functions defined in a set I (called the universe of discourse of the theory) and certain relations with the common domain I. The notions of functions and relations are known to be reducible to each other. To avoid duplication of definitions and proofs we shall eliminate the former notion in favor of the latter and consider only theories in which all the primitive terms are of the type of relations (with an arbitrary number of arguments).It is easy to define for relations the notion of a direct product in the way which is usual in abstract algebra. We are led to this definition in a very natural way when we reflect that relations are but a particular case of functions, and the direct product of an arbitrary number of functions is defined in algebra.A relation which is representable as a product of an arbitrary number of pairwise identical relations is called the power-relation, and the relations which occur as factors are called the base-relations.The content of the present paper can now be characterized more precisely as follows.We shall discuss a theory of which the primitive notions are representable as powers of certain base-relations, and shall try to reduce all the problems concerning this theory (in particular the decision problem) to problems concerning the theory of the base-relations. It will be seen that this reduction is in fact possible, and that several particular cases of the decision problem, the solutions of which are known from the literature, can be included in the general scheme established in the present paper.
A minimum calculus for logic
Authors: Fitch Frederic B. 
Published: May 2006

A logical calculus will be presented which not only is a formulation of a “basic logic” in the sense of the writer's previous papers, but which has the additional property that no weaker calculus can be a formulation of a basic logic. A sort of minimum logical calculus is thus attained, which has nothing superfluous about it for achieving the purpose for which it is designed.In the case of some logical calculi the question can arise as to whether certain of the postulates are really logically valid and necessary. Sometimes a test is available, such as the truth-table test, enabling us to distinguish between logically valid sentences and others, but often no such test is available, especially where quantifiers are involved. Is or is not the axiom of infinity, for example, to be regarded as logically valid? Or is the principle of double negation really acceptable, even though it satisfies the truth-table test?
A system of axiomatic set theory. Part IV. General set theory
Authors: Bernays Paul 
Published: May 2006

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)
Weak simplest normal truth functions
Authors: Nelson Raymond J. 
Published: May 2006

In certain applications of truth-functional logic it is of interest to determine classes of formulas equivalent to a given formula Φ under the hypothesis that certain conjunctions of letters of Φ are always false. Of especial interest is the case where the class to be determined is that of simplest normal truth functions. The problem of giving a calculation procedure for this question is evidently a more general form of the simplification problem as considered in [1] and [2]. The purpose of this note is to indicate how the procedure of [1] applies.Let Π, Π′, Π″, … be fundamental formulas, in the sense of [2], such that every literal of Π, etc., is a literal of Φ and such that Π, Π′, Π″, … are all false for all values of the constituent literals. Then we say that Ψ is a weak simplest disjunctive normal equivalent of Φ if and only if Ψ ≡ Φ, under the hypothesis, and there is no Ψ′ such that Ψ′ is simpler than Ψ and Ψ′ ≡ Φ, under the hypothesis. Similarly, under exactly the same hypothesis, we say that X is a weak simplest conjunctive normal equivalent of Φ. When the class of always false fundamental formulas is empty, as in [1] and [2], we may speak of strong simplest forms and hence of the more general problem of strong simplification.
The þ-function in λ-K-conversion
Authors: Turing A. M. 
Published: May 2006

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.
Modalities and Quantification
Authors: Carnap Rudolf 
Published: May 2006

The purpose of this article is to give a survey of some results I have found in investigations concerning logical modalities. The results refer: (1) to semantical systems, i.e., symbolic language systems for which semantical rules of interpretation are laid down; (2) to corresponding calculi, i.e., syntactical systems with primitive sentences and a rule of inference; (3) to relations between a semantical system and the corresponding calculus.The semantical systems to be dealt with are the following: pro positional logic (PL), functional logic (FL), and the corresponding modal systems, viz. modal propositional logic (MPL) and modal functional logic (MFL).
A partial model for Quine's “New foundations”
Authors: Beneš Václav Edvard 
Published: May 2006

1. In this paper we construct a model for part of the system NF of [4]. Specifically, we define a relation R of natural numbers such that the R-relativiseds of all the axioms except P9 of Hailperin's finitization [2] of NF become theorems of say Zermelo set theory. We start with an informal explanation of the model.2. Scrutiny of P1-P8 of [2] suggests that a model for these axioms might be constructed by so to speak starting with a universe that contained a “universe set” and a “cardinal 1”, and passing to its closure under the operations implicit in P1-P7, viz., the Boolean, the domain, the direct product, the converse, and the mixtures of product and inverse operations represented by P3 and P4. To obtain such closure we must find a way of representing the operations that involve ordered pairs and triples.We take as universe of the model the set of natural numbers ω; we let 0 represent the “universe set” and 1 represent “cardinal 1”. Then, in order to be able to refer in the model to the unordered pair of two sets, we determine all representatives of unordered pairs in advance by assigning them the even numbers in unique fashion (see d3 and d25); we can now define the operations that involve ordered pairs and triples, and obtain closure under them using the odd numbers. It remains to weed out, as in d26, the unnecessary sets so as to satisfy the axiom of extensionality.
Report on some investigations concerning the consistency of the axiom of reducibility
Authors: Myhill John 
Published: May 2006

The purpose of this paper is to construct a non-finitary system of logic S, based on the theory of types, in which classical quantification theory holds without restriction and the axiom of reducibility holds with such slight restriction as is perhaps unlikely to interfere with the construction of classical mathematics. The system will be shown to be consistent.For purposes of orientation, a rough description of the ontology of the system will first be given. The ‘individuals’ are expressions built up in the following way: ‘0’ is an individual, and if ϕ and ψ are individuals, so is ˹*ϕψ˺. The usual apparatus of truth functional connectives and quantifiers is employed, and one primitive four place predicate ‘S’. ˹Sϕψχω˺ says that ϕ is the result of writing ψ for all occurrences of χ in ω, where ϕ, ψ, χ and ω may be formulae, individuals, or members of a wider category called ‘expressions’ which includes individuals, formulae, and other things as well. (At first sight this looks like a confusion of use and mention, but the formation rules and rules of inference are so phrased that no ambi
guity arises.)The individuals are divided into types concentrically, so that type 0 has them all as members and type n includes type m for m > n. There is no highest type, but we shall find it convenient from the point of view of exposition to construct an infinity of ‘auxiliary’ systems Q(k), restricted to types ≦ k, before presenting the system S.
On the use of dots as brackets in logical expressions
Authors: Curry H. B. 
Published: May 2006

The Peanese convention for the use of dots as brackets has the disadvantage that it gives only an awkward method for representing chains of indefinite length, such as the compound implicationSuch chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other operation appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine, is at variance with our ordinary algebraic usage, in which we write a − b + c − d for ((a − b) + c) − d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.Such a modification is easily obtained by generalizing somewhat a procedure of Church. The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation.
Emch's calculus and strict implication
Authors: Lewis C. I. 
Published: May 2006

In a previous number of this Journal, Dr. Arnold F. Emch argues that the mathematical properties of the system of strict implication preclude the interpretation of p⊰q as synonymous with “q is deducible from p” and of ◊(pq), or p o q, as “p and q are consistent.” He proposes a new calculus of propositions in which the primitive ideas of the system of strict implication are retained but an additional idea, “logical consistency,” Op, is introduced, and two new relations, “logical implication,” p∾q, and “logical equivalence,” p = q, are denned. He believes that the properties of this new system are in accord with the facts about deducibility, consistency, and independence of propositions where those of p⊰q and ◊(pq) fail of such accord.Not all of the objections which Dr. Emch advances against strict implication are given due consideration in what follows. Concerning one especially important point, something will be said at the end of this paper. For the rest, discussion will here be limited to certain formal properties of the two systems, and certain consequences for the interpretation of them.For brevity, the system of strict implication will be referred to, in what follows, as the “system S,” or merely as S; and Dr. Emch's calculus as the “system L,” or as L.
A system of formal logic without an analogue to the curry W operator
Authors: Fitch Frederic Brenton 
Published: May 2006

The system of formal logic to be presented in this paper has the following main properties:(1). The Mengenlehre paradoxes do not arise in it.(2). It seems to be free from other contradictions.(3). It does not resort to a theory of types.(4). It uses negation and material implication and does not weaken the principle of excluded middle.(5). It contains no analogue to the Curry W operator; consequently, given a propositional function ϕ, a propositional function Ψ cannot always be found such that, for all x, ϕ(x, x) and Ψ(x) are equivalent propositions.(6). It employs a universal quantifier, G, such that (x)f(x), (x)f(x, x), (x)f(x, x, x), …, would be respectively expressed thus, G(f, f), G(f, f, f), G(f, f, f, f), ….(7). It employs Schönfinkel's functional notation, so that f(x1, x2, …, xn) will be written as (( … ((fx1)x2) …)xn) and abbreviated to (fx1x2… xn) and, when no ambiguity results, to fx1x2…xn.(8). A theory of integers and real numbers cannot, apparently, be deduced from it.(9). A form of the axiom of choice is provable in it.