Correction to A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (3) ◽  
pp. 101-102 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Propositional Function ◽  
Free Variables ◽  
System L ◽  
Universal Quantifiers ◽  
Individual Variables ◽  
The Individual ◽  
Image Position

In A note on the Entscheidungsproblem the author gave a proof of the unsolvability of the general case of the Entscheidungsproblem of the engere Funktionenkalkül. This proof, however, contains an error, in order to correct which it is necessary to modify the “additional axioms” of the system L so that they contain no free variables (either free individual variables or free propositional function variables).The additional axioms of L other than x=y→[F(x)→F(y)] contain no free propositional function variables, and hence it is sufficient to replace each one by an expression obtained from it by quantifying all the individual variables by means of universal quantifiers initially placed—thus, in particular, x = x is replaced by (x)[x=x]. Moreover the axiom x=y→[F(x)→F(y)] may be replaced by the following set of axioms:and similar axioms for each of the functions b1, b2, …, bk.

On the rules of proof in the pure functional calculus of the first order

1951 ◽  
Vol 16 (2) ◽  
pp. 107-111 ◽  
Author(s):  
Andrzej Mostowski
Keyword(s):  
Functional Calculus ◽  
Free Variable ◽  
Propositional Function ◽  
Cartesian Power ◽  
First Order ◽  
A Value ◽  
Free Variables ◽  
Individual Variables ◽  
Definition Of

We consider here the pure functional calculus of first order as formulated by Church.Church, l.c., p. 79, gives the definition of the validity of a formula in a given set I of individuals and shows that a formula is provable in if and only if it is valid in every non-empty set I. The definition of validity is preceded by the definition of a value of a formula; the notion of a value is the basic “semantical” notion in terms of which all other semantical notions are definable.The notion of a value of a formula retains its meaning also in the case when the set I is empty. We have only to remember that if I is empty, then an m-ary propositional function (i.e. a function from the m-th cartesian power Im to the set {f, t}) is the empty set. It then follows easily that the value of each well-formed formula with free individual variables is the empty set. The values of wffs without free variables are on the contrary either f or t. Indeed, if B has the unique free variable c and ϕ is the value of B, then the value of (c)B according to the definition given by Church is the propositional constant f or t according as ϕ(j) is f for at least one j in I or not. Since, however, there is no j in I, the condition ϕ(j) = t for all j in I is vacuously satisfied and hence the value of (c)B is t.

Development of a German-Language Questionnaire to Measure Collective Orientation as an Individual Attitude

2017 ◽  
Vol 76 (3) ◽  
pp. 91-105 ◽  
Author(s):  
Vera Hagemann
Keyword(s):  
Team Performance ◽  
Reliability And Validity ◽  
Future Research ◽  
German Language ◽  
Team Members ◽  
Collective Orientation ◽  
Team Research ◽  
German Speaking ◽  
Individual Variables ◽  
The Individual

Abstract. The individual attitudes of every single team member are important for team performance. Studies show that each team member’s collective orientation – that is, propensity to work in a collective manner in team settings – enhances the team’s interdependent teamwork. In the German-speaking countries, there was previously no instrument to measure collective orientation. So, I developed and validated a German-language instrument to measure collective orientation. In three studies (N = 1028), I tested the validity of the instrument in terms of its internal structure and relationships with other variables. The results confirm the reliability and validity of the instrument. The instrument also predicts team performance in terms of interdependent teamwork. I discuss differences in established individual variables in team research and the role of collective orientation in teams. In future research, the instrument can be applied to diagnose teamwork deficiencies and evaluate interventions for developing team members’ collective orientation.

scholarly journals Eight Simple Guidelines for Improved Understanding of Transformations and Nonlinear Effects

2021 ◽  
pp. 109442812199190
Author(s):  
Mikko Rönkkö ◽  
Eero Aalto ◽  
Henni Tenhunen ◽  
Miguel I. Aguirre-Urreta
Keyword(s):  
Generalized Linear Model ◽  
Functional Form ◽  
Nonlinear Models ◽  
Nonlinear Effects ◽  
Recent Past ◽  
Organizational Research ◽  
Common Practices ◽  
Individual Variables ◽  
The Individual ◽  
The Relationship

Transforming variables before analysis or applying a transformation as a part of a generalized linear model are common practices in organizational research. Several methodological articles addressing the topic, either directly or indirectly, have been published in the recent past. In this article, we point out a few misconceptions about transformations and propose a set of eight simple guidelines for addressing them. Our main argument is that transformations should not be chosen based on the nature or distribution of the individual variables but based on the functional form of the relationship between two or more variables that is expected from theory or discovered empirically. Building on a systematic review of six leading management journals, we point to several ways the specification and interpretation of nonlinear models can be improved.

A dynamic programming approach to evolutionarily stable strategy theory

1980 ◽  
Vol 12 (1) ◽  
pp. 3-5 ◽  
Author(s):  
C. Cannings ◽  
D. Gardiner
Keyword(s):  
Dynamic Programming ◽  
Density Function ◽  
Evolutionarily Stable Strategy ◽  
Programming Approach ◽  
Stable Strategy ◽  
War Of Attrition ◽  
Dynamic Programming Approach ◽  
The Individual ◽  
Image Position ◽  
Evolutionarily Stable

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

Error Analysis of a Mobile Terrestrial LiDAR System

GEOMATICA ◽  
2014 ◽  
Vol 68 (3) ◽  
pp. 183-194 ◽  
Author(s):  
M. Leslar ◽  
B. Hu ◽  
J.G. Wang
Keyword(s):  
Point Cloud ◽  
Point Clouds ◽  
Sensitivity Study ◽  
Error Sources ◽  
Terrestrial Lidar ◽  
Lever Arm ◽  
Ideal Situation ◽  
Individual Variables ◽  
The Individual ◽  
Source Of Error

The understanding of the effects of error on Mobile Terrestrial LiDAR (MTL) point clouds has not increased with their popularity. In this study, comprehensive error analyses based on error propagation theory and global sensitivity study were carried out to quantitatively describe the effects of various error sources in a MTL system on the point cloud. Two scenarios were envisioned; the first using the uncertainties for measurement and calibration variables that are normally expected for MTL systems as they exist today, and the second using an ideal situation where measurement and calibration values have been well adjusted. It was found that the highest proportion of error in the point cloud can be attributed to the boresight and lever arm parameters for MTL systems calibrated using non-rigours methods. In particular, under a loosely controlled error condition, the LiDAR to INS Z lever arm and the LiDAR to INS roll angle contributed more error in the output point cloud than any other parameter, including the INS position. Under tightly controlled error conditions, the INS position became the dominant source of error in the point cloud. In addition, conditional variance analysis has shown that the majority of the error in a point cloud can be attributed to the individual variables. Errors caused by the interactions between the diverse variables are minimal and can be regarded as insignificant.

An extension of Ackermann's set theory

1972 ◽  
Vol 37 (4) ◽  
pp. 703-704
Author(s):  
Donald Perlis
Keyword(s):  
Set Theory ◽  
Free Variables ◽  
Image Position

Ackermann's set theory [1], called here A, involves a schemawhere φ is an ∈-formula with free variables among y1, …, yn and w does not appear in φ. Variables are thought of as ranging over classes and V is intended as the class of all sets.S is a kind of comprehension principle, perhaps most simply motivated by the following idea: The familiar paradoxes seem to arise when the class CP of all P-sets is claimed to be a set, while there exists some P-object x not in CP such that x would have to be a set if CP were. Clearly this cannot happen if all P-objects are sets.Now, Levy [2] and Reinhardt [3] showed that A* (A with regularity) is in some sense equivalent to ZF. But the strong replacement axiom of Gödel-Bernays set theory intuitively ought to be a theorem of A* although in fact it is not (Levy's work shows this). Strong replacement can be formulated asThis lack of A* can be remedied by replacing S above bywhere ψ and φ are ∈-formulas and x is not in ψ and w is not in φ. ψv is ψ with quantifiers relativized to V, and y and z stand for y1, …, yn and z1, …, zm.

Comparison of Russell's resolution of the semantical antinomies with that of Tarski

1976 ◽  
Vol 41 (4) ◽  
pp. 747-760 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Type Theory ◽  
Simple Theory ◽  
Vicious Circle ◽  
New Members ◽  
Domain Type ◽  
Individual Variables ◽  
The Individual ◽  
Essential Restriction

In this paper we treat the ramified type theory of Russell [6], afterwards adopted by Whitehead and Russell in Principia mathematica [12], so that we may compare Russell's resolution of the semantical antinomies by ramified type theory with the now widely accepted resolution of them by the method of Tarski in [7], [8], [9].To avoid impredicativity the essential restriction is that quantification over any domain (type) must not be allowed to add new members to the domain, as it is held that adding new members changes the meaning of quantification over the domain in such a way that a vicious circle results. As Whitehead and Russell point out, there is no one particular form of the doctrine of types that is indispensable to accomplishing this restriction, and they have themselves offered two different versions of the ramified hierarchy in the first edition of Principia (see Preface, p. vii). The version in §§58–59 of the writer's [1], which will be followed in this paper, is still slightly different.To distinguish Russellian types or types in the sense of the ramified hierarchy from types in the sense of the simple theory of types, let us call the former r-types.There is an r-type i to which the individual variables belong. If β1, β2, …, βm are any given r-types, m ≧ 0, there is an r-type (β1, β2, …, βm)/n to which there belong m-ary functional variables of level n, n ≧ 1. The r-type (α1, α2, …, αm)/k is said to be directly lower than the r-type (β1, β2, …, βm)/n if α1 = β1, α2 = β2, …, αm = βm, k < n.

scholarly journals Individual Predictors of the Radical Right-Wing Vote in Europe: A Meta-Analysis of Articles in Peer-Reviewed Journals (1995–2016)

2018 ◽  
Vol 53 (3) ◽  
pp. 569-593 ◽  
Author(s):  
Daniel Stockemer ◽  
Tobias Lentz ◽  
Danielle Mayer
Keyword(s):  
Meta Analysis ◽  
Radical Right ◽  
Political Regime ◽  
Right Wing ◽  
Quantitative Methodology ◽  
Success Rates ◽  
Quantitative Studies ◽  
Individual Vote ◽  
Individual Variables ◽  
The Individual

In this article, we summarize the individual demand-level factors explaining the radical right-wing vote in European countries. To do so, we first review 46 quantitative peer-reviewed articles featuring the individual vote choice in favour of a radical right-wing party as the dependent variable. To identify relevant articles, we use Kai Arzheimer’s bibliography on the radical right and employ the following inclusion criterion: the articles must be written in English, they must use the individual vote for a radical right-wing party as the dependent variable, they must use a quantitative methodology and they must include some type of regression analysis. Using this strategy, we conduct a meta-analysis of 329 relevant models and find that over 20 individual variables are tested. Because many variables such as attitudes towards immigration, employment, age, education and gender only show moderate success rates in attempting to explain an individual’s propensity to vote for the radical right, we complement the review of quantitative studies with an analysis of 14 qualitative publications. The review of these qualitative works shows that the processes through which somebody becomes a voter, supporter or activist of the radical right are often more complex than the commonly used surveys can portray them. Frequently, feelings of relative economic deprivation and dissatisfaction with the political regime trigger an awakening that makes individuals seek engagement. However, the processes behind this awakening are complex and can only be partially captured by quantitative studies.

scholarly journals Absolute Magnitudes by Statistical Parallaxes

1978 ◽  
Vol 80 ◽  
pp. 49-52
Author(s):  
André Heck
Keyword(s):  
Maximum Likelihood ◽  
Absolute Magnitude ◽  
List Type ◽  
Solar Motion ◽  
The Mean ◽  
Absolute Magnitudes ◽  
The Individual ◽  
Image Position ◽  
Principle Of Maximum

Our algorithm for stellar luminosity calibrations (based on the principle of maximum likelihood) allows the calibration of relations of the type:Where n is the size of the sample at hand,Mi, are the individual absolute magnitudes,Cijare observational quantities (j = 1, …, N), andqjare the coefficients to be determined.If we put N = 1 and CiN= 1, we havethe mean absolute magnitude of the sample. As additional output, the algorithm provides us also with the dispersion in magnitude of the sample σM, the mean solar motion (U, V, W) and the corresponding velocity ellipsoid (σu, σV, σw).

On the determination of intermolecular energies by inductive analysis

1942 ◽  
Vol 38 (2) ◽  
pp. 224-230
Author(s):  
William J. C. Orr
Keyword(s):  
Virial Coefficients ◽  
Intermolecular Potential ◽  
Rare Gases ◽  
Second Virial Coefficients ◽  
Classical Expression ◽  
The Individual ◽  
Image Position ◽  
Range Of Values ◽  
Inductive Analysis

For a direct comparison of the individual attractive and repulsive terms of an intermolecular potential determined by the inductive analysis of themodynamic data with the same terms calculated by quantal methods it is desirable to carry out the analyses, in the first approximation, with an intermolecular potential of the form ø(R) = Pe−aR − A1/R6 − A2/R8. For mathematical convenience, in place of the above expression, two potential functions,andare considered, the first being taken to be adequate in the range of values of R between 0 and R0 (the minimum of the potential function) and the second, in the range from R0 to ∞. By dividing the problem in this way it is possible to find substitutions which permit the integration of the classical expression for the second virial coefficients (and other appropriate thermodynamic data) directly in terms of fairly simple series in | ψ0 |, R0, a and r. Finally it is pointed out that for such simple atoms or molecules as the rare gases, oxygen, nitrogen and methane r may be taken as 0·15 throughout, which considerably simplifies the application of the method to the experimental data.

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