An extension of basic logic

1948 ◽  
Vol 13 (2) ◽  
pp. 95-106 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Subsequent Paper ◽  
Basic Logic ◽  
Infinite Class ◽  
Logical Calculus ◽  
Recursive Enumerability ◽  
Basic Calculus

A logical calculus Κ was defined in a previous paper and then shown in a subsequent paper to contain within itself a representation of every constructively definable subclass of expressions of a certain infinite class U of expressions, where Κ itself is one such subclass. (The former paper will be referred to as BL and the latter paper as RC.) The calculus Κ was called a “basic calculus” and its theorems were thought of as expressing the asserted propositions of a “basic logic,” that is, of a logic within which is definable every constructively definable system of logic and indeed every constructively definable class or relation. The notion of constructive definability was essentially equated with the notion of recursive enumerability.

A minimum calculus for logic

1944 ◽  
Vol 9 (4) ◽  
pp. 89-94 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Truth Table ◽  
Additional Property ◽  
Basic Logic ◽  
Double Negation ◽  
Logical Calculus ◽  
Logical Calculi

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 simplification of basic logic

1953 ◽  
Vol 18 (4) ◽  
pp. 317-325 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Fundamental Importance ◽  
Basic System ◽  
Basic Logic ◽  
Infinite Class ◽  
Single Class ◽  
System A ◽  
Definition Of

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.

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

Gas Phase Spin Relaxation of ZX3Y Molecules in the Dipolar Approximation

1975 ◽  
Vol 53 (7) ◽  
pp. 723-738 ◽  
Author(s):  
B. C. Sanctuary ◽  
R. F. Snider
Keyword(s):  
Kinetic Theory ◽  
Gas Phase ◽  
Magnetic Relaxation ◽  
Dipolar Coupling ◽  
Point Group ◽  
Group Symmetry ◽  
Subsequent Paper ◽  
Point Group Symmetry ◽  
Proper Treatment ◽  
Gas Kinetic Theory

The gas kinetic theory of nuclear magnetic relaxation of a polyatomic gas, as formulated in the previous paper, is evaluated for ZX3Y molecules relaxing via a dipolar coupling Hamiltonian. Stress is given to a proper treatment of point group symmetry, here C3v, and the possibility of molecular inversion is included. The detailed formula for the spin traces is however restricted to X nuclei with spin 1/2. A subsequent paper uses these results to elucidate the structure of the high density dependence of T1 forCF3H.

scholarly journals The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Wenxia Chen ◽  
Danping Ding ◽  
Xiaoyan Deng ◽  
Gang Xu
Keyword(s):  
Soliton Solution ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Evolution Process ◽  
Solution Curve ◽  
Solitary Solutions ◽  
Basic Calculus

The evolution process of four class soliton solutions is investigated by basic calculus theory. For any given x, we describe the special curvature evolution following time t for the curve of soliton solution and also study the fluctuation of solution curve.

Reduction of Residual Stresses in Medium Density Fibreboard. Part 1. Taguchi Analysis

Holzforschung ◽  
2001 ◽  
Vol 55 (1) ◽  
pp. 67-72 ◽  
Author(s):  
J. van Houts ◽  
D. Bhattacharyya ◽  
K. Jayaraman
Keyword(s):  
Surface Layer ◽  
Residual Stresses ◽  
Internal Bond ◽  
Tensile Modulus ◽  
Thickness Swell ◽  
Subsequent Paper ◽  
Medium Density ◽  
Medium Density Fibreboard ◽  
Taguchi Analysis ◽  
Dissection Method

Summary This paper demonstrates how the Taguchi method of experimental design can be utilised to investigate methods for relieving the residual stresses present in medium density fibreboard (MDF). Panels have been subjected to heat, moisture and pressure, and after equilibration to room conditions, the changes in residual stresses through various layers have been measured using the dissection method. The application of heat and/or moisture has reduced the magnitude of residual stresses while generally the application of pressure has no effect on these stresses. The subsequent paper in this series uses Taguchi analysis to investigate how other board properties such as thickness swell, internal bond strength, surface layer tensile modulus and surface layer tensile strength are affected by the different treatment methods.

Taxonomic studies in wall barley (Hordeum murinum) and sea barley (Hordeum marinum). 1. Character investigation: assessment of new and traditional characters

1984 ◽  
Vol 62 (4) ◽  
pp. 753-762 ◽  
Author(s):  
Bernard R. Baum ◽  
L. Grant Bailey
Keyword(s):  
Data Analysis ◽  
Numerical Analysis ◽  
Exploratory Data Analysis ◽  
Subsequent Paper ◽  
Natural Habitats ◽  
Group Characters ◽  
Hordeum Murinum ◽  
Exploratory Data ◽  
Taxonomic Studies

In the literature about 60 taxa have been described at various taxonomic levels in the wall barley – sea barley group. Considerably less confusion exists between wall and sea barley than within. It appears that five taxa have been recognized and are likely to be valid, but there has been disagreement as to their status. The authors have investigated new micromorphologic characters and reexamined and critically defined traditional characters of material collected throughout the area of the natural habitats of the group. Characters taken from the seed epiblast and the floral lodicules were found useful for conclusively distinguishing between Hordeum murinum (sensu lato), H. glaucum, and H. marinum (sensu lato). Exploratory data analysis of 35 other characters investigated supports this distinction. Thus, in this particular paper three taxa are tentatively recognized. In the subsequent paper the validity of the three taxa will be tested by numerical analysis, and furthermore the possibility of five taxa will be investigated and similarly tested.

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