Computational logic

1949 ◽  
Vol 14 (3) ◽  
pp. 167-172 ◽  
Author(s):  
Nathan P. Levin
Keyword(s):  
Computational Logic ◽  
Elementary Algebra ◽  
Sentential Calculus ◽  
Primitive Term ◽  
Arbitrary Sign ◽  
Image Position ◽  
Binary Combination

We make certain notational conventions. These are referred to as CL (Computational Logic). The relation of interchangeability is introduced as the basic connection between logical formulas. This approach lends perspicuity to the results of sentential calculus. With its technical devices CL is able to rephrase logical theorems in rather succinct manner. Our exposition tries to steer a middle course between informality and strict rigor.It is felt that the method of CL offers advantages for the teaching of logic. Proofs are algorithmic and resemble those of elementary algebra. All inferences are reversible and practically non-tentative.Sign is a primitive term of CL. Its denning property is a capacity for entering into binary combination with other signs, or with itself, according to this convention:x and y are signs if, and only if, (x y) is a sign. Bracketing is looked upon as an operation on signs in terms of which other operations are definable. The practice of some authors in classifying brackets under the heading of symbols, seems to us questionable; for unlike symbols or signs brackets are never used to denote anything. Brackets enter into the composition of signs, not to denote a grouping, but rather to exhibit it, in the manner of a diagram.An unending list of letters, with or without subscripts,serve to denote arbitrary signs. An arbitrary sign may or may not have other signs as parts. The numeral “2” is used as a constant. Bracketing abbreviation is as follows:and so on. Outermost brackets will ordinarily be omitted. This kind of bracketing may be termed left-associative. For convenience, these bracketing conventions are crystallized into a rule.

Inferential equivalence and natural deduction

1957 ◽  
Vol 22 (3) ◽  
pp. 237-240
Author(s):  
H. Hiż
Keyword(s):  
Natural Deduction ◽  
Identity Relation ◽  
Sentential Calculus ◽  
Image Position

There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists ofAnother one, closely connected with A1–A3, is comprised ofB1–B3 jointly assert the inferential equivalence of a sentence of the form (p ⊃ q) ⊃ r and two sentences of the forms q ⊃ r and ∼p ⊃ r respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by ‘⊦’. ‘H’ will stand here for inferential equivalence. It may be looked upon as a combination of signs ‘⊦’ and ‘⊣’. ‘H’ is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role ‘H’ plays is determined by **1 –*5 below. Besides that ‘H’ remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.

Reviews - J. H. Woodger. Translator's preface. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. vii–ix. - Alfred Tarski. Author's acknowledgments.Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. xi–xii. - Alfred Tarski. On the primitive term of logistic. Modified English translation based on 2852–4. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 1–23. - Alfred Tarski. Foundations of the geometry of solids.Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 24–29. (Translated, with additions, from Księga Pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego, supplement to Annales de la Société Polonaise de Mathématique, Cracow 1929, pp. 29-33.) - Alfred Tarski. On some fundamental concepts of metamathematics. English translation of 2857. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, 30–37. - Jan Łukasiewicz and Alfred Tarski. Investigations into the sentential calculus. English translation of 4077, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 38–59. - Alfred Tarski. Fundamental concepts of the methodology of the deductive sciences. English translation of 2858, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 60–109. - Alfred Tarski. On definable sets of real numbers. English translation of 28510, with additions in the text by the author as well as added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 110–142. - Kazimierz Kuratowski and Alfred Tarski. Logical operations and projective sets. English translation of 4321, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 143–151. - Alfred Tarski. The concept of truth in formalized languages. English translation of 28516, with added footnotes. Logic, semantics, metamathematics, papers from 1923 to 1938.Oxford at the Clarendon Press, London1956, pp. 152–278.

1969 ◽  
Vol 34 (1) ◽  
pp. 99-106 ◽  
Author(s):  
W. A. Pogorzelski ◽  
S. J. Surma
Keyword(s):  
English Translation ◽  
Real Numbers ◽  
Sentential Calculus ◽  
Alfred Tarski ◽  
Primitive Term ◽  
Logical Operations ◽  
Definable Sets ◽  
Concept Of Truth

scholarly journals Oscillation and comparison theorems for certain neutral difference equations

1992 ◽  
Vol 34 (2) ◽  
pp. 245-256 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Comparison Theorems ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
Arbitrary Sign ◽  
Image Position ◽  
Mixed Arguments

AbstractSome comparison theorems and oscillation criteria are established for the neutral difference equationas well as for certain neutral difference equations with coefficients of arbitrary sign. Neutral difference equations with mixed arguments are also considered.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

X-ray microanalysis of thin specimens in the transmission electron microscope at voltages up to 1000 Kv.

1978 ◽  
Vol 36 (1) ◽  
pp. 540-541
Author(s):  
G. Cliff ◽  
M.J. Nasir ◽  
G.W. Lorimer ◽  
N. Ridley
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Weight Fraction ◽  
Present Series ◽  
Constant Independent ◽  
X Ray ◽  
Transmission Electron ◽  
Series Of Experiments ◽  
Image Position ◽  
X Ray Absorption

In a specimen which is transmission thin to 100 kV electrons - a sample in which X-ray absorption is so insignificant that it can be neglected and where fluorescence effects can generally be ignored (1,2) - a ratio of characteristic X-ray intensities, I1/I2 can be converted into a weight fraction ratio, C1/C2, using the equationwhere k12 is, at a given voltage, a constant independent of composition or thickness, k12 values can be determined experimentally from thin standards (3) or calculated (4,6). Both experimental and calculated k12 values have been obtained for K(11<Z>19),kα(Z>19) and some Lα radiation (3,6) at 100 kV. The object of the present series of experiments was to experimentally determine k12 values at voltages between 200 and 1000 kV and to compare these with calculated values.The experiments were carried out on an AEI-EM7 HVEM fitted with an energy dispersive X-ray detector.

SEM analysis of the change in the reaction rates with deposit structures in the copper-iron cementation system

1978 ◽  
Vol 36 (1) ◽  
pp. 456-457
Author(s):  
V. Annamalai ◽  
L.E. Murr
Keyword(s):  
Structural Characteristics ◽  
Secondary Electron Emission ◽  
Copper Deposit ◽  
Reaction Rates ◽  
Sem Analysis ◽  
Experimental Conditions ◽  
Major Drawback ◽  
Emission Mode ◽  
Scrap Iron ◽  
Image Position

Economical recovery of copper metal from leach liquors has been carried out by the simple process of cementing copper onto a suitable substrate metal, such as scrap-iron, since the 16th century. The process has, however, a major drawback of consuming more iron than stoichiometrically needed by the reaction.Therefore, many research groups started looking into the process more closely. Though it is accepted that the structural characteristics of the resultant copper deposit cause changes in reaction rates for various experimental conditions, not many systems have been systematically investigated. This paper examines the deposit structures and the kinetic data, and explains the correlations between them.A simple cementation cell along with rotating discs of pure iron (99.9%) were employed in this study to obtain the kinetic results The resultant copper deposits were studied in a Hitachi Perkin-Elmer HHS-2R scanning electron microscope operated at 25kV in the secondary electron emission mode.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

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