Strict implication, deducibility and the deduction theorem

1953 ◽  
Vol 18 (3) ◽  
pp. 234-236 ◽  
Author(s):  
Ruth Barcan Marcus
Keyword(s):  
Deductive Inference ◽  
Logical Investigation ◽  
Deduction Theorem ◽  
Strict Implication ◽  
Suitable Definition ◽  
Definition Of

Lewis and Langford state, “… it appears that the relation of strict implication expresses precisely that relation which holds when valid deduction is possible. It fails to hold when valid deduction is not possible. In that sense, the system of strict implication may be said to provide that canon and critique of deductive inference which is the desideratum of logical investigation.” Neglecting for the present other possible criticisms of this assertion, it is plausible to maintain that if strict implication is intended to systematize the familiar concept of deducibility or entailment, then some form of the deduction theorem should hold for it. The purpose of this paper is to analyze and extend some results previously established which bear on the problem.We will begin with a rough statement of some relevent considerations. Let the system S contain among its connectives an implication connective ‘I’ and a conjunction connective ‘&’. Let A1, A2, …, An ⊦ B abbreviate that B is provable on the hypotheses A1, A2, …, An for a suitable definition of “proof on hypotheses”, where A1, A2, …, An, B are well-formed expressions of S.

An LQ Approach to Active Control of Vibrations in Helicopters

1996 ◽  
Vol 118 (3) ◽  
pp. 482-488 ◽  
Author(s):  
Sergio Bittanti ◽  
Fabrizio Lorito ◽  
Silvia Strada
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Linear Quadratic ◽  
Control Effort ◽  
Vibration Effect ◽  
Suitable Definition ◽  
Lq Optimal Control ◽  
The One ◽  
Definition Of ◽  
And Control

In this paper, Linear Quadratic (LQ) optimal control concepts are applied for the active control of vibrations in helicopters. The study is based on an identified dynamic model of the rotor. The vibration effect is captured by suitably augmenting the state vector of the rotor model. Then, Kalman filtering concepts can be used to obtain a real-time estimate of the vibration, which is then fed back to form a suitable compensation signal. This design rationale is derived here starting from a rigorous problem position in an optimal control context. Among other things, this calls for a suitable definition of the performance index, of nonstandard type. The application of these ideas to a test helicopter, by means of computer simulations, shows good performances both in terms of disturbance rejection effectiveness and control effort limitation. The performance of the obtained controller is compared with the one achievable by the so called Higher Harmonic Control (HHC) approach, well known within the helicopter community.

A Note on Bronstein's and Tarter's Definition of Strict Implication

1934 ◽  
Vol 43 (5) ◽  
pp. 518 ◽  
Author(s):  
J. C. Chenoweth McKinsey
Keyword(s):  
Strict Implication ◽  
Definition Of

scholarly journals On the Inversion of the Gauss Transformation

1957 ◽  
Vol 9 ◽  
pp. 459-464 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Differential Operator ◽  
Recent Work ◽  
Suitable Definition ◽  
Inversion Theory ◽  
The Subject ◽  
Definition Of ◽  
Image Position ◽  
Gauss Transformation ◽  
Operational Methods

The inversion theory of the Gauss transformation has been the subject of recent work by several authors. If the transformation is defined by1.1,then operational methods indicate that,under a suitable definition of the differential operator.

The Algebra of Dimension-Linking Operators

1965 ◽  
Vol 8 (2) ◽  
pp. 203-222 ◽  
Author(s):  
R. H. Bruck
Keyword(s):  
Group Theory ◽  
Special Reference ◽  
Mathematical Society ◽  
Summer Institute ◽  
Burnside Problem ◽  
The Third ◽  
Australian Mathematical Society ◽  
Suitable Definition ◽  
Definition Of ◽  
Restricted Burnside Problem

In the course of preparing a book on group theory [1] with special reference to the Restricted Burnside Problem and allied problems I stumbled upon the concept of a dimension-linking operator. Later, when I lectured to the Third Summer Institute of the Australian Mathematical Society [2], G. E. Wall raised the question whether the dimension-linking operators could be made into a ring by introduction of a suitable definition of multiplication. The answer was easily found to be affirmative; the result wasthat the theory of dimen sion-linking operators became exceedingly simple.

scholarly journals On Near-Rings of Quotients

1979 ◽  
Vol 22 (2) ◽  
pp. 77-86 ◽  
Author(s):  
A. Oswald
Keyword(s):  
Finite Rank ◽  
Minimum Condition ◽  
Suitable Definition ◽  
Definition Of ◽  
Rings Of Quotients

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank

Extensions of the Lewis system S5

1951 ◽  
Vol 16 (2) ◽  
pp. 112-120 ◽  
Author(s):  
Schiller Joe Scroggs
Keyword(s):  
Broad Class ◽  
Characteristic Matrix ◽  
Normal Extension ◽  
Strict Implication ◽  
Finite Characteristic ◽  
Material Implication ◽  
Sentential Calculus ◽  
The Third ◽  
Simple Class ◽  
Definition Of

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.

scholarly journals The Köthe Dual of an Abstract Banach Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Jiménez Fernández ◽  
M. A. Juan ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Broad Class ◽  
Banach Lattices ◽  
Vector Measures ◽  
Integrable Functions ◽  
Suitable Definition ◽  
Integral Structures ◽  
Definition Of ◽  
Köthe Dual

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.

The possibility of comparing line standards of length photographically

1953 ◽  
Vol 219 (1139) ◽  
pp. 500-515 ◽  
Keyword(s):  
Standard Deviation ◽  
Sufficient Accuracy ◽  
Visual Methods ◽  
Line Position ◽  
Single Measurement ◽  
Photographic Images ◽  
Suitable Definition ◽  
Definition Of ◽  
Better Than

Line standards of length, such as metres or yards, can be compared visually using micrometer microscopes to about one in ten millions in the most precise work. It seems possible that in a photographic comparison appreciably higher precision could be attained with less labour. Photographs of the lines on some line standards have been examined with a densitometer to determine the accuracy with which the distance between two photographic images of such lines could be measured. With suitable definition of line position a single measurement of this distance should have a standard deviation corresponding to less than 0.05 μ . Provided the temperature of the bars is known with sufficient accuracy it should be possible to compare two line standards to much better than one in ten millions in less than half the time taken by present visual methods. A machine for measuring the photographs is suggested. The characteristics of photographs of some lines are given in an appendix.

scholarly journals Prefrontal Cortex Definitions and Their Use in Distinguishing Pornography Addicted Juveniles

2021 ◽  
Author(s):  
Pukovisa Prawiroharjo ◽  
Rizki Edmi Edison ◽  
Hainah Ellydar ◽  
Peter Pratama ◽  
Sitti Evangeline Imelda Suaidy ◽  
...  
Keyword(s):  
Prefrontal Cortex ◽  
Inferior Frontal Gyrus ◽  
Functional Brain Imaging ◽  
Anterior Cingulate ◽  
Brain Segmentation ◽  
Volumetric Mri ◽  
Suitable Definition ◽  
Pornography Addiction ◽  
Mean Differences ◽  
Definition Of

Background and aims: Increasing popularity of Internet has exposed our children pornography addiction. As in other types of addiction, it affects a brain region known as prefrontal cortex (PFC), which is important in executive functions and inhibitory control. However, this region was loosely defined, and there was no consensus for that definition. We aimed to use volumetric MRI in finding the defining region of PFC which would be suitable in distinguishing pornography addicted juveniles. Methods: We enrolled 30 juveniles (12-16 y.o.) consisting of 15 pornography addiction and 15 non-addiction subjects. We proposed several models of PFC definition from mix-and-matched subregions, consisting of orbitofrontal (OFC), inferior frontal gyrus (IFG; pars orbitalis, opercularis, and triangularis), dorsolateral PFC (DLPFC), and anterior cingulate (ACC). Suitable PFC definition was defined as models which volume statistically different between both groups. Brain volumetric was measured using 3D-T1 3T MRI images and analyzed using FreeSurfer for automatic cortical reconstruction and brain segmentation (recon-all command). Results: We found significant differences between groups in 6 models, which mainly included OFC, ACC, and DLPFC, with models devoid of DLPFC had lowest mean differences. Conclusion: The most suitable definition of PFC for pornography addiction study should consist of OFC, ACC, and especially DLPFC. Inferior frontal gyrus pars orbitalis was not necessary for this purpose, but may increase effect size if it is included. Keywords: Addiction, Juvenile, Pornography, Functional Brain Imaging, Defining Area.

ATTRACTORS FOR A CAGINALP MODEL WITH A LOGARITHMIC POTENTIAL AND COUPLED DYNAMIC BOUNDARY CONDITIONS

2013 ◽  
Vol 11 (06) ◽  
pp. 1350024 ◽  
Author(s):  
MONICA CONTI ◽  
STEFANIA GATTI ◽  
ALAIN MIRANVILLE
Keyword(s):  
Boundary Conditions ◽  
Phase Field Model ◽  
Logarithmic Potential ◽  
Dynamic Boundary Conditions ◽  
Exponential Attractors ◽  
Dynamic Boundary ◽  
Suitable Definition ◽  
Finite Fractal Dimension ◽  
Definition Of ◽  
Longtime Behavior

We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both the order parameter and the temperature. Due to the possible lack of distributional solutions, we deal with a suitable definition of solutions based on variational inequalities, for which we prove well-posedness and the existence of global and exponential attractors with finite fractal dimension.

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