Axiom schemes for m-valued functional calculi of first order. Part II

1951 ◽  
Vol 16 (1) ◽  
pp. 22-34 ◽  
Author(s):  
J. B. Rosser ◽  
A. R. Turquette
Keyword(s):  
Analytic Formula ◽  
Truth Values ◽  
First Order ◽  
Order Part ◽  
Elegant Proof

In part I of the present paper axiom schemes and rules of inference were defined for m-valued functional calculi of first order with s(m > s > 1) designated truth-values. A proof of plausibility was given, and it was shown that it is not difficult to extend to m-valued functional calculi of first order certain concepts that are closely analogous to the ordinary 2-valued notions of “consistency with respect to an operator” and “absolute consistency.” The purpose of the present paper is to show that the concept of “deductive completeness” may be extended to m-valued functional calculi of first order. For this purpose we define “analytic formula” for the m-valued case and show that if a formula is analytic, then it is provable in our formalization of m-valued functional calculi of first order.In proving deductive completeness for the m-valued case, it is possible to use a method which is analogous to that used by Gödel in establishing the completeness of 2-valued functional calculi of first order. However, in the present paper we will indicate only very briefly how the Gödel procedure may be extended to the m-valued case. Our chief concern will be the problem of extending to our formalization of m-valued functional calculi the more elegant proof of deductive completeness for the 2-valued case which has recently been developed by Leon Henkin.

Axiom schemes for m-valued functional calculi of first order. Part I. Definition of axiom schemes and proof of plausibility

1948 ◽  
Vol 13 (4) ◽  
pp. 177-192 ◽  
Author(s):  
J. B. Rosser ◽  
A. R. Turquette
Keyword(s):  
Truth Values ◽  
First Order ◽  
Order Part ◽  
Definition Of

A calculus, or its formalization, is m-valued when its truth-functions are allowed to take truth-values ranging from 1 through m. It is customary to divide the m truth-values that are possible into those that are “designated” and those that are “undesignated.” Furthermore, it is usually desired of a formalization that provable formulas and only provable formulas take designated truth-values exclusively. For our present purposes, we shall suppose that the truth-values 1, …, 8 are designated and the truth-values 8+1, …, m are undesignated. When calculi differ in respect to the number of their designated truth-values, we shall consider them different calculi, even if they are otherwise similar.

scholarly journals First order normalization in the generalized photo gravitational non-planar restricted three body problems

2015 ◽  
Vol 3 (2) ◽  
pp. 46
Author(s):  
Nirbhay Kumar Sinha
Keyword(s):  
Oblate Spheroid ◽  
Second Order ◽  
Elliptic Motion ◽  
First Order ◽  
Short Period ◽  
Order Part ◽  
Three Body

<p>In this paper, we normalised the second-order part of the Hamiltonian of the problem. The problem is generalised in the sense that fewer massive primary is supposed to be an oblate spheroid. By photogravitational we mean that both primaries are radiating. With the help of Mathematica, H<sub>2</sub> is normalised to H<sub>2</sub> = a<sub>1</sub>b<sub>1</sub>w<sub>1</sub> + a<sub>2</sub>b<sub>2</sub>w<sub>2</sub>. The resulting motion is composed of elliptic motion with a short period (2p/w<sub>1</sub>), completed by an oscillation along the z-axis with a short period (2p/w<sub>2</sub>).</p>

scholarly journals Numerical description of mode coupled waves in inhomogeneous magnetized plasmas

2019 ◽  
Vol 203 ◽  
pp. 01011
Author(s):  
Kota Yanagihara ◽  
Shin Kubo ◽  
Toru Tsujimura
Keyword(s):  
Ray Tracing ◽  
Electron Cyclotron Resonance ◽  
Numerical Approach ◽  
Geometrical Optics ◽  
Order Theory ◽  
Complex Amplitude ◽  
First Order ◽  
First Order Theory ◽  
Full Wave Analysis ◽  
Order Part

Geometrical optics (GO) ray tracing has been widely used for a description of electron cyclotron resonance waves in inhomogeneous magnetized fusion plasmas. However, this reduced approach is not correct in sufficient low density plasmas with a sheared magnetic field, where mode coupling between two electromagnetic-like cold plasma modes can occur. Here, we extend a ray tracing method based on the first-order theory of extended geometrical optics (XGO), which captures mode coupled complex amplitude between O and X mode along the ray trajectory. In our approach, reference ray is calculated with ray equation to satisfy the lowest-order part of XGO theory and an evolution of complex amplitude profile along the reference ray is calculated with partial differential equation derived from first-order terms. Calculation results performed by extended ray tracing are in good agreement with 1D full wave analysis. By introducing second-order terms into our numerical approach, diffraction will be treated.

A NOTE ON THE ASYMPTOTIC PROBABILITIES OF EXISTENTIAL SECOND-ORDER MINIMAL GÖDEL SENTENCES WITH EQUALITY

1995 ◽  
Vol 06 (04) ◽  
pp. 339-351
Author(s):  
WIESŁAW SZWAST
Keyword(s):  
Dense Subset ◽  
Unit Interval ◽  
Second Order ◽  
Existential Quantifier ◽  
First Order ◽  
Order Part ◽  
Universal Quantifiers

The minimal Gödel class is the class of first-order prenex sentences whose quantifier prefix consists of two universal quantifiers followed by just one existential quantifier. We prove that asymptotic probabilities of existential second-order sentences, whose first-order part is in the minimal Gödel class, form a dense subset of the unit interval.

scholarly journals ON EXISTENTIAL DECLARATIONS OF INDEPENDENCE IN IF LOGIC

2013 ◽  
Vol 6 (2) ◽  
pp. 254-280 ◽  
Author(s):  
FAUSTO BARBERO
Keyword(s):  
Expressive Power ◽  
Inference Rules ◽  
Declaration Of Independence ◽  
Truth Values ◽  
First Order ◽  
If Logic ◽  
Knowledge Memory ◽  
Special Classes

AbstractWe analyze the behaviour of declarations of independence between existential quantifiers in quantifier prefixes of Independence-Friendly (IF) sentences; we give a syntactical criterion to decide whether a sentence beginning with such prefix exists, such that its truth values may be affected by removal of the declaration of independence. We extend the result also to equilibrium semantics values for undetermined IF sentences.The main theorem defines a schema of sound and recursive inference rules; we show more explicitly what happens for some simple special classes of sentences.In the last section, we extend the main result beyond the scope of prenex sentences, in order to give a proof of the fact that the fragment of IF sentences with knowledge memory has only first-order expressive power.

Axiom Schemes for m-Valued Functional Calculi of First Order. Part II. Deductive Completeness.

1951 ◽  
Vol 16 (4) ◽  
pp. 269
Author(s):  
Burton Spencer Dreben ◽  
J. B. Rosser ◽  
A. R. Turquette
Keyword(s):  
First Order ◽  
Order Part

3D Interlock Composite Preforming Simulation

2012 ◽  
Vol 504-506 ◽  
pp. 261-266 ◽  
Author(s):  
Jean Guillaume Orliac ◽  
Adrien Charmetant ◽  
Fabrice Morestin ◽  
Philippe Boisse ◽  
Stephane Otin
Keyword(s):  
Virtual Work ◽  
Experimental Testing ◽  
Second Order ◽  
Beam Model ◽  
Forming Simulation ◽  
First Order ◽  
Material Stiffness ◽  
Material Formulation ◽  
Order Part ◽  
Continuous Material

In order to simulate 3D interlock composite reinforcement behavior in forming processes like Resin Transfer Molding (RTM), it is necessary to predict yarns positions in the fabric during the preforming stage of the process. The present paper deals about thick 3D interlock fabric forming simulation using a specific hexahedral semi-discrete finite elements simulation tool : Plast4. Using the virtual work principle, we distinguish the virtual internal work due to tensions in yarns from other internal virtual works. The part of material stiffness relative to yarns tension is described as "first order stiffness" by a 3D discrete beam model. The rest of the rigidities - like transverse compression, shear strains or friction between yarns - are depicted by a continuous quad-based discretization designated in our work as "second order stiffness". A combination of this "first order" discrete model and a continuous orthotropic hyperelastic "second order" material formulation will enables us to simulate interlock preforming process. Jointly to the simulation work, we also had to specify and perform experimental testing identification of material's parameters. Thoses parameters concern both parts of the model. A bilinear tension approach for the yarns discrete modelization and an orthotropic continuous material for the "second order" part.

First-Order Extensional Inference with Distributional Truth Values

2008 ◽  
pp. 1-7
Author(s):  
Ben Goertzel ◽  
Matthew Iklé ◽  
Izabela Freire Goertzel ◽  
Ari Heljakka
Keyword(s):  
Truth Values ◽  
First Order

First-Order Extensional Inference with Indefinite Truth Values

2008 ◽  
pp. 1-10
Author(s):  
Ben Goertzel ◽  
Matthew Iklé ◽  
Izabela Freire Goertzel ◽  
Ari Heljakka
Keyword(s):  
Truth Values ◽  
First Order
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