Satisfaction for n-th order languages defined in n-th order languages

1965 ◽  
Vol 30 (1) ◽  
pp. 13-25 ◽  
Author(s):  
William Craig
Keyword(s):  
Interpolation Theorem ◽  
The Other ◽  
Present Formulation ◽  
Finite Axiomatizability ◽  
First Order ◽  
Order Language

In this paper we show that for n-th order languages L′ with p nonlogical constants, 2 ≦ n < ω, 0 ≦ p < ω, a notion of satisfaction can be defined in an n-th order language containing one additional nonlogical constant, say S. By the usual methods we also show that this notion cannot be defined in L′. Hence, in its present formulation, Beth's Theorem in [1] for first order languages has no analogue for L′.Our defining expression is such that, given any values of the other nonlogical constants and any appropriate m-tuple, it allows us to determine whether or not the m-tuple belongs to the value S of S without considering the totality of objects which are of the same type as S. Whether every definition in an n-th order language is equivalent to one thus “predicative”, and hence whether there is a formulation of Beth's Theorem which generalizes to higher orders, we do not know.The falsehood for L′ of the analogue of Beth's Theorem implies the falsehood of an analogue of an interpolation theorem for first order languages. The above definability of satisfaction for L′ implies a result on finite axiomatizability in slightly richer languages. Details are given in § 4.

On Axiomatizability of Non-Commutative Lp-Spaces

2007 ◽  
Vol 50 (4) ◽  
pp. 519-534
Author(s):  
C. Ward Henson ◽  
Yves Raynaud ◽  
Andrew Rizzo
Keyword(s):  
Hilbert Spaces ◽  
Normed Space ◽  
The Other ◽  
Space Structures ◽  
Operator Spaces ◽  
Infinite Dimensions ◽  
Schatten Classes ◽  
Lp Spaces ◽  
First Order ◽  
Order Language

AbstractIt is shown that Schatten p-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative Lp-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative Lp spaces. As a consequence, the class of non-commutative Lp-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative Lp-spaces. For p = 1 this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

A direct proof of the Feferman-Vaught theorem and other preservation theorems in products

1991 ◽  
Vol 56 (2) ◽  
pp. 632-636 ◽  
Author(s):  
Yiannis Vourtsanis
Keyword(s):  
Direct Proof ◽  
The Other ◽  
Direct Products ◽  
First Order ◽  
New Methods ◽  
Preservation Theorems ◽  
Order Language ◽  
Reduced Products ◽  
The One ◽  
By Products

Here we give short and direct proofs of the Feferman-Vaught theorem and other preservation theorems in products of structures. In 1952, Mostowski [5] first showed the preservation of ≡ωω by direct powers of structures. Subsequently, in 1959, Feferman and Vaught [2] proved the preservation of ≡ωω by arbitrary direct products and also by reduced products with respect to cofinite filters. In 1962, Frayne, Morel and Scott [3] noticed that the results extend to arbitrary reduced products. In 1970, Barwise and Eklof [1] showed the preservation of ≡∞λ by products and in 1971 Malitz [4] showed the preservation of ≡κλ with κ strongly inaccessible (or ∞) by products. Below, we give short proofs of the above results. The ideas used here have initiated, in [7], [8], [9], [10], [11], the introduction of several new methods in the theory of products, which on the one hand give new, direct proofs of the known results in the area, including generalizations or strengthenings of some of those, and, on the other hand, give several new results as well in the theory of products and related areas.Below, L denotes a (first order) language, and by a structure we mean an L-structure. 0 and 1 denote the logically valid and false sentence, respectively. We may write ā ∈ A for ā ∈ An for some n. Also, the values 1 and 2 of a parameter l in the definitions below express a duality corresponding to disjunctive and conjunctive forms.

Finite Axiomatizability using additional predicates

1958 ◽  
Vol 23 (3) ◽  
pp. 289-308 ◽  
Author(s):  
W. Craig ◽  
R. L. Vaught
Keyword(s):  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Logical Constant ◽  
Finite Axiomatizability ◽  
Logical Constants ◽  
First Order

By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ.For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.

A CAYLEY THEOREM FOR THE MULTIPLICATIVE SEMIGROUP OF A FIELD

2012 ◽  
Vol 11 (02) ◽  
pp. 1250042 ◽  
Author(s):  
E. NAZARI ◽  
Yu. M. MOVSISYAN
Keyword(s):  
Type Theorem ◽  
The Other ◽  
Binary Representation ◽  
Multiplicative Semigroup ◽  
First Order ◽  
Order Language ◽  
Multiplicative Semigroups

Since there exist two commutative elementarily equivalent semigroups of which one is the multiplicative semigroup of a field and the other is not a multiplicative semigroup of any field, it is impossible to characterize multiplicative semigroups of fields by formulas of the first order language (logic). In this work we characterize the multiplicative semigroup of a field by its binary representation (Cayley type theorem).

Interpolation theorems for Lk,k2+

1978 ◽  
Vol 43 (3) ◽  
pp. 535-549 ◽  
Author(s):  
Ruggero Ferro
Keyword(s):  
Natural Deduction ◽  
Interpolation Theorem ◽  
Positive Answer ◽  
Strong Limit ◽  
First Order ◽  
Limit Cardinal ◽  
Saturated Models ◽  
Order Language ◽  
Craig’S Interpolation Theorem

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].

On the Craig-Lyndon interpolation theorem

1968 ◽  
Vol 33 (2) ◽  
pp. 271-274
Author(s):  
Arnold Oberschelp
Keyword(s):  
Interpolation Theorem ◽  
Relation Symbol ◽  
First Order ◽  
Order Language ◽  
Identity Symbol

In his paper [3] Henkin proved for a first order language with identity symbol but without operation symbols the following version of the Craig-Lyndon interpolation theorem:Theorem 1. If Γ╞Δ then there is a formula θ such that Γ ├Δand(i) any relation symbol with a positive (negative) occurrence in θ has a positive (negative) occurrence in some formula of Γ.

scholarly journals Thematic Maps for the Variation of Bearing Capacity of Soil Using SPTs and MATLAB

Geosciences ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 329
Author(s):  
Mahdi O. Karkush ◽  
Mahmood D. Ahmed ◽  
Ammar Abdul-Hassan Sheikha ◽  
Ayad Al-Rumaithi
Keyword(s):  
Bearing Capacity ◽  
Mean Squared Error ◽  
Ground Level ◽  
The Other ◽  
Thematic Maps ◽  
First Order ◽  
Squared Error ◽  
Order Polynomial ◽  
Interpolation Polynomials ◽  
Penetration Tests

The current study involves placing 135 boreholes drilled to a depth of 10 m below the existing ground level. Three standard penetration tests (SPT) are performed at depths of 1.5, 6, and 9.5 m for each borehole. To produce thematic maps with coordinates and depths for the bearing capacity variation of the soil, a numerical analysis was conducted using MATLAB software. Despite several-order interpolation polynomials being used to estimate the bearing capacity of soil, the first-order polynomial was the best among the other trials due to its simplicity and fast calculations. Additionally, the root mean squared error (RMSE) was almost the same for the all of the tried models. The results of the study can be summarized by the production of thematic maps showing the variation of the bearing capacity of the soil over the whole area of Al-Basrah city correlated with several depths. The bearing capacity of soil obtained from the suggested first-order polynomial matches well with those calculated from the results of SPTs with a deviation of ±30% at a 95% confidence interval.

Neural computation of motion in the fly visual system: quadratic nonlinearity of responses induced by picrotoxin in the HS and CH cells

1995 ◽  
Vol 74 (6) ◽  
pp. 2665-2684 ◽  
Author(s):  
Y. Kondoh ◽  
Y. Hasegawa ◽  
J. Okuma ◽  
F. Takahashi
Keyword(s):  
Mean Square Error ◽  
Order Term ◽  
Mirror Image ◽  
Second Order ◽  
The Other ◽  
Linear Component ◽  
Mean Square ◽  
Nonlinear Component ◽  
First Order ◽  
Order Kernel

1. A computational model accounting for motion detection in the fly was examined by comparing responses in motion-sensitive horizontal system (HS) and centrifugal horizontal (CH) cells in the fly's lobula plate with a computer simulation implemented on a motion detector of the correlation type, the Reichardt detector. First-order (linear) and second-order (quadratic nonlinear) Wiener kernels from intracellularly recorded responses to moving patterns were computed by cross correlating with the time-dependent position of the stimulus, and were used to characterize response to motion in those cells. 2. When the fly was stimulated with moving vertical stripes with a spatial wavelength of 5-40 degrees, the HS and CH cells showed basically a biphasic first-order kernel, having an initial depolarization that was followed by hyperpolarization. The linear model matched well with the actual response, with a mean square error of 27% at best, indicating that the linear component comprises a major part of responses in these cells. The second-order nonlinearity was insignificant. When stimulated at a spatial wavelength of 2.5 degrees, the first-order kernel showed a significant decrease in amplitude, and was initially hyperpolarized; the second-order kernel was, on the other hand, well defined, having two hyperpolarizing valleys on the diagonal with two off-diagonal peaks. 3. The blockage of inhibitory interactions in the visual system by application of 10-4 M picrotoxin, however, evoked a nonlinear response that could be decomposed into the sum of the first-order (linear) and second-order (quadratic nonlinear) terms with a mean square error of 30-50%. The first-order term, comprising 10-20% of the picrotoxin-evoked response, is characterized by a differentiating first-order kernel. It thus codes the velocity of motion. The second-order term, comprising 30-40% of the response, is defined by a second-order kernel with two depolarizing peaks on the diagonal and two off-diagonal hyperpolarizing valleys, suggesting that the nonlinear component represents the power of motion. 4. Responses in the Reichardt detector, consisting of two mirror-image subunits with spatiotemporal low-pass filters followed by a multiplication stage, were computer simulated and then analyzed by the Wiener kernel method. The simulated responses were linearly related to the pattern velocity (with a mean square error of 13% for the linear model) and matched well with the observed responses in the HS and CH cells. After the multiplication stage, the linear component comprised 15-25% and the quadratic nonlinear component comprised 60-70% of the simulated response, which was similar to the picrotoxin-induced response in the HS cells. The quadratic nonlinear components were balanced between the right and left sides, and could be eliminated completely by their contralateral counterpart via a subtraction process. On the other hand, the linear component on one side was the mirror image of that on the other side, as expected from the kernel configurations. 5. These results suggest that responses to motion in the HS and CH cells depend on the multiplication process in which both the velocity and power components of motion are computed, and that a putative subtraction process selectively eliminates the nonlinear components but amplifies the linear component. The nonlinear component is directionally insensitive because of its quadratic non-linearity. Therefore the subtraction process allows the subsequent cells integrating motion (such as the HS cells) to tune the direction of motion more sharply.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

scholarly journals Ganglioside incorporation and release by the isolated perfused rat liver

1991 ◽  
Vol 274 (2) ◽  
pp. 581-585 ◽  
Author(s):  
S C Kivatinitz ◽  
A Miglio ◽  
R Ghidoni
Keyword(s):  
Rat Liver ◽  
The Other ◽  
Regulatory Role ◽  
Isolated Perfused Rat Liver ◽  
Order Kinetic ◽  
Perfused Rat Liver ◽  
Ganglioside Gm1 ◽  
First Order ◽  
Pseudo First Order

The fate of exogenous ganglioside GM1 labelled in the sphingosine moiety, [Sph-3H]GM1, administered as a pulse, in the isolated perfused rat liver was investigated. When a non-recirculating protocol was employed, the amount of radioactivity in the liver and perfusates was found to be dependent on the presence of BSA in the perfusion liquid and on the time elapsed after the administration of the ganglioside. When BSA was added to the perfusion liquid, less radioactivity was found in the liver and more in the perfusate at each time tested, for up to 1 h. The recovery of radioactivity in the perfusates followed a complex course which can be described by three pseudo-first-order kinetic constants. The constants, in order of decreasing velocity, are interpreted as: (a) the dilution of the labelled GM1 by the constant influx of perfusion liquid; (b) the washing off of GM1 loosely bound to the surface of liver cells; (c) the release of gangliosides from the liver. Process (b) was found to be faster in the presence of BSA, probably owing to the ability of BSA to bind gangliosides. The [Sph-3H]GM1 in the liver underwent metabolism, leading to the appearance of products of anabolic (GD1a, GD1b) and catabolic (GM2, GM3) origin; GD1a appeared before GM2 and GM3 but, at times longer than 10 min, GM2 and GM3 showed more radioactivity than GD1a. At a given time the distribution of the radioactivity in the perfusates was quite different from that of the liver. In fact, after 60 min GD1a was the only metabolite present in any amount, the other being GM3, the quantity of which was small. This indicates that the liver is able to release newly synthesized gangliosides quite specifically. When a recirculating protocol was used, there were more catabolites and less GD1a than with the non-recirculating protocol. A possible regulatory role of ganglioside re-internalization on their own metabolism in the liver is postulated.

Sign in / Sign up

Export Citation Format

Share Document