Compactness of a supervaluational language

1983 ◽  
Vol 48 (2) ◽  
pp. 384-386 ◽  
Author(s):  
E. Bencivenga
Keyword(s):  
Open Problem ◽  
Present Note ◽  
Complete Solution ◽  
The Other ◽  
Unary Function ◽  
Power Set ◽  
Binary Predicate

Compactness of supervaluational semantics has long been an open problem. Recently, Peter Woodruff [2] showed that the full quantificational language is not compact. In the present note, I will show that the quantifier-free fragment of the language is compact. Since this result can be easily extended to the monadic quantificational language, and since Woodruff's result only requires the presence of one binary predicate, the two results combined give a complete solution of the problem.My language L contains infinitely many individual constants, infinitely many n-ary predicates (for every n > 0), the usual connectives, the symbols E! and =, and parentheses. (Atomic) sentences are defined as usual, and Ab/a is the result of uniformly substituting b for a in the sentence A.An interpretation of L is an ordered pair I = 〈D, φ〉, where D is a set (possibly empty) and φ is a unary function, total from the set of n-ary predicates into the power set of Dn, and partial from the set of constants into D.A classical valuation for an interpretation I = 〈D, φ〉 is a total unary function V from the set of sentences into {T, F} such that(a) if φ is defined for all of a1, …, am, then V(Pa1 … an) = T iff 〈φ(a1), …, φ(an)〉 ∈ φ(P);(b) if φ(a) and φ(b) are both defined, then V(a = b) = T iff φ(a) = φ(b;(c) if exactly one of φ(a) and φ(b) is defined, then V(a = b) = F;(d) V(a = a) = T;(e) if V(a = b) = T and A is atomic, then V(A) = V(Ab/a);(f)V(E!a) = T iff φ(a) is defined;(g) V(~ A) = T iff V(A) = F, and similarly for the other connectives.

A Romano-British Interment, with Bucket and Sceptres, from Brough, East Yorkshire

1938 ◽  
Vol 18 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Philip Corder ◽  
I. A. Richmond
Keyword(s):  
Present Note ◽  
Fourth Century ◽  
The Other ◽  
The West ◽  
West Side ◽  
The Road ◽  
The Subject ◽  
Chief Interest ◽  
Roman Age ◽  
The Way

The Roman Ermine Street, having crossed the Humber on the way to York from Lincoln, leaves Brough Haven on its west side, and the little town of Petuaria to the east. For the first half-mile northwards from the Haven its course is not certainly known: then, followed by the modern road, it runs northwards through South Cave towards Market Weighton. In the area thus traversed by the Roman road burials of the Roman age have already been noted in sufficient quantity to suggest an extensive cemetery. The interment which is the subject of the present note was found on 10th October 1936, when men laying pipes at right angles to the modern road, in the carriage-drive of Mr. J. G. Southam, having cut through some 4 ft. of blown sand, came upon a mass of mixed Roman pottery, dating from the late first to the fourth century A.D. Bones of pig, dog, sheep, and ox were also represented. Presently, at a depth of about 5 ft., something attracted closer attention. A layer of thin limestone slabs was found, covering two human skeletons, one lying a few feet from the west margin of the modern road, the other parallel with the road and some 8 ft. from its edge. The objects described below were found with the second skeleton, and the first to be discovered was submitted by Mr. Southam to Mr. T. Sheppard, F.S.A.Scot., Director of the Hull Museums, who visited the site with his staff. All that can be recorded of the circumstances of the discovery is contained in the observations then made, under difficult conditions. ‘Slabs of hard limestone’, it was reported, ‘taken from a local quarry of millepore oolite and forming the original Roman road, were distinctly visible beneath the present roadway—one of the few points where the precise site of the old road has been located. On the side of this… a burial-place has been constructed. What it was like originally it is difficult to say, beyond that a layer of thin … slabs of limestone occurred over the skeletons. This had probably been kept in place or supported by some structure of wood, as several large iron nails, some bent at right angles, were among the bones.’ If this were all that could be said about the burials, they would hardly merit a place in these pages. The chief interest of the record would be its apparent identification of the exact course of the Roman road at a point where this had hitherto been uncertain. Three objects associated with the second skeleton are, however, of exceptional interest.

scholarly journals On the mesodermic origin and the fate of the so-called mesectoderm in petromyzon

1915 ◽  
Vol 88 (606) ◽  
pp. 457-475 ◽  
Keyword(s):  
Present Note ◽  
Scientific Journal ◽  
The Other ◽  
Great War ◽  
Long Time ◽  
Epithelial Structure ◽  
Principal Points ◽  
The Great War ◽  
Branchial Region

About 20 years ago v. Kupffer (85) described in the embryos of Petromyzon an epithelial structure extending, between the ectoderm and the somatic plate of the mesoderm, from the head to the posterior boundary of the branchial region, and described it under the name of the neurodermis; subsequently, he bestowed on it the name branchiodermis. Seventeen years later the same structure was again discovered by Koltzoff (02), who identified it with the mesectoderm which was described by Miss Piatt (94) in Necturus embryos. Subsequently, so far as Petromyzon is concerned, nothing was published until last year, when a paper by Sehalk (13) appeared, although the corresponding layer of cells was described by A. Dohrn (02) in Selachii and by Brauer (04) in Gymnophiona. For a long time the origin and fate of the layer in question engaged my attention. Last summer I was able to re-examine my sections and to confirm observations which I had previously published in a paper entitled “Die Bildungsweise und erste Differenzierung des Mesoderms beim Neunauge ( Lampetra mitsukurii , Hatta),” in which, the origin and differentiation of the so-called mesectoderm are described and illustrated by a series of microphotographs. To my regret the paper, which was ready for press when the great war broke out, could not be sent to the editor of a certain scientific journal in Belgium, who had promised to publish it in his journal. The present note is an attempt to communicate some of the principal points of that paper which relate to the mesectoderm. The other organs dealt with in the above-mentioned paper have already been described in preliminary notes or in my previous papers.

A note on disturbances in a layered medium containing a magnetic field

1964 ◽  
Vol 54 (6A) ◽  
pp. 1771-1777
Author(s):  
D. K. Sinha
Keyword(s):  
Magnetic Field ◽  
Free Surface ◽  
Layered Medium ◽  
Present Note ◽  
Infinite Medium ◽  
The Other ◽  
Time Dependent ◽  
The Earth ◽  
Propagation Of Waves ◽  
Initial Magnetic Field

abstract In recent years, Kaliski has contributed a series of papers on the interaction of elastic and magnetic fields and some of them, [1], [2], [3] are concerned with the propagation of waves in a semi-infinite medium either loaded or conditioned otherwise, at its free surface. Such problems, as Kaliski [1] has remarked, may have relevance in the practical seismic problem of detecting the mechanical explosions inside the earth. Moreover, their geophysical implications have also been examined by Knopoff [4[, Cagniard [5], Banos [6], and Rikitake [7]. The present note seeks to investigate disturbances in a medium consisting of two layers (one finite and the other infinite) of elastic medium intervened by a thin layer of vacuum. The vacuum is traversed by an initial magnetic field. The disturbances in the medium are assumed to have been produced by a time-dependent load on the free surface of the medium. The method of Laplace transform has been used to facilitate the solution of the problem.

Ven. Robert Sutton of Stafford a Note on his Family and Early Life

1953 ◽  
Vol 2 (02) ◽  
pp. 150-166
Author(s):  
D.M. Rogers
Keyword(s):  
Sixteenth Century ◽  
Early Life ◽  
Present Note ◽  
Early Years ◽  
The Other ◽  
Elizabeth I ◽  
Catholic Faith

Robert Sutton is a name that occurs quite often in sixteenth century records. It was borne by two of the English martyrs under Elizabeth I, the only two, among the three hundred and sixty martyrs at present officially listed, to bear identical names. One of these was a layman, a school-master, hanged at Clerkenwell in October 1588 for being reconciled to the Catholic faith (1). The other was a secular priest hanged, drawn and quartered at Stafford a year earlier (2). The present note concerns the priest, but since further contemporaries of these two martyrs also had the same name, others, too, will be mentioned in the course of investigating the early years of the Ven. Robert Sutton, the priest martyr of 1587.

scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

Notes on Ptolemy

1952 ◽  
Vol 14 (1) ◽  
pp. 78-86
Author(s):  
J. Ph. Vogel
Keyword(s):  
Present Note ◽  
The Other ◽  
Trade Routes ◽  
Guiding Principle ◽  
Saint Martin ◽  
The One

It has been recognized long ago that Ptolemy's topography of Cis-Gangetic India was based on trade-routes. Nearly a century ago Vivien de Saint Martin spoke of “the almost exclusive employment of itineraries of merchants and caravans indicating on each route the series of daily stations”. We may compare the lists of stations inserted in books on India of the seventeenth and eighteenth centuries. One of the earliest examples we find in Joannes de Laet's little volume De Imperio Magni Mogolis (Leiden, 1631), p. 57. The author evidently derived his lists from the itineraries of the English merchants Richard Steel and John Crowther. The French jeweller and traveller Jean-Baptiste Tavernier has included full lists of stations along the principal trade-routes in his Indian Travels (ch. iv–xii). In the first place he deals with the two routes from Surat to Agra, all-important to European traders, the one by the Tāptī valley and Mālwā, the other by way of Aḥmadābād and Rājpūtānā. If we keep in mind that Ptolemy must have used similar lists, it will go far to explain the disconcerting fact that so many among the localities in his tables are not known from indigenous sources, either literary or epigraphical, whereas famous towns have been omitted. His tables are fundamentally lists of stages, and this must be our guiding principle in unravelling the riddle of Ptolemy's topography. The present note is an attempt to demonstrate this in some detail.

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

Edward Jenner, F. R. S. and the cuckoo

1974 ◽  
Vol 28 (2) ◽  
pp. 235-240 ◽  
Keyword(s):  
Small Town ◽  
Present Note ◽  
The Other ◽  
150Th Anniversary ◽  
The Subject ◽  
The One ◽  
Medical Historian ◽  
National Biography

In January 1973 the 150th anniversary of the death of Edward Jenner, the originator of vaccination, was modestly commemorated in the small town of Berkeley, Gloucestershire, where he was born and lived for most of his life (1). Jenner was himself the subject of controversy during and after his lifetime, on the one hand lauded as a selfless saviour of mankind, and on the other denounced as a self-seeking charlatan; his discovery likewise has virtually never ceased to be a subject of debate (2). There is no dearth of books and articles about Jenner and vaccination (3) and the subject of the present note is an unconnected and very minor discovery of Jenner’s—nevertheless one which is not only intrinsically interesting to naturalists, but also pertinent to any study of the character and integrity of its author, for the discovery was still being disputed more than a century after is was made. Edward Jenner was the first to publish, in 1788, an accurate account of what happens to the unfortunate young of the parent bird in whose nest a cuckoo deposits one of her eggs. Yet to consult the Dictionary of National Biography for details of Jenner’s life is to read, in an article published in 1892, that the ‘absurdity’ of this account had been demonstrated by the naturalist Charles Waterton. A few years earlier Charles Creighton (1847-1927), the epidemiologist and medical historian, whose intense opposition to vaccination led him to denigrate Jenner in every conceivable way, had described his paper on the cuckoo as mainly ‘a tissue of inconsistencies and absurdities’ (4).

TϒΓxanΩ for TϒΓxanΩ Ωn in Attic Prose

1926 ◽  
Vol 20 (3-4) ◽  
pp. 195-200
Author(s):  
W. L. Lorimer
Keyword(s):  
Present Note ◽  
The Other ◽  
Modern View

The question whether τυγχ⋯νω can be used forτυγχ⋯νω ὥν in Attic Prose has been differently answered by different scholars. Phrynichus (p. 277 Lob., p. 342 Ruth.) held that it could not, and Porson (ad Eur. Hec. 788 [782]) followed him. The generality of modern scholars, however, have taken the other view—so, e.g., Locella, Heindorf, Lobeck, Ast, Schneider, Madvig, Stallbaum, Krüger, W. H. Thompson, Rutherford, Jebb, Adam, Kühner-Gerth. The object of the present note is to show that the ‘modern’ view, if it is to be maintained, must be based on other evidence than that hitherto given for it.

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