scholarly journals The modal logic of affine planes is not finitely axiomatisable

2008 ◽  
Vol 73 (3) ◽  
pp. 940-952
Author(s):  
Ian Hodkinson ◽  
Altaf Hussain
Keyword(s):  
Modal Logic ◽  
Affine Plane ◽  
Kripke Frame ◽  
Affine Planes ◽  
Modal Language ◽  
Line Point ◽  
Point Line

AbstractWe consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes

1964 ◽  
Vol 16 ◽  
pp. 443-472 ◽  
Author(s):  
David A. Foulser
Keyword(s):  
Positive Integer ◽  
Collineation Group ◽  
Affine Plane ◽  
Affine Planes ◽  
Collineation Groups ◽  
Point Line ◽  
Explicit Determination ◽  
Desarguesian Affine Plane

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).

scholarly journals BRUCK NETS AND PARTIAL SHERK PLANES

2017 ◽  
Vol 104 (1) ◽  
pp. 1-12
Author(s):  
JOHN BAMBERG ◽  
JOANNA B. FAWCETT ◽  
JESSE LANSDOWN
Keyword(s):  
Affine Plane ◽  
Finite Geometries ◽  
Affine Planes ◽  
Incidence Structures ◽  
Finite Affine ◽  
Odd Order ◽  
Finite Incidence ◽  
Desarguesian Affine Plane

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

scholarly journals The modal logic of inequality

1992 ◽  
Vol 57 (2) ◽  
pp. 566-584 ◽  
Author(s):  
Maarten de Rijke
Keyword(s):  
Modal Logic ◽  
Expressive Power ◽  
Modal Operator ◽  
Modal Language

AbstractWe consider some modal languages with a modal operator D whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.

Provability logic

2018 ◽  
Author(s):  
Albert Visser
Keyword(s):  
Modal Logic ◽  
Formal System ◽  
Formal Theory ◽  
Incompleteness Theorem ◽  
Provability Logic ◽  
Modal System ◽  
Modal Language ◽  
Modal Operators ◽  
Syntactic Approach

Central to Gödel’s second incompleteness theorem is his discovery that, in a sense, a formal system can talk about itself. Provability logic is a branch of modal logic specifically directed at exploring this phenomenon. Consider a sufficiently rich formal theory T. By Gödel’s methods we can construct a predicate in the language of T representing the predicate ‘is formally provable in T’. It turns out that T is able to prove statements of the form - (1) If A is provable in T, then it is provable in T that A is provable in T. In modal logic, predicates such as ‘it is unavoidable that’ or ‘I know that’ are considered as modal operators, that is, as non-truth-functional propositional connectives. In provability logic, ‘is provable in T’ is similarly treated. We write □A for ‘A is provable in T’. This enables us to rephrase (1) as follows: - (1′) □A →□□A. This is a well-known modal principle amenable to study by the methods of modal logic. Provability logic produces manageable systems of modal logic precisely describing all modal principles for □A that T itself can prove. The language of the modal system will be different from the language of the system T under study. Thus the provability logic of T (that is, the insights T has about its own provability predicate as far as visible in the modal language) is decidable and can be studied by finitistic methods. T, in contrast, is highly undecidable. The advantages of provability logic are: (1) it yields a very perspicuous representation of certain arguments in a formal theory T about provability in T; (2) it gives us a great deal of control of the principles for provability in so far as these can be formulated in the modal language at all; (3) it gives us a direct way to compare notions such as knowledge with the notion of formal provability; and (4) it is a fully worked-out syntactic approach to necessity in the sense of Quine.

Investigation and Research of the Ancient City Wall of Xunxian Countyarea Characteristic Update Strategy

2013 ◽  
Vol 438-439 ◽  
pp. 1912-1916
Author(s):  
Chang Zheng Gao
Keyword(s):  
Field Investigation ◽  
Construction Technology ◽  
City Wall ◽  
Residential Architecture ◽  
Original Culture ◽  
Ancient City ◽  
The Status ◽  
Line Point ◽  
Point Line ◽  
Update Strategy

Through the use of investigation method of line-point, and the field investigation and study of the ancient city wall of Xunxian County historical area of traditional residential architecture, key historical relics and characteristics of the status quo, this paper summarizes the courtyard layout, material, construction technology and geographical environment of Xunxian County traditional residential area to adapt to the historical characteristics and cultural value. The ancient city wall of historical district is discussed to explore a retains for its original culture, and meet the needs of the development of city while protecting the history and the ancient city of the whole area of the long history of cultural by "Point-Line-Face" update strategy.

On axiomatising products of Kripke frames

2000 ◽  
Vol 65 (2) ◽  
pp. 923-945 ◽  
Author(s):  
Ágnes Kurucz
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Modal Language ◽  
First Order ◽  
Kripke Frames ◽  
Other Hand ◽  
The Many ◽  
By Products

AbstractIt is shown that the many-dimensional modal logic Kn, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any n > 2. On the other hand, Kn is determined by a class of frames satisfying a single first-order sentence.

A ‘point–line–point’ hybrid electrocatalyst for bi-functional catalysis of oxygen evolution and reduction reactions

2016 ◽  
Vol 4 (9) ◽  
pp. 3379-3385 ◽  
Author(s):  
Hao-Fan Wang ◽  
Cheng Tang ◽  
Xiaolin Zhu ◽  
Qiang Zhang
Keyword(s):  
Oxygen Reduction ◽  
Oxygen Evolution ◽  
Reduction Reactions ◽  
Active Point ◽  
Conductive Line ◽  
Line Point ◽  
Point Line

A hybrid electrocatalyst with ‘active point–conductive line–active point’ connections was proposed and exhibited superb bi-functional reactivity for both oxygen reduction and oxygen evolution reactions.

The Steiner system S(5,8, 24) constructed from dual affine planes

1986 ◽  
Vol 103 (1-2) ◽  
pp. 147-160
Author(s):  
G. A. Kadir ◽  
J. D. Key
Keyword(s):  
Affine Plane ◽  
Steiner System ◽  
Maximal Subgroups ◽  
Affine Planes ◽  
Tactical Configuration ◽  
Dual Affine ◽  
Mathieu Groups

SynopsisWe construct firstly a single tactical configuration which has the structure of the dual of the affine plane of order 4, and show how to obtain a further set of 3 such dual planes which, together with , satisfy a certain set of intersection properties. This set of 4 dual planes is used to extend the 20 points of to the Steiner system = S(5, 8, 24). The construction leads to the production of involutions of the type which fix the points of an octad. It is shown that 3 involutions each of this type suffice to generate M24, each of the simple Mathieu groups inside M24, the Todd group, and all the intransitive maximal subgroups of M24.

scholarly journals Undecidability of First-Order Intuitionistic and Modal Logics with Two variables

2005 ◽  
Vol 11 (3) ◽  
pp. 428-438 ◽  
Author(s):  
Roman Kontchakov ◽  
Agi Kurucz ◽  
Michael Zakharyaschev
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Modal Logics ◽  
Kripke Frame ◽  
Standard Nomenclature ◽  
Quantified Modal Logic ◽  
First Order

AbstractWe prove that the two-variable fragment of first-order intuitionistic logic is undecidable, even without constants and equality. We also show that the two-variable fragment of a quantified modal logic L with expanding first-order domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those in the standard nomenclature above, even the monadic two-variable fragments turn out to be undecidable.

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