scholarly journals On Equivalence Relations Between Interpreted Languages, with an Application to Modal and First-Order Language

Erkenntnis ◽  
2021 ◽  
Author(s):  
Kai F. Wehmeier
Keyword(s):  
Equivalence Relations ◽  
Modal Operator ◽  
First Order ◽  
Order Language ◽  
Bona Fide

AbstractI examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of bona fide first-order languages.

An axiomatization of the logic with the rough quantifier

1991 ◽  
Vol 56 (2) ◽  
pp. 608-617 ◽  
Author(s):  
Michał Krynicki ◽  
Hans-Peter Tuschik
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Order Structure ◽  
Partition Model ◽  
Weak Model ◽  
First Order ◽  
Basic Properties ◽  
Generalized Quantifier ◽  
Order Language ◽  
The Right

We consider the language L(Q), where L is a countable first-order language and Q is an additional generalized quantifier. A weak model for L(Q) is a pair 〈, q〉 where is a first-order structure for L and q is a family of subsets of its universe. In case that q is the set of classes of some equivalence relation the weak model 〈, q〉 is called a partition model. The interpretation of Q in partition models was studied by Szczerba [3], who was inspired by Pawlak's paper [2]. The corresponding set of tautologies in L(Q) is called rough logic. In the following we will give a set of axioms of rough logic and prove its completeness. Rough logic is designed for creating partition models.The partition models are the weak models arising from equivalence relations. For the basic properties of the logic of weak models the reader is referred to Keisler's paper [1]. In a weak model 〈, q〉 the formulas of L(Q) are interpreted as usual with the additional clause for the quantifier Q: 〈, q〉 ⊨ Qx φ(x) iff there is some X ∊ q such that 〈, q〉 ⊨ φ(a) for all a ∊ X.In case X satisfies the right side of the above equivalence we say that X is contained in φ(x) or, equivalently, φ(x) contains X.

Some remarks on definable equivalence relations in O-minimal structures

1986 ◽  
Vol 51 (3) ◽  
pp. 709-714 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Equivalence Relation ◽  
Linear Order ◽  
Finite Union ◽  
Equivalence Relations ◽  
Complete Theory ◽  
First Order ◽  
Definable Set ◽  
Order Language ◽  
Minimal Structure ◽  
Definable Sets

Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n-tuples in M, to study definable (in M) equivalence relations on Mn. In particular, we show that if E is an A-definable equivalence relation on Mn (A ⊂ M) then E has only finitely many classes with nonempty interior in Mn, each such class being moreover also A-definable. As a consequence, we are able to give some conditions under which an O-minimal theory T eliminates imaginaries (in the sense of Poizat [P]).If L is a first order language and M an L-structure, then by a definable set in M, we mean something of the form X ⊂ Mn, n ≥ 1, where X = {(a1…,an) ∈ Mn: M ⊨ϕ(ā)} for some formula ∈ L(M). (Here L(M) means L together with names for the elements of M.) If the parameters from come from a subset A of M, we say that X is A-definable.M is said to be O-minimal if M = (M, <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals (a, b) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th(M) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M.

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 ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

scholarly journals Word order in Latin locative constructions: a corpus study in Caesar’s De bello gallico

2011 ◽  
Vol 64 (2) ◽  
Author(s):  
Stavros Skopeteas
Keyword(s):  
Word Order ◽  
Information Structure ◽  
Motion Verbs ◽  
Structural Domains ◽  
Corpus Study ◽  
First Order ◽  
Order Language ◽  
Free Word

AbstractClassical Latin is a free word order language, i.e., the order of the constituents is determined by information structure rather than by syntactic rules. This article presents a corpus study on the word order of locative constructions and shows that the choice between a Theme-first and a Locative-first order is influenced by the discourse status of the referents. Furthermore, the corpus findings reveal a striking impact of the syntactic construction: complements of motion verbs do not have the same ordering preferences with complements of static verbs and adjuncts. This finding supports the view that the influence of discourse status on word order is indirect, i.e., it is mediated by information structural domains.

Small substructures and decidability issues for first-order logic with two variables

2012 ◽  
Vol 77 (3) ◽  
pp. 729-765 ◽  
Author(s):  
Emanuel Kieroński ◽  
Martin Otto
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Equivalence Relations ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
First Order ◽  
Surrounding Structure ◽  
Small Model ◽  
Exponential Size

AbstractWe study first-order logic with two variables FO2 and establish a small substructure property. Similar to the small model property for FO2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO2 has the finite model property and is complete for non-deterministic exponential time, just as for plain FO2. With two equivalence relations, FO2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO2 is undecidable.

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.

Characterizations of Axiomatic Categories of Models Canonically Isomorphic to (Quasi-)Varieties

1988 ◽  
Vol 31 (3) ◽  
pp. 287-300 ◽  
Author(s):  
Michel Hébert
Keyword(s):  
Forgetful Functor ◽  
Canonical Isomorphism ◽  
First Order ◽  
Order Language ◽  
Algebraic Language ◽  
Algebraic Constructions

AbstractLet be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion of L which assigns to a model its (unique) expansion. This solves a problem of H. Volger.In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.

On the undecidability of finite planar graphs

1971 ◽  
Vol 36 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Solomon Garfunkel ◽  
Herbert Shank
Keyword(s):  
Planar Graphs ◽  
Equivalence Relations ◽  
Original Graph ◽  
First Order ◽  
Semantic Embedding ◽  
Different Types ◽  
Hereditary Undecidability

In this paper we demonstrate the hereditary undecidability of finite planar graphs. In §2 we introduce the preliminary logical notions used and outline the Rabin–Scott method of semantic embedding. This method is illustrated in §3 by proving the undecidability of the theory of two finite equivalence relations of a special type. In §4 we give a proof of the main theorem by embedding these equivalence relations into finite planar graphs.The basic idea is first to form a graph which codes a pair of these relations and then to take a representative of it and “squish” it to the plane. This “squishing” requires the introduction of crossings; and edges of the original graph become paths in the new one. To distinguish the original edges we place two different types of “diamonds” about crossing points. We can then uncode our new graphs to recover the equivalence relations by means of simple first-order incidence properties.

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