On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

1980 ◽  
Vol 45 (2) ◽  
pp. 265-283 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Automorphism Groups ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

Notions of locality and their logical characterizations over finite models

1999 ◽  
Vol 64 (4) ◽  
pp. 1751-1773 ◽  
Author(s):  
Lauri Hella ◽  
Leonid Libkin ◽  
Juha Nurmonen
Keyword(s):  
Small Degree ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Finite Models ◽  
Order Case ◽  
The Relationship

AbstractMany known tools for proving expressibility bounds for first-ordér logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

On amalgamations of languages with Magidor-Malitz quantifiers

1979 ◽  
Vol 44 (4) ◽  
pp. 549-558
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Universal Algebra ◽  
Model Theory ◽  
Natural Extension ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Free Variables ◽  
Countably Compact

In this paper we indicate how compact languages containing the Magidor-Malitz quantifiers Qκn in different cardinalities can be amalgamated to yield more expressive, compact languages.The language Lκ<ω, originally introduced by Magidor and Malitz [9], is a natural extension of the language L(Q) introduced by Mostowski and investigated by Fuhrken [6], [7], Keisler [8] and Vaught [13]. Intuitively, Lκ<ω is first-order logic together with quantifiers Qκn (n ∈ ω) binding n free variables which express “there is a set X of cardinality κ such than any n distinct elements of X satisfy …”, or in other words, iff the relation on determined by φ contains an n-cube of cardinality κ. With these languages one can express a variety of combinatorial statements of the type considered by Erdös and his colleagues, as well as concepts in universal algebra which are beyond the scope of first-order logic. The model theory of Lκ<ω has been further developed by Badger [1], Magidor and Malitz [10] and Shelah [12].We refer to a language as being < κ compact if, given any set of sentences Σ of the language, if Σ is finitely satisfiable and ∣Σ∣ < κ, then Σ has a model. The phrase countably compact is used in place of <ℵ1 compact.

scholarly journals A dichotomy in classifying quantifiers for finite models

2005 ◽  
Vol 70 (4) ◽  
pp. 1297-1324
Author(s):  
Saharon Shelah ◽  
Mor Doron
Keyword(s):  
Partial Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Partial Order ◽  
Existential Quantifier ◽  
First Order ◽  
Finite Models ◽  
One To One

AbstractWe consider a family of finite universes. The second order existential quantifier Qℜ means for each U Є quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Qℜ, either Qℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Qℜ) (first order logic plus the quantifier Qℜ) is undecidable.

Fraïssé–Hintikka theorem in institutions

2020 ◽  
Vol 30 (7) ◽  
pp. 1377-1399
Author(s):  
Daniel Găină ◽  
Tomasz Kowalski
Keyword(s):  
Classical Case ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Specification Languages ◽  
Higher Order Logic ◽  
Elementary Equivalence ◽  
First Order

Abstract We generalize the characterization of elementary equivalence by Ehrenfeucht–Fraïssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that the characterization is proved directly for all signatures, including infinite ones.

scholarly journals THE LINDENBAUM ALGEBRA OF THE THEORY OF THE CLASS OF ALL FINITE MODELS

2002 ◽  
Vol 02 (02) ◽  
pp. 145-225 ◽  
Author(s):  
STEFFEN LEMPP ◽  
MIKHAIL PERETYAT'KIN ◽  
REED SOLOMON
Keyword(s):  
Boolean Algebra ◽  
Order Theory ◽  
Boolean Algebras ◽  
Quotient Algebra ◽  
First Order ◽  
Finite Models ◽  
First Order Theory ◽  
Recursive Isomorphism ◽  
Atomic Boolean Algebra

In this paper, we investigate the Lindenbaum algebra ℒ(T fin ) of the theory T fin = Th (M fin ) of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra (ℒ(T fin )/ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ(T fin ); moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra (ℒ(T fin )/ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature.

Generalisation of first-order logic to nonatomic domains

1985 ◽  
Vol 50 (3) ◽  
pp. 815-838 ◽  
Author(s):  
P. Roeper
Keyword(s):  
Predicate Logic ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Quantification Theory ◽  
First Order ◽  
General Kind ◽  
Existential Quantification

The quantifiers of standard predicate logic are interpreted as ranging over domains of individuals, and interpreted formulae beginning with a quantifier make claims to the effect that something is true of every individual, i.e. of the whole domain, or of some individuals, i.e. of part of the domain. To state that something is true of all or part of a totality seems to be the basic significance of universal and existential quantification, and this by itself does not involve a specification of the structure of the totality. This means that the notion of quantification by itself does not demand totalities of individuals, i.e. atomic totalities, as domains of quantification. Nonatomic domains, such as volumes of space, or surfaces, are equally in order. So one might say that a certain predicate applies “everywhere” or “somewhere” in such a domain. All that the concept of quantification requires is a totality which is structured in terms of a part-to-whole relation, and appropriate properties that apply to part or all of the totality. Quantification does not demand that the totality have smallest parts, or atoms. There is no conflict with the sense of universal or existential quantification if the domain is nonatomic, if every one of its parts has itself proper parts.The most general kind of quantification theory must then deal with totalities of any kind, atomic or not. The relationships among the parts of a domain are described by the theory of Boolean algebras, which we can regard as the most general characterisation of a totality, of a domain of quantification.In this paper I shall be concerned with this generalised theory of quantification, which encompasses nonatomic domains as well as atomic and mixed domains, i.e. totalities consisting entirely or partly of individuals.

scholarly journals Stone duality for first-order logic: a nominal approach to logic and topology

10.29007/tp3z ◽  
2018 ◽  
Author(s):  
Murdoch J. Gabbay
Keyword(s):  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Specific Class ◽  
Stone Duality ◽  
First Order ◽  
And Topology ◽  
Universal Quantification ◽  
Stone Spaces ◽  
Nominal Sets

What are variables, and what is universal quantification over a variable?Nominal sets are a notion of `sets with names', and using equational axioms in nominal algebra these names can be given substitution and quantification actions.So we can axiomatise first-order logic as a nominal logical theory.We can then seek a nominal sets representation theorem in which predicates are interpreted as sets; logical conjunction is interpreted as sets intersection; negation as complement.Now what about substitution; what is it for substitution to act on a predicate-interpreted-as-a-set, in which case universal quantification becomes an infinite sets intersection?Given answers to these questions, we can seek notions of topology.What is the general notion of topological space of which our sets representation of predicates makes predicates into `open sets'; and what specific class of topological spaces corresponds to the image of nominal algebras for first-order logic?The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces.Nominal algebra lets us extend Boolean algebras to `FOL-algebras', and nominal sets let us correspondingly extend Stone spaces to `∀-Stone spaces'.These extensions reveal a wealth of structure, and we obtain an attractive and self-contained account of logic and topology in which variables directly populate the denotation, and open predicates are interpreted as sets rather than functions from valuations to sets.

scholarly journals A Definability Theorem for First Order Logic

1997 ◽  
Vol 4 (3) ◽  
Author(s):  
Carsten Butz ◽  
Ieke Moerdijk
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Definable Subset ◽  
Language L ◽  
The One

In this paper, we will present a definability theorem for first order logic.<br />This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S M (i.e., a subset S = {a | M |= phi(a)} defined by some formula phi) is invariant under all<br />automorphisms of M. The same is of course true for subsets of M" defined<br />by formulas with n free variables.<br /> Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L.<br />Our presentation is entirely selfcontained, and only requires familiarity<br />with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning<br />Boolean valued models.<br />The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models.

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