scholarly journals Finite-Variable Logics Do Not Have Weak Beth Definability Property

2015 ◽  
pp. 125-133
Author(s):  
Hajnal Andréka ◽  
István Németi
Keyword(s):  
Beth Definability ◽  
Beth Definability Property

scholarly journals INFINITARY PROPOSITIONAL RELEVANT LANGUAGES WITH ABSURDITY

2017 ◽  
Vol 10 (4) ◽  
pp. 663-681
Author(s):  
GUILLERMO BADIA
Keyword(s):  
Expressive Power ◽  
Interpolation Theorem ◽  
Boolean Logic ◽  
Isomorphism Theorem ◽  
Strong Completeness ◽  
Lack Of Compactness ◽  
Relevant Logics ◽  
Beth Definability ◽  
Beth Definability Property

AbstractAnalogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic L∞ω holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

Beth definability, interpolation and language splitting

Synthese ◽  
2010 ◽  
Vol 179 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Rohit Parikh
Keyword(s):  
Beth Definability

Beth definability in infinitary languages

1974 ◽  
Vol 39 (1) ◽  
pp. 22-26 ◽  
Author(s):  
John Gregory
Keyword(s):  
Unpublished Result ◽  
Negative Results ◽  
Beth Definability ◽  
Image Position

Some negative results will be proved concerning the following for certain infinitary languages ℒ1 and ℒ2.Definition. Beth(ℒ1, ℒ2) iff, for every sentence ϕ(R) of ℒ1, and n-place relation symbols R and S such that S does not occur in ϕ(R), ifthen there is an ℒ2 formula θ(x1, …, xn) such thatand θ is built up using only those constant and relation symbols of ϕ other than R.That is, Beth(ℒ1, ℒ2) iff for every implicit ℒ1 definition ϕ(R) of relations, there is a corresponding explicit ℒ2 definition θ. Beth(ℒωω, ℒωω) was proved by Beth.Malitz proved that not Beth(ℒω1 ω1, ℒ∞∞) (hence not Beth (ℒ∞∞, ℒ∞∞)), but Beth (ℒ∞ω, ℒ∞∞). In §1, it is shown that Beth(ℒ∞ω, ℒ∞ω) is false. In §2, we strengthen this by showing that, for every cardinal κ, not Beth(ℒ∞ω, ℒ∞κ). In fact, not Beth (ℒκ+ω, ℒ∞κ) follows from property A(κ) defined in §2, and A(κ) is known for regular κ > ω (unpublished result of Morley).More information on infinitary Beth and Craig theorems is given in [2] and [3]. We assume that the reader is acquainted with the languages ℒκλ which allow conjunctions over ≺κ formulas and quantifiers over ≺λ variables. Thus, we assume that the reader is acquainted with the back and forth argument for showing that two structures are ≡∞κ (ℒ∞κ-elementarily equivalent). Our notation is fairly standard.

Properties of first-order logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Model Theory ◽  
Propositional Logic ◽  
Final Section ◽  
Order Logic ◽  
Countable Model ◽  
First Order Logic ◽  
First Order ◽  
Beth Definability ◽  
The Subject ◽  
Infinite Model

We show that first-order logic, like propositional logic, has both completeness and compactness. We prove a countable version of these theorems in Section 4.1. We further show that these two properties have many useful consequences for first-order logic. For example, compactness implies that if a set of first-order sentences has an infinite model, then it has arbitrarily large infinite models. To fully understand completeness, compactness, and their consequences we must understand the nature of infinite numbers. In Section 4.2, we return to our discussion of infinite numbers that we left in Section 2.5. This digression allows us to properly state and prove completeness and compactness along with the Upward and Downward Löwenhiem–Skolem theorems. These are the four central theorems of first-order logic referred to in the title of Section 4.3. We discuss consequences of these theorems in Sections 4.4–4.6. These consequences include amalgamation theorems, preservation theorems, and the Beth Definability theorem. Each of the properties studied in this chapter restrict the language of first-order logic. First-order logic is, in some sense, weak. There are many concepts that cannot be expressed in this language. For example, whereas first-order logic can express “there exist n elements” for any finite n, it cannot express “there exist countably many elements.” Any sentence having a countable model necessarily has uncountable models. As we previously mentioned, this follows from compactness. In the final section of this chapter, using graphs as an illustration, we discuss the limitations of first-order logic. Ironically, the weakness of first-order logic makes it the fruitful logic that it is. The properties discussed in this chapter, and the limitations that follow from them, make possible the subject of model theory. All formulas in this chapter are first-order unless stated otherwise. Many of the properties of first-order logic, including completeness and compactness, are consequences of the following fact: Every model has a theory and every theory has a model. Recall that a set of sentences is a “theory” if it is consistent (i.e. if we cannot derive a contradiction). “Every theory has a model” means that if a set of sentences is consistent, then it is satisfiable.

scholarly journals SOME MODEL THEORY OF GUARDED NEGATION

2018 ◽  
Vol 83 (04) ◽  
pp. 1307-1344
Author(s):  
VINCE BÁRÁNY ◽  
MICHAEL BENEDIKT ◽  
BALDER TEN CATE
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Description Logics ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Beth Definability ◽  
Order Translations ◽  
Certain Answers

AbstractThe Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all positive existential formulas, can express the first-order translations of basic modal logic and of many description logics, along with many sentences that arise in databases. It has been shown that the syntax of GNFO is restrictive enough so that computational problems such as validity and satisfiability are still decidable. This suggests that, in spite of its expressive power, GNFO formulas are amenable to novel optimizations. In this article we study the model theory of GNFO formulas. Our results include effective preservation theorems for GNFO, effective Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to a large class of GNFO sentences within very restricted logics.

scholarly journals Two-variable logic has weak, but not strong, Beth definability

2021 ◽  
pp. 1-23
Author(s):  
H. Andréka ◽  
I. Németi
Keyword(s):  
Beth Definability ◽  
Variable Logic

scholarly journals Combined Covers and Beth Definability

2020 ◽  
pp. 181-200 ◽  
Author(s):  
Diego Calvanese ◽  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Marco Montali ◽  
Andrey Rivkin
Keyword(s):  
Beth Definability
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