An Omitting Types Theorem for first order logic with infinitary relation symbols

2007 ◽  
Vol 53 (6) ◽  
pp. 564-570 ◽  
Author(s):  
Tarek Sayed Ahmed ◽  
Basim Samir
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Omitting Types

scholarly journals Omitting types in fragments and extensions of first order logic

2021 ◽  
Author(s):  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Omitting Types

Fix \(2 < n < \omega\). Let \(L_n\) denote first order logic restricted to the first $n$ variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for \(L_n\) and for infinitary variants and extensions of \(L_{\omega, \omega}\).

Filter Logics on ω

1984 ◽  
Vol 49 (1) ◽  
pp. 241-256
Author(s):  
Matt Kaufmann
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Basic Properties ◽  
Countable Compactness ◽  
Omitting Types

AbstractLogics LF(M) are considered, in which M (“most”) is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form {(⋂F) ∪ X: ∣ω − X∣ < ω} with ∣ω − ⋂F∣ = ω. Moreover the set of validities of LF (M) and even of depends only on a few basic properties of F. Similar characterizations are given of the class of filters F for which LF (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (, where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindström's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindström's argument is applied to characterize the invariant sentences as just those of first-order logic.

scholarly journals On the multi-dimensional modal logic of substitutions

2019 ◽  
Vol 56 (4) ◽  
pp. 454-481
Author(s):  
Tarek Sayed Ahmed ◽  
Mohammad Assem Mahmoud
Keyword(s):  
Amalgamation Property ◽  
Modal Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Semantics ◽  
Infinite Dimensions ◽  
Finite Axiomatizability ◽  
First Order ◽  
Abstract Entities ◽  
Omitting Types

Abstract We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

Omitting types for finite variable fragments and complete representations of algebras

2008 ◽  
Vol 73 (1) ◽  
pp. 65-89 ◽  
Author(s):  
Hajnal Andréka ◽  
István Németi ◽  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Complete Representations ◽  
Atomic Relation ◽  
Representations Of Algebras ◽  
Omitting Types ◽  
Binary Relation Symbol

AbstractWe give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

Sign in / Sign up

Export Citation Format

Share Document