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.

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.

An Interpolation Theorem

2000 ◽  
Vol 6 (4) ◽  
pp. 447-462 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Elementary Model ◽  
Relation Symbol ◽  
First Order ◽  
Theoretic Proof ◽  
Universal Quantification ◽  
The Given ◽  
Interpolation Result

AbstractLyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U.It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts.

Linear Spaces, Boolean Algebras, and First Order Logic

Semantics ◽  
2019 ◽  
pp. 17-47
Author(s):  
András Kornai
Keyword(s):  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Linear Spaces ◽  
First Order

scholarly journals Expressive Power of Abstract Meaning Representations

2016 ◽  
Vol 42 (3) ◽  
pp. 527-535 ◽  
Author(s):  
Johan Bos
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Simple Extension ◽  
Universal Quantifier ◽  
Proper Treatment ◽  
First Order ◽  
Current Definition ◽  
Universal Quantification ◽  
Definition Of

The syntax of abstract meaning representations (AMRs) can be defined recursively, and a systematic translation to first-order logic (FOL) can be specified, including a proper treatment of negation. AMRs without recurrent variables are in the decidable two-variable fragment of FOL. The current definition of AMRs has limited expressive power for universal quantification (up to one universal quantifier per sentence). A simple extension of the AMR syntax and translation to FOL provides the means to represent projection and scope phenomena.

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

1987 ◽  
Vol 65 (2) ◽  
pp. 97-170 ◽  
Author(s):  
M. Makkai
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Stone Duality ◽  
First Order

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 GAMES AND CARDINALITIES IN INQUISITIVE FIRST-ORDER LOGIC

2021 ◽  
pp. 1-28
Author(s):  
IVANO CIARDELLI ◽  
GIANLUCA GRILLETTI
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order
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