scholarly journals Multisequent Gentzen Deduction Systems For B2 2-Valued First-Order Logic

2018 ◽  
Vol 7 (1) ◽  
pp. 53
Author(s):  
Wei Li ◽  
Yuefei Sui
Keyword(s):  
Boolean Algebra ◽  
Order Logic ◽  
First Order Logic ◽  
Deduction System ◽  
Truth Values ◽  
First Order ◽  
Soundness And Completeness ◽  
Completeness Theorems

For the four-element Boolean algebra B22, a multisequent Г|Δ|∑|∏ is a generalization of sequent Г→Δ in traditional B22 valued first-order logic. By defining the truth-values of quantified formulas, a Gentzen deduction system G22 for B22-valued first-order logic will be built and its soundness and completeness theorems will be proved.

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.

scholarly journals Completeness theorems for first-order logic analysed in constructive type theory

2021 ◽  
Vol 31 (1) ◽  
pp. 112-151
Author(s):  
Yannick Forster ◽  
Dominik Kirst ◽  
Dominik Wehr
Keyword(s):  
Type Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
The Standard Model ◽  
Coq Proof Assistant ◽  
Game Theoretic ◽  
Completeness Theorems

Abstract We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov’s Principle and Weak K̋nig’s Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.

scholarly journals CIRCULARITY IN SOUNDNESS AND COMPLETENESS

2014 ◽  
Vol 20 (1) ◽  
pp. 24-38
Author(s):  
RICHARD KAYE
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Foundational Research ◽  
Soundness And Completeness

AbstractWe raise an issue of circularity in the argument for the completeness of first-order logic. An analysis of the problem sheds light on the development of mathematics, and suggests other possible directions for foundational research.

Natural deduction based set theories: a new resolution of the old paradoxes

1986 ◽  
Vol 51 (2) ◽  
pp. 393-411 ◽  
Author(s):  
Paul C. Gilmore
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Truth Values ◽  
First Order ◽  
Axiom Scheme ◽  
Universe Of Sets ◽  
Comprehension Axiom

AbstractThe comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

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.

scholarly journals Formalizing Kant’s Rules

2019 ◽  
Vol 49 (4) ◽  
pp. 613-680
Author(s):  
R. Evans ◽  
M. Sergot ◽  
A. Stephenson
Keyword(s):  
General Procedure ◽  
Cognitive Architecture ◽  
Order Logic ◽  
First Order Logic ◽  
Pure Reason ◽  
Critique Of Pure Reason ◽  
Input Output ◽  
Truth Values ◽  
First Order ◽  
Natural Way

AbstractThis paper formalizes part of the cognitive architecture that Kant develops in the Critique of Pure Reason. The central Kantian notion that we formalize is the rule. As we interpret Kant, a rule is not a declarative conditional stating what would be true if such and such conditions hold. Rather, a Kantian rule is a general procedure, represented by a conditional imperative or permissive, indicating which acts must or may be performed, given certain acts that are already being performed. These acts are not propositions; they do not have truth-values. Our formalization is related to the input/ output logics, a family of logics designed to capture relations between elements that need not have truth-values. In this paper, we introduce KL3 as a formalization of Kant’s conception of rules as conditional imperatives and permissives. We explain how it differs from standard input/output logics, geometric logic, and first-order logic, as well as how it translates natural language sentences not well captured by first-order logic. Finally, we show how the various distinctions in Kant’s much-maligned Table of Judgements emerge as the most natural way of dividing up the various types and sub-types of rule in KL3. Our analysis sheds new light on the way in which normative notions play a fundamental role in the conception of logic at the heart of Kant’s theoretical philosophy.

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
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