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Continuous Logic and Proof Theory fo...

Wei, Jin,

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Continuous Logic and Proof Theory for First-Order Lukasiewicz Logic /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Continuous Logic and Proof Theory for First-Order Lukasiewicz Logic // Jin Wei.
Author:	Wei, Jin,
Description:	1 electronic resource (74 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31994830
ISBN:	9798280759312

Continuous Logic and Proof Theory for First-Order Lukasiewicz Logic /
Wei, Jin,

Continuous Logic and Proof Theory for First-Order Lukasiewicz Logic /Jin Wei. - 1 electronic resource (74 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

Continuous model theory traces back to Chang and Keisler (1966) and was recently redeveloped by Yaacov et al. (2008) to study complete metric structures with applications in operator algebras and many other fields. The underlying logic is called continuous logic, which is a natural generalization of first-order classical logic by extending the truth values to the continuous set r0, 1s. From a syntactic point of view, continuous logic is an extension of (first-order) Lukasiewicz logic, one of the famous fuzzy logics.The motivations for continuous logic were mostly semantic, while Yaacov and Pedersen (2010) provided a Hilbert-style proof system with proofs of its soundness and (approximate) completeness. In this thesis, we will discuss further about syntactic properties of continuous logic and proof theory for first-order Lukasiewicz logic. We will present a hypersequent calculus GL∀ for first-order Lukasiewicz logic, first discovered by Baaz and Metcalfe (2010). The main result will be our proof of approximate completeness of GL∀ with respect to the r0, 1s-semantics for arbitrary first-order formulas, while the original proof only applies to prenex formulas as pointed out in Gerasimov (2020). We will also explore some of the consequences and potential applications to continuous model theory, particularly in term of proof mining and extracting quantitative information from proofs using continuous ultraproducts.

English

ISBN: 9798280759312Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Continuous model theory

Continuous Logic and Proof Theory for First-Order Lukasiewicz Logic /
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520 $a Continuous model theory traces back to Chang and Keisler (1966) and was recently redeveloped by Yaacov et al. (2008) to study complete metric structures with applications in operator algebras and many other fields. The underlying logic is called continuous logic, which is a natural generalization of first-order classical logic by extending the truth values to the continuous set r0, 1s. From a syntactic point of view, continuous logic is an extension of (first-order) Lukasiewicz logic, one of the famous fuzzy logics.The motivations for continuous logic were mostly semantic, while Yaacov and Pedersen (2010) provided a Hilbert-style proof system with proofs of its soundness and (approximate) completeness. In this thesis, we will discuss further about syntactic properties of continuous logic and proof theory for first-order Lukasiewicz logic. We will present a hypersequent calculus GL∀ for first-order Lukasiewicz logic, first discovered by Baaz and Metcalfe (2010). The main result will be our proof of approximate completeness of GL∀ with respect to the r0, 1s-semantics for arbitrary first-order formulas, while the original proof only applies to prenex formulas as pointed out in Gerasimov (2020). We will also explore some of the consequences and potential applications to continuous model theory, particularly in term of proof mining and extracting quantitative information from proofs using continuous ultraproducts.
546 $a English
590 $a School code: 0175
650 4 $a Mathematics. $3 515831
650 4 $a Logic. $3 529544
650 4 $a Applied mathematics. $3 2122814
653 $a Continuous model theory
653 $a First-order logic
653 $a Fuzzy logic
653 $a Hypersequent calculus
653 $a Proof theory
653 $a Substructural logic
690 $a 0405
690 $a 0395
690 $a 0364
710 2 $a University of Pennsylvania. $b Mathematics. $e degree granting institution. $3 3823085
720 1 $a Towsner, Henry $e degree supervisor.
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791 $a Ph.D.
792 $a 2025
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