Boolean algebras with an automorphism group: a framework for Łukasiewicz logic
Authors: |
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Title: | Boolean algebras with an automorphism group: a framework for Łukasiewicz logic | |
Journal: | Journal of Multiple-Valued Logic and Soft Computing | |
Volume | 14 | |
Number | ||
Pages: | 51 - 67 | |
Year: | 2008 |
Abstract
We introduce a framework within which reasoning according to Lukasiewicz logic can be represented. We consider a separable Boolean algebra Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal B} endowed with a (certain type of) group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle G} of automorphisms; the pair Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle ({\mathcal B},G)} will be called a Boolean ambiguity algebra. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal B} is meant to model a system of crisp properties; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle G} is meant to express uncertainty about these properties.
We define fuzzy propositions as subsets of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal B} which are, most importantly, closed under the action of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle G} . By defining a conjunction and implication for pairs of fuzzy propositions in an appropriate manner, we are led to the algebraic structure characteristic for Lukasiewicz logic.