NDLI: Set Venn Diagrams Applied to Inclusions and Non-inclusions

Content Provider	Springer Nature Link
Author	Freitas, Renata Viana, Petrucio
Copyright Year	2015
Abstract	In this work, formulas are inclusions $$t_1 \subseteq t_2$$ and non-inclusions $$t_1\not \subseteq t_2$$ between Boolean terms $$t_1$$ and $$t_2$$ . We present a set of rules through which one can transform a term t in a diagram $$\Delta t$$ and, consequently, each inclusion $$t_1 \subseteq t_2$$ (non-inclusion $$t_1\not \subseteq t_2$$ ) in an inclusion $$\varDelta t_1 \subseteq \varDelta t_2$$ (non-inclusion $$\varDelta t_1 \not \subseteq \varDelta t_2$$ ) between diagrams. Also, by applying the rules just to the diagrams we are able to solve the problem of verifying if a formula $$\varphi $$ is consequence of a, possibly empty, set $$\varSigma $$ of formulas taken as hypotheses. Our system has a diagrammatic language based on Venn diagrams that are read as sets, and not as statements about sets, as usual. We present syntax and semantics of the diagrammatic language, define a set of rules for proving consequence, and prove that our set of rules is strongly sound and complete in the following sense: given a set $$\varSigma \cup \varphi $$ of formulas, $$\varphi $$ is a consequence of $$\varSigma $$ iff there is a proof of this fact that is based only on the rules of the system and involves only diagrams associated to $$\varphi $$ and to the members of $$\varSigma $$ .
Starting Page	457
Ending Page	485
Page Count	29
File Format	PDF
ISSN	09258531
Journal	Journal of Logic, Language and Information
Volume Number	24
Issue Number	4
e-ISSN	15729583
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-10-06
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Algebra of sets Inclusions Non-inclusions Diagrammatic systems Venn diagrams Soundness Completeness Decidability Logic Semantics Information Systems Applications (incl. Internet)
Content Type	Text
Resource Type	Article
Subject	Philosophy Computer Science Linguistics and Language

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