Set Venn Diagrams Applied to Inclusions and Non-inclusions

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 $\varSigma$.