Set Venn Diagrams Applied to Inclusions and Non-inclusions

Abstract

In this work, formulas are inclusions $$t_1 \subseteq t_2$$t1⊆t2 and non-inclusions $$t_1\not \subseteq t_2$$t1?t2 between Boolean terms $$t_1$$t1 and $$t_2$$t2. We present a set of rules through which one can transform a term t in a diagram $$\Delta t$$Δt and, consequently, each inclusion $$t_1 \subseteq t_2$$t1⊆t2 (non-inclusion $$t_1\not \subseteq t_2$$t1?t2) in an inclusion $$\varDelta t_1 \subseteq \varDelta t_2$$Δt1⊆Δt2 (non-inclusion $$\varDelta t_1 \not \subseteq \varDelta t_2$$Δt1?Δt2) 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 $$\varSigma $$Σ 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 $$\varSigma \cup \varphi $$Σ?? of formulas, $$\varphi $$? is a consequence of $$\varSigma $$Σ 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 $$\varSigma $$Σ.