#DifferentialPropositionalCalculus • 5.6
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

\(\text{Figure 8. Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagrams-e280a2-p-q-r-e280a2-linear-propositions.jpg

At the bottom of Figure 8 is #VennDiagram for the #LinearProposition of rank 0, the constant \(0\) function or the everywhere false proposition, expressed in #CactusSyntax by the form \(\texttt{(}~\texttt{)}\) or in algebraic form by a simple \(0.\)

\(\text{Figure 8.4 Venn Diagram for}~\texttt{(}~\texttt{)}\)

#DifferentialPropositionalCalculus • 5.3
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

At the top of Figure 8 is the #VennDiagram for the #LinearProposition of rank 3, which may be expressed by any one of the following 3 forms:

\[\texttt{(}p\texttt{,(}q\texttt{,}r\texttt{))}, \quad \texttt{((}p\texttt{,}q\texttt{),}r\texttt{)}, \quad p+q+r.\]

\(\text{Figure 8.1. Rank 3 Linear}\, f : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagram-e280a2-p-q-r-e280a2-p-q-r.jpg

#Logic #LogicalGraphs
#PaintedAndRootedCacti
#MinimalNegationOperators

#DifferentialPropositionalCalculus • 4.9
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

In each family the rank \(k\) ranges from \(0\) to \(n\) and counts the number of positive appearances of #CoordinatePropositions \(a_1, \ldots, a_n\) in the resulting expression. For example, when \(n=3\) the #LinearProposition of rank \(0\) is \(0,\) the #PositiveProposition of rank \(0\) is \(1,\) and the #SingularProposition of rank \(0\) is \(\texttt{(}a_1\texttt{)} \texttt{(}a_2\texttt{)} \texttt{(}a_3\texttt{)}.\)

#Logic