#LinearProposition

2022-11-23

#DifferentialPropositionalCalculus • 5.6
inquiryintoinquiry.com/2020/02

\(\text{Figure 8. Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
inquiryintoinquiry.files.wordp

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 \(0.\)

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

This is a venn diagram for a particular boolean function on 3 boolean variables p, q, r.

The venn diagram consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas representing 3 subsets of that universe.  The circular areas are labeled p, q, r respectively to indicate the subsets where the corresponding boolean variables are equal to 1, in other words, true.

The function of interest in this case is given as an example of a “rank 0 linear function on 3 variables” but it's more familiar to us as the constant 0 function or the everywhere false proposition.

The present series of venn diagrams represents boolean functions and logical propositions by means of the subsets where the functions evaluate to 1 or the propositions are true.  Each diagram indicates that subset by means of a conventional shading, in this case, a shade of blue, leaving the remaining regions white.  In the present case the indicated subset is the empty set and so the venn diagram is nowhere blue and everywhere white.
2022-11-22

#DifferentialPropositionalCalculus • 5.3
inquiryintoinquiry.com/2020/02

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:

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

\(\text{Figure 8.1. Rank 3 Linear}\, f : \mathbb{B}^3 \to \mathbb{B}\)
inquiryintoinquiry.files.wordp

#Logic #LogicalGraphs
#PaintedAndRootedCacti
#MinimalNegationOperators

This is a venn diagram for a particular boolean function on 3 boolean variables p, q, r.

The venn diagram consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas representing 3 subsets of that universe.  The circular areas are labeled p, q, r respectively to indicate the subsets where the corresponding boolean variables are equal to 1, in other words, true.

The function of interest in this case is called the rank 3 linear function on 3 variables and it evaluates to 1 on just 4 subsets, namely, the following:

(1) p and q and r are all true.
(2) p alone is true, but neither q or r.
(3) q alone is true, but neither p or r
(4) r alone is true, but neither p or q.

The venn diagram indicates those 4 subsets with a conventional shading, in this case, a shade of blue, leaving the remaining regions white.
2022-11-19

#DifferentialPropositionalCalculus • 4.9
inquiryintoinquiry.com/2020/02

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

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