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#DifferentialPropositionalCalculus • 6.2
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

Figure 9. #VennDiagrams for #PositivePropositions on 3 Variables
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

Rank 3 (Top). The #BooleanProduct or #LogicalConjunction \(pqr.\)

Rank 2. The 3 #BooleanProducts \(pr, qr, pq.\)

Rank 1. The 3 #BasicPropositions \(q, p, r.\)

Rank 0 (Bottom). The #ConstantFunction \(1 : \mathbb{B}^3 \to \mathbb{B}\) or the #ConstantProposition expressed in #CactusSyntax by the form \(\texttt{((}~\texttt{))}.\)

#Logic

This Figure is repeated from the previous post on this thread. Please refer to the previous post for the full descriptive text.

#DifferentialPropositionalCalculus • 6.1
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p, q, r\) there are \(2^3 = 8\) #PositivePropositions. Their #VennDiagrams are shown in Figure 9.

\(\text{Figure 9. Positive Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

The Figure shows the 8 venn diagrams corresponding to the 8 positive propositions or positive boolean functions on 3 variables p, q, r. There is 1 venn diagram at the top of the Figure, 3 in the 2nd row, 3 in the 3rd row, and 1 venn diagram at the bottom. These represent the positive propositions of rank 3, 2, 1, and 0, respectively.

Each of the 8 venn diagrams consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas P, Q, R representing 3 subsets of that universe. The circular areas are labeled p, q, r respectively to indicate the subsets P, Q, R where the corresponding boolean variables are equal to 1, in other words, true.

The positive proposition of rank 3 corresponds to the boolean product or logical conjunction pqr. The function equals 1 or the proposition is true in the intersection of the 3 circular areas & the venn diagram indicates that by coloring that cell a shade of blue, leaving the remaining area white.

The 3 positive propositions of rank 2 correspond to the 3 boolean products, pr, qr, pq, respectively. Their diagrams shade the intersections P∩R, Q∩R, P∩Q, respectively.

The 3 positive propositions of rank 1 are just the 3 basic propositions, p, q, r and so their diagrams shade the circular areas P, Q, R, respectively.

The positive proposition of rank 0 is just the everywhere true proposition or the constant 1 function, so its diagram shades the whole rectangular area representing the universe of discourse.

#DifferentialPropositionalCalculus • 6
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

The #PositivePropositions \(\{p:\mathbb{B}^n\to \mathbb{B}\}=(\mathbb{B}^n \xrightarrow{p}\mathbb{B})\) may be written as products:

\[\prod_{i=1}^n e_i~=~e_1 \cdot\ldots\cdot e_n~\text{where}~\left\{\begin{matrix}e_i=a_i\\ \text{or}\\e_i=1\end{matrix}\right\}~\text{for}~i=1~\text{to}~n.\]

To get a sense of this family's structure we'll next draw the #VennDiagrams for the 3 variable case.

#Logic #LogicalGraphs #DifferentialLogic

#DifferentialPropositionalCalculus • 4.11
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

Linearity, Positivity, Singularity are relative to the basis \(\mathcal{A}.\) #SingularPropositions on one basis do not remain so if new features are added to the basis. #BasisChanges even within the same pairwise options \(\{a_i\}\cup\{\texttt{(}a_i\texttt{)}\}\) change the sets of #LinearPropositions and #PositivePropositions. Both are fixed by the choice of #BasicPropositions which amounts to taking a cell as origin.

#Logic

#DifferentialPropositionalCalculus • 4.7
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

The #PositivePropositions \(\{p : \mathbb{B}^n \to \mathbb{B}\} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B})\) may be written as products:

\[\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n ~\text{where}~ \left\{ \begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix} \right\} ~\text{for}~ i = 1 ~\text{to}~ n.\]

#DifferentialPropositionalCalculus • 4.5
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

Each of the families — #LinearPropositions, #PositivePropositions, #SingularPrpositions — is naturally parameterized by the coordinate \(n\)-tuples in \(\mathbb{B}^n\) and falls into \(n+1\) ranks, with a #BinomialCoefficient \(\tbinom{n}{k}\) giving the number of propositions having rank or weight \(k\) in their class.

#DifferentialPropositionalCalculus • 4.4
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

Among the \(2^{2^n}\) propositions in \([a_1, \ldots, a_n]\) are several families numbering \(2^n\) propositions each which take on special forms with respect to the basis \(\{a_1, \ldots, a_n \}.\) Three families are especially prominent in the present context, the #LinearPropositions, the #PositivePropositions, and the #SingularPropositions.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

#PositivePropositions

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