#DifferentialPropositionalCalculus • 7.3
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

Figure 10. #SingularPropositions on 3 Variables
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-singular-propositions.jpg

Rank 3. The cell \(pqr.\)

Rank 2. The 3 cells \(pr\texttt{(}q\texttt{)},~qr\texttt{(}p\texttt{)},~pq\texttt{(}r\texttt{)}.\)

Rank 1. The 3 cells \(q\texttt{(}p\texttt{)(}r\texttt{)},~p\texttt{(}q\texttt{)(}r\texttt{)},~r\texttt{(}p\texttt{)(}q\texttt{)}.\)

Rank 0. The cell \(\texttt{(}p\texttt{)(}q\texttt{)(}r\texttt{)}.\)

#Logic #DifferentialLogic

#DifferentialPropositionalCalculus • 7.2
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p,q,r\) there are \(2^3=8\) #SingularPropositions. Their #VennDiagrams are shown in Figure 10.

\(\text{Figure 10. Singular Propositions} : \mathbb{B}^3 \to\mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-singular-propositions.jpg

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

#DifferentialPropositionalCalculus • 7.1
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

The #SingularPropositions \(\{\mathbf{x}:\mathbb{B}^n\to\mathbb{B}\}=(\mathbb{B}^n\xrightarrow{s}\mathbb{B})\) may be written as products:

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

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#MinimalNegationOperators

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

#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 • 4.10
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

The #BasicPropositions \(a_i : \mathbb{B}^n \to \mathbb{B}\) are both linear and positive. So those two families of propositions, the linear & the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Related Subjects —

#CoordinatePropositions #SimplePropositions
#LinearPropositions #SingularPropositions

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

#DifferentialPropositionalCalculus • 4.8
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

The #SingularPropositions \(\{\mathbf{x} : \mathbb{B}^n \to \mathbb{B}\} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B})\) may be written as products:

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

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

#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