#linearpropositions — Public Fediverse posts

#DifferentialPropositionalCalculus • 5.5
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

The third row of Figure 8 shows #VennDiagrams for the 3 #LinearPropositions of rank 1, which are none other than the 3 #BasicPropositions, \(p, q, r.\)

For example —

\(\text{Figure 8.3. Venn Diagram for}~p\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagram-e280a2-p-q-r-e280a2-p.jpg

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

#DifferentialPropositionalCalculus • 5.4
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

The second row of Figure 8 shows #VennDiagrams for the 3 #LinearPropositions of rank 2, expressed in terms of #MinimalNegationOperators by the following 3 forms, respectively:

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

For example —

\(\text{Figure 8.2. Venn Diagram for}~\texttt{(}p\texttt{,}q\texttt{)}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagram-e280a2-p-q-r-e280a2-p-q.jpg

#Logic #LogicalGraphs

#DifferentialPropositionalCalculus • 5.2
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

In a #UniverseOfDiscourse based on three #BooleanVariables \(p, q, r\) the #LinearPropositions take the shapes of the #VennDiagrams shown in Figure 8. Equivalent verbal & variant logical expressions are given in the next few posts.

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

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

#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 • 4.5
• 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.

Related Subjects —
#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