#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\)
#BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #logic #basicpropositions #linearpropositions #venndiagrams #DifferentialPropositionalCalculus
#DifferentialPropositionalCalculus • 5.4
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/
The 2nd row of Figure 8 gives #VennDiagrams for the 3 #LinearPropositions of rank 2, expressed in terms of #MinimalNegationOperators by the following 3 forms, respectively:
\[\texttt{(}p\texttt{,}r\texttt{)}, \quad \texttt{(}q\texttt{,}r\texttt{)}, \quad \texttt{(}p\texttt{,}q\texttt{)}.\]
\(\text{Figure 8.2. Venn Diagram for}~\(\texttt{(}p\texttt{,}q\texttt{)}\)
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#MinimalNegationOperators #linearpropositions #venndiagrams #DifferentialPropositionalCalculus
#DifferentialPropositionalCalculus • 5.2
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/
In a #UniverseOfDiscourse based on 3 #BooleanVariables, \(p, q, r,\) the #LinearPropositions take the shapes shown in Figure 8.
\(\text{Figure 8. Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagrams-e280a2-p-q-r-e280a2-linear-propositions.jpg
Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #logic #linearpropositions #booleanvariables #UniverseOfDiscourse #DifferentialPropositionalCalculus
#DifferentialPropositionalCalculus • 5.1
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/
The #LinearPropositions \(\{\ell : \mathbb{B}^n \to \mathbb{B}\} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B})\) may be written as sums:
\[\sum_{i=1}^n e_i~=~e_1+\ldots+e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.\]
One thing to remember — the values in \(\mathbb{B}=\{0,1\}\) are added “mod 2”, so that \(1+1=0.\)
#DifferentialLogic #logic #linearpropositions #DifferentialPropositionalCalculus
#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. A #BasisChange even within the same pairwise options \(\{a_i\}\cup\{\texttt{(}a_i\texttt{)}\}\) changes the sets of #LinearPropositions & #PositivePropositions as both are decided by the choice of #BasicPropositions, in effect choosing a cell as origin.
#logic #basicpropositions #positivepropositions #linearpropositions #basischange #singularpropositions #DifferentialPropositionalCalculus
#DifferentialPropositionalCalculus • 4.6
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/
The #LinearPropositions \(\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B})\) may be written as sums:
\[\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.\]
Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #logic #linearpropositions #DifferentialPropositionalCalculus
#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.
#binomialcoefficient #singularprpositions #positivepropositions #linearpropositions #DifferentialPropositionalCalculus
#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
#BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #logic #singularpropositions #positivepropositions #linearpropositions #DifferentialPropositionalCalculus