Jon Awbrey · @Inquiry
51 followers · 165 posts · Server mathstodon.xyz
Jon Awbrey · @Inquiry
46 followers · 158 posts · Server mathstodon.xyz

• 5.4
inquiryintoinquiry.com/2020/02

The 2nd row of Figure 8 gives for the 3 of rank 2, expressed in terms of 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}~\(\texttt{(}p\texttt{,}q\texttt{)}\)
inquiryintoinquiry.files.wordp

#MinimalNegationOperators #linearpropositions #venndiagrams #DifferentialPropositionalCalculus

Last updated 2 years ago

Jon Awbrey · @Inquiry
39 followers · 143 posts · Server mathstodon.xyz
Jon Awbrey · @Inquiry
38 followers · 138 posts · Server mathstodon.xyz

• 5.1
inquiryintoinquiry.com/2020/02

The \(\{\ell : \mathbb{B}^n \to \mathbb{B}\} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B})\) 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

Last updated 2 years ago

Jon Awbrey · @Inquiry
38 followers · 131 posts · Server mathstodon.xyz

• 4.11
inquiryintoinquiry.com/2020/02

Linearity, Positivity, Singularity are relative to the basis \(\mathcal{A}.\) on one basis do not remain so if new features are added to the basis. A even within the same pairwise options \(\{a_i\}\cup\{\texttt{(}a_i\texttt{)}\}\) changes the sets of & as both are decided by the choice of , in effect choosing a cell as origin.

#logic #basicpropositions #positivepropositions #linearpropositions #basischange #singularpropositions #DifferentialPropositionalCalculus

Last updated 2 years ago

Jon Awbrey · @Inquiry
31 followers · 112 posts · Server mathstodon.xyz

• 4.6
inquiryintoinquiry.com/2020/02

The \(\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B})\) 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 —

#BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #logic #linearpropositions #DifferentialPropositionalCalculus

Last updated 2 years ago

Jon Awbrey · @Inquiry
31 followers · 110 posts · Server mathstodon.xyz

• 4.5
inquiryintoinquiry.com/2020/02

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

#binomialcoefficient #singularprpositions #positivepropositions #linearpropositions #DifferentialPropositionalCalculus

Last updated 2 years ago

Jon Awbrey · @Inquiry
30 followers · 103 posts · Server mathstodon.xyz

• 4.4
inquiryintoinquiry.com/2020/02

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 , the , and the .


#BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #logic #singularpropositions #positivepropositions #linearpropositions #DifferentialPropositionalCalculus

Last updated 2 years ago