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

Jon Awbrey

Special Classes of Propositions —

Before moving on, let’s unpack some of the assumptions, conventions, & implications involved in the array of concepts & notations introduced above.

A universe \(A^\bullet = [a_1, \ldots, a_n]\) based on the logical features \(a_1, \ldots, a_n\) is a set \(A\) plus the set of all possible functions from the space \(A\) to the #BooleanDomain \(\mathbb{B} = \{0, 1\}.\)

#Logic #BooleanFunctions

Differential Propositional Calculus • 4

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#Peirce #Semiotics
#Logic #PropositionalCalculus
#BooleanDomain #BooleanFunctions
#LogicalGraphs #DifferentialLogic

#DifferentialPropositionalCalculus • 4.1
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

There are \(2^n\) elements in \(A,\) often pictured as the cells of a #VennDiagram or the nodes of a #HyperCube.

There are \(2^{2^n}\) functions from \(A\) to \(\mathbb{B},\) accordingly pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.

Differential Propositional Calculus • 4

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#DifferentialPropositionalCalculus • 4.2
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

A logical proposition about the elements of \(A\) is either true or false of each element in \(A,\) while a function \(f : A \to \mathbb{B}\) evaluates to \(1\) or \(0\) on each element of \(A.\) The analogy between logical propositions and boolean-valued functions is close enough to adopt the latter as models of the former and simply refer to the functions \(f : A \to \mathbb{B}\) as propositions about the elements of \(A.\)

Differential Propositional Calculus • 4

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#DifferentialPropositionalCalculus • 4.3
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

The full set of propositions \(f : A \to \mathbb{B}\) contains a number of smaller classes deserving of special attention.

A #BasicProposition in the universe of discourse \([a_1, \ldots, a_n]\) is one of the propositions in the set \(\{a_1, \ldots, a_n\}.\) There are of course exactly \(n\) of these. Depending on the context, #BasicPropositions may also be called #CoordinatePropositions or #SimplePropositions.

#LogicalGraphs

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#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.

Differential Propositional Calculus • 4

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Jon Awbrey Nov 18, 2022

#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.

Differential Propositional Calculus • 4

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Jon Awbrey Nov 18, 2022

#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:

\[\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.\]

Differential Propositional Calculus • 4

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#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:

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

Differential Propositional Calculus • 4

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#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:

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

Differential Propositional Calculus • 4

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#DifferentialPropositionalCalculus • 4.9
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

In each family the rank \(k\) ranges from \(0\) to \(n\) and counts the number of positive appearances of #CoordinatePropositions \(a_1, \ldots, a_n\) in the resulting expression. For example, when \(n=3\) the #LinearProposition of rank \(0\) is \(0,\) the #PositiveProposition of rank \(0\) is \(1,\) and the #SingularProposition of rank \(0\) is \(\texttt{(}a_1\texttt{)} \texttt{(}a_2\texttt{)} \texttt{(}a_3\texttt{)}.\)

#Logic

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#CoordinatePropositions #SimplePropositions
#LinearPropositions #SingularPropositions

#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 —

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

Differential Propositional Calculus • 4

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