Logic Syllabus • 5
• https://inquiryintoinquiry.com/logic-syllabus/

Related Articles
• https://oeis.org/wiki/Logic_Syllabus#Related_articles

Cactus Language • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Futures Of Logical Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs
Differential Propositional Calculus • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
Differential Logic • https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Propositions As Types Analogy • https://oeis.org/wiki/Propositions_As_Types_Analogy
Propositional Equation Reasoning Systems • https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems
Prospects for Inquiry Driven Systems • https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems
Introduction to Inquiry Driven Systems • https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems
Inquiry Driven Systems • Inquiry Into Inquiry • https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

#Logic #LogicSyllabus #CactusLanguage #LogicalGraphs #DifferentialLogic
#DifferentialPropositionalCalculus #DifferentialLogicAndDynamicSystems
#PropositionsAsTypesAnalogy #PropositionalEquationReasoningSystems
#Inquiry #InquiryDrivenSystems #InquiryIntoInquiry #DynamicalSystems

Logic Syllabus • 5
• https://inquiryintoinquiry.com/logic-syllabus/

Related Articles
• https://oeis.org/wiki/Logic_Syllabus#Related_articles

Cactus Language • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Futures Of Logical Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs
Differential Propositional Calculus • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
Differential Logic • https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Propositions As Types Analogy • https://oeis.org/wiki/Propositions_As_Types_Analogy
Propositional Equation Reasoning Systems • https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems
Prospects for Inquiry Driven Systems • https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems
Introduction to Inquiry Driven Systems • https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems
Inquiry Driven Systems • Inquiry Into Inquiry • https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

#Logic #LogicSyllabus #CactusLanguage #LogicalGraphs #DifferentialLogic
#DifferentialPropositionalCalculus #DifferentialLogicAndDynamicSystems
#PropositionsAsTypesAnalogy #PropositionalEquationReasoningSystems
#Inquiry #InquiryDrivenSystems #InquiryIntoInquiry #DynamicalSystems

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

Figure 10. #VennDiagrams for #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

#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

#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:

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

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

#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 two values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the minimal distinct regions of the universe, represented by single cells of the corresponding #VennDiagram.

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

#DifferentialPropositionalCalculus • 6.2
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

Figure 9. #VennDiagrams for the #PositivePropositions on 3 Variables
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

Rank 3 (Top). #VennDiagram for the #BooleanProduct or #LogicalConjunction \(pqr.\)

Rank 2. Venn Diagrams for the 3 #BooleanProducts \(pr,\) \(qr,\) \(pq.\)

Rank 1. Venn Diagrams for the 3 #BasicPropositions \(p,\) \(q,\) \(r.\)

Rank 0 (Bottom). Venn Diagram for the #ConstantFunction or the #ConstantProposition \(1.\)

#DifferentialPropositionalCalculus • 6.1
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p, q, r\) there are \(2^3 = 8\) #PositivePropositions. Their #VennDiagrams are shown in Figure 9.

\(\text{Figure 9. Positive Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

#DifferentialPropositionalCalculus • 6
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

The #PositivePropositions \(\{p:\mathbb{B}^n\to \mathbb{B}\}=(\mathbb{B}^n \xrightarrow{p}\mathbb{B})\) may be written as products:

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

To get a sense of this family's place we'll next draw the #VennDiagrams for the 3 variable case.

#Logic #LogicalGraphs
#DifferentialLogic

#DifferentialPropositionalCalculus • 5.6
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

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

At the bottom of Figure 8 is #VennDiagram for the #LinearProposition of rank 0, the constant \(0\) function or the everywhere false proposition, expressed in #CactusSyntax by the form \(\texttt{(}~\texttt{)}\) or in algebraic form by a simple \(0.\)

\(\text{Figure 8.4 Venn Diagram for}~\texttt{(}~\texttt{)}\)

#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\)

#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}~\(\texttt{(}p\texttt{,}q\texttt{)}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagram-e280a2-p-q-r-e280a2-p-q.jpg

#DifferentialPropositionalCalculus • 5.3
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

At the top of Figure 8 is the #VennDiagram for the #LinearProposition of rank 3, which may be expressed by any one of the following 3 forms:

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

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

#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

#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 \(1+1=0.\)

#Logic #DifferentialLogic

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

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general. We can do this by recruiting our visual imaginations and drawing up a sufficient budget of #VennDiagrams for each family of propositions. The case for 3 variables is exemplary enough for a start.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

Survey of #DifferentialLogic
• https://inquiryintoinquiry.com/2022/11/20/survey-of-differential-logic-4/

This is a Survey of blog and wiki posts on Differential Logic, material I plan to develop toward a more compact and systematic account.

#Logic
#LogicalGraphs
#CactusLanguage
#QualitativeDynamics
#PropositionalCalculus
#LogicalTransformations
#MinimalNegationOperators
#DiscreteDynamicalSystems
#TransformationsOfDiscourse
#DifferentialPropositionalCalculus
#DifferentialAnalyticTuringAutomata

#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

#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 these two kinds of propositions, the linear and 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 • 4.9
• 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