#differentialpropositionalcalculus — Public Fediverse posts

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

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

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to 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.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → 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 → B as propositions about the elements of A.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

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

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to 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.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → 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 → B as propositions about the elements of A.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

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

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to 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.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → 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 → B as propositions about the elements of A.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

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

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to 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.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → 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 → B as propositions about the elements of A.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

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

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to 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.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → 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 → B as propositions about the elements of A.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • Overview
• https://inquiryintoinquiry.com/2023/11/12/differential-propositional-calculus-overview-a/

❝The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.❞

— W. Ross #Ashby • An Introduction to #Cybernetics

Here's the outline of a sketch I wrote on “differential propositional calculi”, which extend propositional calculi by adding terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe. I wrote this as an intuitive introduction to differential logic, which is my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms. I'll be looking at ways to improve this draft as I serialize it to my blog.

Part 1 —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1

Casual Introduction
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction

Cactus Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Cactus_Calculus

Part 2 —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2

Formal_Development
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Formal_Development

Elementary Notions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Elementary_Notions

Special Classes of Propositions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Special_Classes_of_Propositions

Differential Extensions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions

Appendices —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices

References —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

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

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

#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 • 4.1
• 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.

#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.

#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.

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

#DifferentialPropositionalCalculus • 3.3
• https://inquiryintoinquiry.com/2020/02/24/differential-propositional-calculus-3/

The universe \(A^\bullet\) may be regarded as an ordered pair \((A, A^\uparrow)\) having the type \((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))\) and this last type designation may be abbreviated as \(\mathbb{B}^n\ +\!\!\to \mathbb{B}\) or even more succinctly as \([ \mathbb{B}^n ].\) For convenience, the #DataType of a finite set on \(n\) elements may be indicated by either of the notations \([n]\) or \(\mathbf{n}.\)