Jon AwbreyNov 14, 2022

#DifferentialPropositionalCalculus • Overview
• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

By way of introduction to #DifferentialLogic we begin a chapter on #DifferentialPropositionalCalculi, which augment #PropositionalCalculi with terms for describing aspects of change and difference, for example, processes taking place in a #UniverseOfDiscourse or #LogicalTransformations mapping a #SourceUniverse to a #TargetUniverse.

#Logic #LogicalGraphs #DifferentialLogic
#DiscreteDynamics #QualitativeDynamics

Differential Propositional Calculus • Overview

Inquiry Into Inquiry
Show threadJon AwbreyNov 15, 2022

#DifferentialPropositionalCalculus • 2
https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

#CactusCalculus —

Table 6 introduces a #Syntax for #PropositionalCalculus based on 2 families of #MultigradeLogicalConnectives.

Table 6. #Syntax and #Semantics of a Calculus for #PropositionalLogic
https://inquiryintoinquiry.files.wordpress.com/2020/02/syntax-and-semantics-of-a-calculus-for-propositional-logic.png

Related Subjects —
#Peirce #Semiotics #BooleanFunctions
#Logic #LogicalGraphs #DifferentialLogic
#GraphTheory #CactusGraphs #CactusLanguage
#MinimalNegationOperators #PaintedAndRootedCacti

Differential Propositional Calculus • 2

Inquiry Into Inquiry
Show threadJon Awbrey

#DifferentialPropositionalCalculus • 2.1
• https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

Table 6 outlines a #Syntax for #PropositionalCalculus based on two types of #LogicalConnectives, both of variable \(k\)-ary scope.
• https://inquiryintoinquiry.files.wordpress.com/2020/02/syntax-and-semantics-of-a-calculus-for-propositional-logic.png

In the first type of connective a bracketed list of propositional expressions in the form \(\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\) indicates exactly one of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) is false.

Differential Propositional Calculus • 2

Inquiry Into Inquiry
Nov 16, 2022 at 1:40pmWeb
Show threadJon AwbreyNov 16, 2022

#DifferentialPropositionalCalculus • 2.2
• https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

Table 6 outlines a #Syntax for #PropositionalCalculus based on two types of #LogicalConnectives, both of variable \(k\)-ary scope.
• https://inquiryintoinquiry.files.wordpress.com/2020/02/syntax-and-semantics-of-a-calculus-for-propositional-logic.png

In the second type of connective a concatenation of propositional expressions in the form \(e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k\) indicates all the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) are true, in other words, their #LogicalConjunction is true.

Differential Propositional Calculus • 2

Inquiry Into Inquiry
Show threadJon AwbreyNov 16, 2022

#DifferentialPropositionalCalculus • 2.3
• https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

All other #PropositionalConnectives can be expressed in terms of the above 2 forms. The concatenation form is dispensable in terms of the bracket form but useful to maintain as an abbreviation for more complicated bracket expressions.

In contexts where parentheses are needed for other purposes “teletype” parentheses \(\texttt{(} \ldots \texttt{)}\) or barred parentheses \( (\!| \ldots |\!) \) may be used for logical operators.

Differential Propositional Calculus • 2

Inquiry Into Inquiry
Show threadJon AwbreyNov 16, 2022

#DifferentialPropositionalCalculus • 2.4
• https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

#CactusCalculus Conventions —

The briefest expression for #LogicalTruth is the #EmptyWord, denoted \(\varepsilon\) or \(\lambda\) in #FormalLanguages, where it forms the #IdentityElement for #Concatenation. It may be given visible expression in this context by means of the logically equivalent form \(\texttt{((} ~ \texttt{))},\) or, especially if operating in an algebraic context, by a simple \(1.\)

Differential Propositional Calculus • 2

Inquiry Into Inquiry