#propositionalequationreasoningsystems — Public Fediverse posts

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

Praeclarum Theorema

Introduction

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus noted and named by G.W. Leibniz, who stated and proved it in the following manner.

If a is b and d is c, then ad will be bc.
This is a fine theorem, which is proved in this way:
a is b, therefore ad is bd (by what precedes),
d is c, therefore bd is bc (again by what precedes),
ad is bd, and bd is bc, therefore ad is bc.  Q.E.D.

— Leibniz • Logical Papers, p. 41.

Expressed in contemporary logical notation, the theorem may be written as follows.

Expressed as a logical graph under the existential interpretation, the theorem takes the shape of the following formal equivalence or propositional equation.

And here’s a neat proof of that nice theorem —

The steps of the proof are replayed in the following animation.

Reference

• Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

Readings

• Jon Awbrey • Propositional Equation Reasoning Systems
• John F. Sowa • Peirce’s Rules of Inference

Resources

• Logical Graphs
• Logical Graphs • First Impressions
• Logical Graphs • Formal Development
• Metamath Proof Explorer • Praeclarum Theorema
• Frithjof Dau • Animated Proof of Leibniz’s Praeclarum Theorema

cc: FB | Logical Graphs • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science