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Logic Syllabus • Discussion 1
• https://inquiryintoinquiry.com/2023/06/02/logic-syllabus-discussion-1/

Re: Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
Re: Laws of Form ( https://groups.io/g/lawsofform/topic/logic_syllabus/99218507 )
Re: John Mingers ( https://groups.io/g/lawsofform/message/2326 )

JM: ❝In a previous post you mentioned the minimal negation operator. Is there also the converse of this, i.e. an operator which is true when exactly one of its arguments is true? Or is this just XOR?❞

Yes, the “just one true” operator is a very handy tool. We discussed it earlier under the headings of “genus and species relations” or “radio button logic”. Viewed in the form of a venn diagram it describes a partition of the universe of discourse into mutually exclusive and exhaustive regions.

Reading \(\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_m \texttt{)}\) to mean just one of \(x_1, \ldots, x_m\) is false, the form \(\texttt{((} x_1 \texttt{),} \ldots \texttt{,(} x_m \texttt{))}\) means just one of \(x_1, \ldots, x_m\) is true.

For two logical variables, though, the cases “condense” or “degenerate” and saying “just one true” is the same thing as saying “just one false”.

\[\texttt{((} x_1 \texttt{),(} x_2 \texttt{))} = \texttt{(} x_1 \texttt{,} x_2 \texttt{)} = x_1 + x_2 = \mathrm{xor} (x_1, x_2).\]

There's more information on the following pages.

Minimal Negation Operators
• https://oeis.org/wiki/Minimal_negation_operator

Genus, Species, Pie Charts, Radio Buttons
• https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/

Related Discussions
• https://inquiryintoinquiry.com/?s=Radio+Buttons

#Logic #LogicSyllabus #BooleanDomain #BooleanFunction #BooleanValuedFunction
#Peirce #LogicalGraph #MinimalNegationOperator #ExclusiveDisjunction #XOR
#CactusLanguage #PropositionalCalculus #RadioButtonLogic #TruthTable

Logic Syllabus • 3
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Concepts
• https://oeis.org/wiki/Logic_Syllabus#Logical_concepts

#Logic #LogicSyllabus #Ampheck #BooleanDomain #BooleanFunction #BooleanValuedFunction
#DifferentialLogic #LogicalGraph #MinimalNegationOperator #MultigradeOperator
#ParametricOperator #PeircesLaw #PropositionalCalculus #SoleSufficientOperator
#TruthTable #UniverseOfDiscourse #ZerothOrderLogic

#LogicSyllabus
• https://inquiryintoinquiry.com/logic-syllabus/

This page serves as a focal node for a collection of related resources.

#Logic #LogicalOperators

#LogicalConjunction #LogicalDisjunction
#ExclusiveDisjunction #LogicalEquality
#LogicalImplication #LogicalNegation
#LogicalNAND #LogicalNNOR

#LogicalConcepts

#Ampheck #SoleSufficientOperator
#BooleanDomain #UniverseOfDiscourse
#LogicalGraph #PropositionalCalculus
#MinimalNegationOperator #TruthTable
#BooleanFunction #BooleanValuedFunction

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

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

#BooleanDomain

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