Jon AwbreyLogic 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
Related Truth Tables
• https://oeis.org/wiki/Minimal_negation_operator#Truth_tables
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
