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 )
❝_❞ John Mingers ( https://groups.io/g/lawsofform/message/2326 )

❝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 as 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”.

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

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

Logic Syllabus • 2
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Operators
• https://oeis.org/wiki/Logic_Syllabus#Logical_operators

Logical Negation • https://oeis.org/wiki/Logical_negation
Logical NAND • https://oeis.org/wiki/Logical_NAND
Logical NNOR • https://oeis.org/wiki/Logical_NNOR
Logical Conjunction • https://oeis.org/wiki/Logical_conjunction
Logical Disjunction • https://oeis.org/wiki/Logical_disjunction
Exclusive Disjunction • https://oeis.org/wiki/Exclusive_disjunction
Logical Implication • https://oeis.org/wiki/Logical_implication
Logical Equality • https://oeis.org/wiki/Logical_equality

#Logic #LogicSyllabus #LogicalOperator #LogicalConnective
#Negation #NAND #NNOR #LogicalConjunction #LogicalDisjunction
#ExclusiveDisjunction #XOR #LogicalImplication #LogicalEquality

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

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

#LogicalOperators

#ExclusiveDisjunction #LogicalImplication
#LogicalConjunction #LogicalNAND
#LogicalDisjunction #Logical#NNOR
#LogicalEquality #Negation

To Be Continued ...

#DifferentialPropositionalCalculus • 2.5
• https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

A plus sign \(+\) may be used for #ExclusiveDisjunction, allowing the following equivalents.

\[\begin{matrix} x + y ~=~ \texttt{(} x \texttt{,} y \texttt{)} \\[6pt] x + y + z ~=~ \texttt{((} x \texttt{,} y \texttt{),} z \texttt{)} ~=~ \texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))} \end{matrix}\]

But note the last expressions are not equivalent to the triple bracket \(\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}.\)