#Truthtable

2023-11-24
2023-06-02

Logic Syllabus • Discussion 1
inquiryintoinquiry.com/2023/06

Re: Logic Syllabus ( inquiryintoinquiry.com/logic-s )
Re: Laws of Form ( groups.io/g/lawsofform/topic/l )
Re: John Mingers ( groups.io/g/lawsofform/message )

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
oeis.org/wiki/Minimal_negation

Related Truth Tables
oeis.org/wiki/Minimal_negation

Genus, Species, Pie Charts, Radio Buttons
inquiryintoinquiry.com/2021/11

Related Discussions
inquiryintoinquiry.com/?s=Radi

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

2022-12-02

#ThemeOneProgram#JetsAndSharks 2.1
inquiryintoinquiry.com/2022/08

Our #CactusGraph bears a vocabulary of \(41\) #LogicalTerms, each denoting a #BooleanVariable, so our proposition, call it \(``q",\) is a #BooleanFunction \(q:\mathbb{B}^{41}\to\mathbb{B}.\) Since \(2^{41}=2,199,023,255,552,\) its #TruthTable has \(>\) 2 trillion rows and its #VennDiagram has that many cells. There are \(2^{2^{41}}\) functions \(f:\mathbb{B}^{41}\to\mathbb{B}\) and \(q\) is just one of them.

#Logic #LogicalGraphs

Client Info

Server: https://mastodon.social
Version: 2025.04
Repository: https://github.com/cyevgeniy/lmst