Logic Syllabus • Discussion

[Jon Awbrey]

Cf: 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 )

<QUOTE 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?
</QUOTE>

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 (x₁, ..., xₘ) to mean just one of x₁, ..., xₘ is false,
the form ((x₁), ..., (xₘ)) means just one of x₁, ..., xₘ 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”.

• ((x₁), (x₂)) = (x₁, x₂) = x₁ + x₂ = XOR(x₁, x₂).

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

Regards,

Jon

[Jon Awbrey]

Cf: Logic Syllabus • Discussion 2
https://inquiryintoinquiry.com/2023/06/04/logic-syllabus-discussion-2/

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/2328 )

<QUOTE JM:>
Is [the “just one true” operator] the same or different to xor?
I have read that xor is true when an odd number of variables are
true which would make it different. But I also read somewhere that
xor was true when only one is true.
</QUOTE>

Here's my syllabus entry on Exclusive Disjunction (xor), also known
as Logical Inequality, Symmetric Difference, and a few other names.
It's my best effort so far at straightening out the reigning confusions
and also at highlighting the links between the various notations and
visualizations we find in practice.

Exclusive Disjunction
• https://oeis.org/wiki/Exclusive_disjunction

Exclusive disjunction, also known as logical inequality or
symmetric difference, is an operation on two logical values,
typically the values of two propositions, which produces a
value of true just in case exactly one of its operands is true.

To say exactly one operand is true is to say the other is false,
which is to say the two operands are different, that is, unequal.

Expressed algebraically, x₁ + x₂ = 1 (mod 2).

Viewed in that light, it is tempting to think a natural extension of xor
to many variables x₁, …, xₘ will take the form x₁ + … + xₘ = 1 (mod 2).
And saying the bit sum of several boolean values is 1 is just another way
of saying an odd number of the values are 1.

Sums of that order form a perfectly good family of boolean functions,
ones we'll revisit in a different light, but their kinship to the family
of logical disjunctions is a bit more strained than uniquely natural.

Regards,

Jon (https://mathstodon.xyz/@Inquiry)