Logic Syllabus • Discussion 2
• https://inquiryintoinquiry.com/2023/06/04/logic-syllabus-discussion-2/
Re: Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
Re: John Mingers ( https://groups.io/g/lawsofform/message/2328 )
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.❞
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, \(x_1 + x_2 = 1 ~ (\text{mod}~ 2).\)
Viewed in that light, it is tempting to think a natural extension of XOR to many variables \(x_1, \ldots, x_m\) will take the form \(x_1 + \ldots + x_m = 1 ~ (\text{mod}~ 2).\) 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.