Talk:Exclusive Or with Itself

I was introduced to the Lindenbaum–Tarski algebra by LF.

I do not have enough experience with it but I will see if this helps (in fact I am making parts of it up):

Let $\left({\mathcal P, \oplus}\right)$ be the set of all WFFs of propositional calculus under $\oplus$.

$\equiv$ (which we use to denote $\iff$ to save space) is a congruence relation for $\left({\mathcal P, \oplus}\right)$.

The Lindenbaum–Tarski algebra of PropCalc under $\oplus$ is the quotient structure:


 * $\left({\mathcal P / \equiv, \oplus_{\equiv} }\right)$

From this point of view the equivalence class of $\bot$ (which may be denoted $\left[\!\left[{\bot}\right]\!\right]_\equiv$) is a zero element.

Now the question is what kind of structure do you know where $x \circ x = 0$?

--Jshflynn (talk) 15:14, 5 February 2013 (UTC)


 * What specific relevance has the above to this page? --prime mover (talk) 15:15, 5 February 2013 (UTC)


 * If you introduce this concept into the wiki then the phrasing on this page is at least one thing that you can change. --Jshflynn (talk) 15:25, 5 February 2013 (UTC)


 * From the point of view of basic propositional logic, it would be a mistake to import the fairly complicated techniques alluded to above. It may be worth adding as a second proof, but deliberately adding that level of sophistication for something which is basically trivial seems the wrong approach. --prime mover (talk) 15:39, 5 February 2013 (UTC)


 * This remark applies to more things, like the comment on Definition talk:Associative earlier. In fact, the whole point of the LT algebra is to make these apparent connections and similarities rigorous. Thus, while relevant, it's best to not burden the page a priori with this information. What Jshflynn seems to intend is to streamline the phrasing according to the algebraic analogues encountered - such can yield a viable approach though I don't know/recall what that's called in abstract algebra. --Lord_Farin (talk) 15:54, 5 February 2013 (UTC)


 * It reminded me of a B-Algebra for a moment.


 * For this page I would only want it to be a remark at the bottom "This may be interpreted as...".


 * Speaking beyond this page, I think the concept would lend itself well to the site because Proofwiki is quite strong in the abstract algebra department. --Jshflynn (talk) 16:01, 5 February 2013 (UTC)


 * The term "annihilator" rang a bell, and I did google for "rule of annihilation" but apparently not. As I say, "destroys copies of itself" is the language I have seen. But when it comes down to it, it doesn't really need a name, it's only a result, and a pretty trivial one at that. I'm beginning to wish I'd never bothered with it, I wasn't expecting it was going to cause this much work. --prime mover (talk) 16:10, 5 February 2013 (UTC)


 * As for Jshflynn's suggestion, I'm not going anywhere near it myself until we have some more bedrock (and I'm speaking beyond workaday extensions of trivial results for group theory being applied to monoids and semigroups). If someone has a source for that Lindenbaum-Tarski algebra, then it would be good to see it here.


 * Oh, and we are *not* strong on abstract algebra, we are pitifully weak. A lot of recent posts have pointed out how utterly rubbish it all is. --prime mover (talk) 16:10, 5 February 2013 (UTC)

Having tracked down the reference where I found "Exclusive Or desroys copies of itself" I have posted it up. Suggestions for rewording this will be considerd as long as a workable alternative is suggested. --prime mover (talk) 22:42, 5 February 2013 (UTC)