The minor premise then presents evidence to establish one or the other of the exclusive elements. If the evidence supports A, then the negation of Not A follows logically (NOT Not A). If the evidence supports Not A, then the negation of A follows (NOT A). It is important to note that the "exclusive or" is not a suitable description of every possible "or" statement. Some situations allow for an "inclusive or". In these cases, the "or" means either A is true, or Not A is true, or BOTH are true. Either Joe lives in Philly, or Joe lives in Arizona, or Joe lives in both places at different times of the year.
It is also worth noting that some situations are not describable by EITHER the inclusive or the exclusive or. In cases where there are more than two options (or three in the case of the inclusive or), neither of the or statements offer a true description. For instance, it makes no sense to say, "Either it is 32 degrees or it is 64 degrees." That statement makes no sense because we know there is a very large number of possible temperatures above, below, and between 32 degrees and 64 degrees. That may seem obvious, but sometimes we see people try to reason as if there are only two alternatives when in fact there are more than two alternatives. That kind of reasoning can be deceptive. It is an example of a logical fallacy: it may appear to have the form of a logical argument, but it has no real logical power because the use of the "or" statement doesn't accurately correspond to the possibilities that exist in the world we inhabit.
Why all this prattle about logic? Because, if you have been paying attention to the news for the past eighteen months or so, you have witnessed the President of the United States in what appears to be multiple, discrete contradictions--situations that are accurately described by the "exclusive or" and in which Donald Trump has stated, or allegedly stated, that both A and Not A. I will focus on just three such situations. First, and most recently, there is the case of whether or not Trump spent a night in Moscow during the Miss Universe Pageant. This became an issue when a former British intelligence operative released a dossier that claimed the Russians had compromising video of Trump cavorting with prostitutes during his stay in Moscow for the Pageant. In response, President Trump allegedly told then-FBI Director James Comey that he did not stay overnight in Moscow for the Miss Universe Pageant. Comey documented these statements in contemporaneous memos prepared and distributed while he was still FBI Director. Just this week, when flight records were released showing Trump arrived in Moscow on Friday and departed on Sunday, the President said publicly that "of course" he stayed overnight in Moscow and that the Comey memos were a lie.  Second, there is the case of the infamous Access Hollywood video, in which Donald Trump (long before he ran for President) was caught on tape describing his practice of sexually assaulting women--grabbing and kissing them--without their consent. At the time this came out, candidate Trump acknowledged the comments and apologized. Later, shortly after becoming president, several reports surfaced in which he allegedly denied that it was his voice on the tape.  Third, in March of 2016 Trump introduce George Papadopoulus as one of five members of his foreign policy team, describing him as "an energy and oil consultant. Excellent guy." After Papadopoulus plead guilty to lying to the FBI about his contacts with Russians during the campaign, Trump tweeted that "few people knew the young, low-level volunteer named George, who has already proved to be a liar." 
I could go on, but the point is clear. If the reports of the President's comments about Moscow and Access Hollywood are true, then he has, at different times, asserted mutually contradictory things about his behavior. And viewers can watch Trump's position regarding George Papadopoulos change from citing him as a key member of his foreign policy team to calling him a "low-level volunteer." The President appears to be either lying, delusional, or horribly misinformed. His supporters appear willing to support him, no matter what. And Republicans in Congress have shown a troubling pattern of putting loyalty to their party and their party's president in front of what is good for the country. It is up to all of us to raise our voices to spotlight behaviors that, in my opinion, make Trump unfit to be President of the United States.