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we're on an assignment from god
part one: I am embarking on a project and/or quest. A document I'm working on at work drove me around the twist this afternoon, and I announced on Facebook that I'm going on a quest to eliminate "and/or". BECAUSE IT MEANS "OR". It looks sloppy and adds no meaning. I delete "and/" every time I see it (unless it's appropriate to delete "/or", of course, but that's rare), and am prepared to go seven rounds with whatever analyst or project manager wants to argue with me that it should go back in. GAH. This job is not helpful if what you want to eliminate is your tendency to obsess over pet peeves.
part two: I am not embarking on other kinds of journey. My first instinct, when I went on Facebook to rant about "and/or", was to announce that I was going on a crusade against it. Naturally I decided not to use that word, because unpleasant implications. The next option that occurred to me was jihad. No better. (The very first person to "like" my post was a former co-worker who happens to be Muslim—from Pakistan—so I had immediate reason to be even gladder I'd avoided both those terms.)
part three: Conflation. A dude I know from curling commented that he'd join my team opposing "and/or" if I would work to bring back the Oxford comma. Of course I had to insist that IT HASN'T GONE ANYWHERE. And then as I was walking home from work it occurred to me that another thing I'm not on vis-a-vis "and/or" is a mission. But who was on a mission? The Blues Brothers. And what else did they say? "It's a hundred and six miles to Chicago, we've got a full tank of gas, a half pack of cigarettes, it's dark, and we're wearing sunglasses." I mean [apart from the coordination of unlike categories or failure of parallel structure—there should actually be an "and" rather than a comma after "gas", but never mind], I can hear the Oxford comma in that. Can't you?
HIT IT.
part two: I am not embarking on other kinds of journey. My first instinct, when I went on Facebook to rant about "and/or", was to announce that I was going on a crusade against it. Naturally I decided not to use that word, because unpleasant implications. The next option that occurred to me was jihad. No better. (The very first person to "like" my post was a former co-worker who happens to be Muslim—from Pakistan—so I had immediate reason to be even gladder I'd avoided both those terms.)
part three: Conflation. A dude I know from curling commented that he'd join my team opposing "and/or" if I would work to bring back the Oxford comma. Of course I had to insist that IT HASN'T GONE ANYWHERE. And then as I was walking home from work it occurred to me that another thing I'm not on vis-a-vis "and/or" is a mission. But who was on a mission? The Blues Brothers. And what else did they say? "It's a hundred and six miles to Chicago, we've got a full tank of gas, a half pack of cigarettes, it's dark, and we're wearing sunglasses." I mean [apart from the coordination of unlike categories or failure of parallel structure—there should actually be an "and" rather than a comma after "gas", but never mind], I can hear the Oxford comma in that. Can't you?
HIT IT.

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Sorry.
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If you got a memo that said "Anyone who is wearing blue or black today will get a gift certificate for a free latte," and you're wearing blue jeans and a black shirt, you wouldn't conclude that you don't get a gift certificate because you're on "and" rather than "or". That's not how "or" works.
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Three options: Jane and Jill, Jane alone, Jill alone. Not two options: Jane alone, Jill alone.
ETA: I don't know what you mean by XOR.
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And the reason the Boolean search needs XOR is that AND entails OR. That is: if I ask for "AUTHOR=Jane OR Jill" and I get a result written by both of these women, the search engine has not failed to give me what I asked for. If I ask you for strawberries or blueberries and you give me a dish of strawberries, blueberries, raspberries, blackberries, and whipped cream, you have not failed to give me what I asked for. If you hope Jane or Jill comes to your party and they both show up, you've got what you hoped for and more (assuming they don't hate each other's guts, in which event you'd probably have used a word like "either" in there somewhere).
When you have two conditions, P and Q, there are four possible outputs:
1. Not P, not Q
2. Yes P, not Q
3. Not P, yes Q
4. Yes P, yes Q
"P or Q" is true if P is true or if Q is true. So in this array, 1 does not satisfy "P or Q", but 2-4 do, just as you say. It's true that 4 happens to also satisfy "P and Q", but that doesn't mean it doesn't satisfy "P or Q". Put another way: "and/or" means "or", because "and" entails "or".
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4. Yes P, yes Q
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CMOS:
and/or. Avoid this Janus-faced term. It can often be replaced by and or or with no loss in meaning. Where it seems needed {take a sleeping pill and/or a warm drink}, try or ... or both {take a sleeping pill or a warm drink or both}. But think of other possibilities {take a sleeping pill with a warm drink}. [See Bryan A. Garner, Legal Writing in Plain English (Chicago: University of Chicago Press, 2001), 112–113.]
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In terms of programming, no one cares about the programming language used if it takes half an hour to run a very verbose code (just because it is so extremely precise) if everyone knows it could be condensed down into something leaner if fuzzier that runs in 10 seconds as opposed to half an hour. Just as some kind of a logic sorter could be added to ensure the faster, more efficient program won't loop on the fuzziness of an and/or type of declaration that isn't strictly true, the and/or declaration is processed by the human brain in much the same way...our brains can and will sort out the inherent fuzziness.
(take a sleeping pill and/or a warm drink)
Let's assume all of the following are strictly true:
You can take a sleeping pill or a warm drink.
You can take a sleeping pill and a warm drink.
So at the minimum a sleeping pill or a warm drink will help but you might want to indulge in both for maximum effect. Rather than *say* all that, and/or is used. Efficiency is increased and the reader can parse out the best meaning as it applies to their own situation. Maybe one person just needs a sleeping pill or a warm drink. But another person in worse shape might just need both. What is wrong with using and/or to simplify the language used around this?
Let's look at the stated alternative...
"But think of other possibilities {take a sleeping pill with a warm drink}."
Really? So a lawyer or someone who tries to simplify law writing thinks someone should be instructed to do both simply to avoid the and/or construct and/or the obvious tediousness of reading a sentence like "take a sleeping pill or a warm drink or both"? Any lawyer worth their salt should know this could result in a really big lawsuit should those instructions happen to be entirely overkill for the individual's particular situation. Why is no one talking about that?