## OR and XOR

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Cartofel
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### OR and XOR

Hi,

I'm posting this here because I don't think it would fit in Coding... Apologies if this has been dissected here many times before...

When I started programming, one of the first things that bugged me (and which didn't seem to be language-specific) was the way OR and XOR are used. In Boolean algebra, the function of OR is called disjunction (which is given the + sign, as that pretty much describes what it does) -- if any or both of the two values is true, their disjunction is is true. So, basically, the only time you can get a false out of a disjunction is if both values are false. XOR is eXclusive OR (or parity in Boolean algebra, given a + in a circle), in that if both values are true their parity is false (I suppose the exclusive part is because the values have to be mutually exclusive).

Now, this is the bit that gets my beef (as a linguist who programs as well) -- I don't think that the conjunction "or" describes disjunction.
Take, for example, this question:

"Would you like cake or ice cream?"

What could your possible answers be? Obviously, "Cake, please" or "Ice cream, please", or words to that effect. Would you expect someone to say, "Both!"? Personally, I would think that that person was breaking the rules, or something. (I remember at a party I was at once, when someone did something similar, someone else said, "What? It's cake or ice cream!", which I think proves my point.) I think that if your host wanted to offer that choice, they would probably say,

"Would you like cake, or ice cream, or both?"

This might only be true for speakers of UK English, but I think not -- have you noticed how common the ugly "and/or" is in legal texts? Doesn't this imply that "or" by itself wouldn't mean that both were an option? (I've also seen this in French texts, where they seem to use "et/ou" in exactly the same way.)

I think that, in fact, "or" as it is used in actual English is really parity, and that the real alien idea is the non-exclusive (can we please say "inclusive", before my eyes start bleeding?) version of "or". So maybe we should have OR for parity, and something like IOR (Inclusive OR) for disjunction.

So... yeah. Thanks for your time...

EDIT: Terminological urk
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Moo
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### Re: OR and XOR

"This function is true if x is true or y is true".

When you phrase it this way there is no ambiguity. The problem is that you are relying on words subject to various interpretations and meanings; words that rely on context to make sense because they are language words, and not bound by the same logic as mathematics.

"Would you like cake or ice cream" implies an exclusivity. If you were allowed both things you might not be offered it in that way. Think of it this way: in English, or is a broad term that can be used for either a Boolean or or xor, depending on context or further clarification.

edit:

Wrt the "and/or", it is to cover legal asses because of the way "or" can be interpreted to mean both "or" or "xor", and legal and other important texts need to eliminate any potential ambiguity.

Which, I suppose, is kind of what the OP is saying, that he dislikes this possible ambiguity; but I'm afraid that is what language is. And rightly so. Without ambiguity you couldn't make puns, entendres or "that's what she said" jokes! We've relied on context and intonation to add meaning beyond words long before Boolean algebra was formalised. I am curious whether this problem has been solved in Lobjan?
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Koboldskind
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### Re: OR and XOR

I don't think the "or" in natural language necessarily implies exclusivity, at least in englishi feel it strongly depends on intonation. (I say feel here because i'm relying on my language intuition, not on any rules that I am aware of):

To the sentence "Would you like cake or icecream?" with the standard intonation for a question, "Both, actually" appears to be a perfectly valid, if maybe impolite, answer. (=> logical OR)

If you change it to "Would you like CAKE or ICECREAM?" with an emphasis on the 2 desserts in question (think the way the flight attendant on a plane would ask), then the speaker expects you to make an either-or choice. (=> logical XOR)

Please excuse me if that example is nonsense, I am not a native english speaker, but this at least is my perception.

So it boils down to (as always) logical expression not really mapping well into natural language, because spoken natural language transports much more information than just the semantic content of a phrase.

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### Re: OR and XOR

Wrt the "and/or", it is to cover legal asses because of the way "or" can be interpreted to mean both "or" or "xor", and legal and other important texts need to eliminate any potential ambiguity.

Reminds me... I had just put down my copy of Strunk and White:

The Elements of Style wrote:And/or. A device, or shortcut, that damages a sentence and often leads to confusion or ambiguity.

Harsh. "___ or ___ or both" is what they recommend.

Anyway, I always relied on intonation to distinguish between the two uses of OR that English has given it. More often than not it was meant as the inclusive.

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### Re: OR and XOR

Linguistically the idea is that "or" really does, semantically, mean inclusive-or, but when people mean "both", they use a conjunction instead, from which we can often infer that people are using "or" exclusively (because they would have said "and" if they meant both, which is simpler).

This sort of pragmatic reasoning, "the sentence is ambiguous, but if they intended meaning B, they would have said it in a different, simpler way, so they must have intended meaning A," has a name in linguistics, but it's early in the morning, so I'm blanking on it.

I'm also blanking on the standard examples used in intro linguistics classes to show people that "or" really is inclusive. Try posting in the the linguistics forum, someone there will remember these things =)
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Rysto
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### Re: OR and XOR

There are plenty of instances in English where inclusive or is intended. For example, 'I drive to work when it is raining or it's below 5 degrees Celcius outside'.

If it's raining and the temperature is 1 degree, does he drive to work? Of course he does.

EternalVortex
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### Re: OR and XOR

Inclusive or is common in English. For example, "the grass is wet if it rains or the sprinker is on".

However, I don't believe any sentence in English uses the exclusive or, because no sentence in English is false if both parts are true.

quintopia
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### Re: OR and XOR

EternalVortex wrote:However, I don't believe any sentence in English uses the exclusive or, because no sentence in English is false if both parts are true.

Sometimes in English "either" or "one of" are used to select the exclusive or. But it's still possible to have an exclusive or without them.

For instance: This car comes in red, black, or white.

It is not meant that the car comes in red AND black, so the statement would be false if there existed a model that was both red and black.

Berengal
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### Re: OR and XOR

Iterestingly, "this car comes in red or white" can mean the same as "this car comes in red and white." Both can be interpreted as "this car comes in either red or white but not both." (Both can also be interpreted to mean different things.)
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### Re: OR and XOR

iirc latin has seperate words for exclusive or and inclusive or, its just an english thing.
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Cartofel
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### Re: OR and XOR

evilbeanfiend wrote:iirc latin has seperate words for exclusive or and inclusive or, its just an english thing.

habet "aut" et "vel", nescio qualem discrepantiam esse.
(io!)
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roc314
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### Re: OR and XOR

quintopia wrote:
EternalVortex wrote:However, I don't believe any sentence in English uses the exclusive or, because no sentence in English is false if both parts are true.

Sometimes in English "either" or "one of" are used to select the exclusive or. But it's still possible to have an exclusive or without them.

For instance: This car comes in red, black, or white.
If this was a logical xor, then one of the possible choices would be red and black and white (xor is true if an odd number of the choices are true--not only 1 (check the truth tables)).

Usually the colloquial exclusive or in English speech means "one (and not more) of the following". If there are only two choices, then this is equivalent to the logical xor.
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### Re: OR and XOR

evilbeanfiend wrote:iirc latin has seperate words for exclusive or and inclusive or, its just an english thing.

The way you put it it sounds like English is unique in having an ambiguous "or". I don't have any statistics at hand, but I find it pretty likely that the majority of all languages has ambiguous "or" at least as an option.

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### Re: OR and XOR

Cartofel wrote:
evilbeanfiend wrote:iirc latin has seperate words for exclusive or and inclusive or, its just an english thing.

habet "aut" et "vel", nescio qualem discrepantiam esse.
(io!)

N.B. "vel" is why the symbol sometimes encountered for logical OR is [imath]a\vee b[/imath] . As for "aut"...

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quintopia
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### Re: OR and XOR

roc314 wrote:
quintopia wrote:
EternalVortex wrote:However, I don't believe any sentence in English uses the exclusive or, because no sentence in English is false if both parts are true.

Sometimes in English "either" or "one of" are used to select the exclusive or. But it's still possible to have an exclusive or without them.

For instance: This car comes in red, black, or white.
If this was a logical xor, then one of the possible choices would be red and black and white (xor is true if an odd number of the choices are true--not only 1 (check the truth tables)).

I can't find my trinary logic tables right now, but I'm pretty sure that a trinary xor works out to mean one (and not more) is true(1) or true(2). Notice that I wrote "red, white, or black," not "(red or white) or black," clearly indicating my use of trinary logic. [/snark]

roc314
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### Re: OR and XOR

quintopia wrote:I can't find my trinary logic tables right now, but I'm pretty sure that a trinary xor works out to mean one (and not more) is true(1) or true(2). Notice that I wrote "red, white, or black," not "(red or white) or black," clearly indicating my use of trinary logic. [/snark]
Point taken, but I was pointing out that xor is just as vague as or. Unless it's made obvious which xor is being used, it's confuses the meaning. The clearest way to express the idea of "one and only" is to say "one and only one". Or, xor, etc. are too easily confused.
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Berengal
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### Re: OR and XOR

You wouldn't use xor in an unclear context. A car salesman, no matter how well versed in logic, and knowing that I too know logic, would never say "This car comes in xor(blue, yellow, red)". Usually when we're using xor, it's a predicate that takes exactly two terms, so xor(blue, red, yellow) would not compute, and this should be pretty clear from context. Note that trinary xor also only takes two terms, but those terms may evaluate to one of three values, not just two.
Xor for multiple arguments does exist, and it's defined as true iff an odd number of arguments are true. Since xor is associative, "xor(a, b, c)" is the the same as "a xor b xor c", so the arity 2 one is pretty much all we need.
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### Re: OR and XOR

And I think we just crossed the point of taking this stuff way too seriously

DubioserKerl
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### Re: OR and XOR

Koboldskind wrote:To the sentence "Would you like cake or icecream?" with the standard intonation for a question, "Both, actually" appears to be a perfectly valid, if maybe impolite, answer. (=> logical OR)

If you change it to "Would you like CAKE or ICECREAM?" with an emphasis on the 2 desserts in question (think the way the flight attendant on a plane would ask), then the speaker expects you to make an either-or choice. (=> logical XOR)

Actually, if you used the "or" and "xor" in the language just like in logic, the only valid answers could be "true" and "false"; represented by "yes" and "no".

So, the answer to "Would you like cake or icecream" would be "yes", if the logical "or" is implied and you want at least one of the deserts. The answer would be "no" if you use the "xor" semantics and actually want both. If you answer "yes" to a xor-choice, no one would know what desert you want until it is clarified by additional information.

Now please (flame XOR praise) me.

Grop
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### Re: OR and XOR

There was an interesting article on Language Log about or being always exclusive or not. The examples given are funny and convincing.

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### Re: OR and XOR

DubioserKerl wrote:If you answer "yes" to a xor-choice, no one would know what desert you want until it is clarified by additional information.

They would know that you don't want both nor none.
Additionally, If you replied true to binary "or" and "xor" you would get either an icecream cake or a sugarless cake with ice topping.

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### Re: OR and XOR

DubioserKerl wrote:
Koboldskind wrote:To the sentence "Would you like cake or icecream?" with the standard intonation for a question, "Both, actually" appears to be a perfectly valid, if maybe impolite, answer. (=> logical OR)

If you change it to "Would you like CAKE or ICECREAM?" with an emphasis on the 2 desserts in question (think the way the flight attendant on a plane would ask), then the speaker expects you to make an either-or choice. (=> logical XOR)

Actually, if you used the "or" and "xor" in the language just like in logic, the only valid answers could be "true" and "false"; represented by "yes" and "no".

So, the answer to "Would you like cake or icecream" would be "yes", if the logical "or" is implied and you want at least one of the deserts. The answer would be "no" if you use the "xor" semantics and actually want both. If you answer "yes" to a xor-choice, no one would know what desert you want until it is clarified by additional information.

Now please (flame XOR praise) me.

When it comes to deserts I personally prefer Arrakis.
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### Re: OR and XOR

Berengal wrote:Iterestingly, "this car comes in red or white" can mean the same as "this car comes in red and white." Both can be interpreted as "this car comes in either red or white but not both." (Both can also be interpreted to mean different things.)

No, if the car salesmen is saying "this car comes in red and white" and he means "or" then it simply means he made a logical error. In proper English "and" should NEVER be used in place of "or".

Goplat
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### Re: OR and XOR

Cartofel wrote:I don't think that the conjunction "or" describes disjunction.
Take, for example, this question:

"Would you like cake or ice cream?"
Here "or" is just being used to delimit a list of choices. It is not being used as any kind of logical operator, because if it were the question would be yes-or-no.

When "or" is used as a logical operator in English, as far as I know it is always inclusive.

monroetransfer
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### Re: OR and XOR

OR: would you like cake or ice-cream?
XOR: would you like cake, or ice-cream?

poirelli
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### Re: OR and XOR

Nobody offers cake or ice cream anyway.

It's always: "Would you like cake and ice cream?"

or perhaps "How much cake and ice cream would you like?"

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### Re: OR and XOR

I haven't read the thread, but this entire discussion is rather ridiculous. Or in natural language is not in one-to-one correspondence with either the logical OR or XOR. The entire idea of representing English with formal logic won't work due to pragmatic factors. You'll always be able to find exceptions to any would-be rule you can come up with.

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### Re: OR and XOR

Berengal wrote:Iterestingly, "this car comes in red or white" can mean the same as "this car comes in red and white." Both can be interpreted as "this car comes in either red or white but not both." (Both can also be interpreted to mean different things.)

No, if the car salesmen is saying "this car comes in red and white" and he means "or" then it simply means he made a logical error. In proper English "and" should NEVER be used in place of "or".

In that case the "and sentase and "or" sentanses actually are syntatically different, but both valid under descriptive rules of english.

It is worth emphasisng is that "This car" does not refer to an indivual car in this context, but a catagory of cars which "this one" exemplifies (probably the year + make). If taken literally you'd obviously just look at the car in front of you.

The "and" sentanse is bascially saying: "This car can come in red, and this car can come in white" Reffering to the catagory.

While the "or" sentanse is saying: "I can sell you this car having paint of the colors red xor white" Reffering to any member of the catagory.
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### Re: OR and XOR

Qoppa wrote:I haven't read the thread, but this entire discussion is rather ridiculous. Or in natural language is not in one-to-one correspondence with either the logical OR or XOR. The entire idea of representing English with formal logic won't work due to pragmatic factors. You'll always be able to find exceptions to any would-be rule you can come up with.

this.

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### Re: OR and XOR

monroetransfer wrote:OR: would you like cake or ice-cream?
XOR: would you like cake, or ice-cream?

I agree with this.

Of course the alternative is that we all start going around asking things like "would you like cake xor ice-cream?"
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### Re: OR and XOR

akashra wrote:
monroetransfer wrote:OR: would you like cake or ice-cream?
XOR: would you like cake, or ice-cream?

I agree with this.

Of course the alternative is that we all start going around asking things like "would you like cake xor ice-cream?"

This is what I would say if someone had just answered the standard version with "yes."

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### Re: OR and XOR

quintopia wrote:This is what I would say if someone had just answered the standard version with "yes."

Yeah. See, I'm the kind of smart-ass that when asked "Would you like cake or ice-cream" would answer "yes". It's like asking "do you know the time?".
You need to be specific, people. Lawyers hate people like us though, so that's a plus
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### Re: OR and XOR

akashra wrote:It's like asking "do you know the time?".
You need to be specific, people. Lawyers hate people like us though, so that's a plus

Another fun example is "Do you know how to get to <place>?" versus "How do you get to <place>?" If someone just asks the second I'l always give them precise directions, regarless if the answer to the first question was yes or no.
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### Re: OR and XOR

akashra wrote:It's like asking "do you know the time?".
You need to be specific, people. Lawyers hate people like us though, so that's a plus

Another fun example is "Do you know how to get to <place>?" versus "How do you get to <place>?" If someone just asks the second I'l always give them precise directions, regarless if the answer to the first question was yes or no.

My friend always gives "precise" direction, even if they asked the first question AND the answer is no.
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### Re: OR and XOR

Obviously the solution here is to start a motion to make "xor" a word. Then you can simply ask "would you like cake xor ice cream?"
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### Re: OR and XOR

'; DROP DATABASE;-- wrote:Obviously the solution here is to start a motion to make "xor" a word. Then you can simply ask "would you like cake xor ice cream?"
"Yes, I would like one of {cake, ice cream}."
"Um... Okay, here's some cake."
"No, thank you."

We also need a new word that asks for the truth values of all the the subexpressions of the expression it's applied to. Or a word that asks that the other person please not be a jerk.

"Multiwould you like cake xor ice cream?"
"True {false, true}."
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### Re: OR and XOR

Vempele wrote:
'; DROP DATABASE;-- wrote:Obviously the solution here is to start a motion to make "xor" a word. Then you can simply ask "would you like cake xor ice cream?"
"Yes, I would like one of {cake, ice cream}."
"Um... Okay, here's some cake."
"No, thank you."

We also need a new word that asks for the truth values of all the the subexpressions of the expression it's applied to. Or a word that asks that the other person please not be a jerk.

"Multiwould you like cake xor ice cream?"
"True {false, true}."

select top up-to(1) from (cake, ice cream) order by preference as to whether you would like to have it right now.

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### Re: OR and XOR

That's doomed to produce a type error on a runtime that doesn't know about the type Desert's instance witnesses in the classes Ord and Num...
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### Re: OR and XOR

Slightly off topic, or possibly back on topic...

I once spent an entire day treating the word "or" the same way you treat it in logic. For example: "Are you riding the bus home or are you being picked up?" "Yes." The problem, I realized, is that that response will often make people assume the latter, so it could make you miss your bus here.

I got two detentions and a lot of bad words.