English "or", logical OR/XOR

mzrowan's post, "Linguists: XOR?" asked whether there were languages which have different words corresponding to the OR and XOR operators in formal logic, since the English word "or" can be either. A common rumor has it that Latin had words "vel" and "aut" corresponding to these two logical operators, but blk debunked that.

In fact, the English word "or" is not a logical operator. It rarely plays that role when used in natural language. However, it can be used in ways that resemble almost any operator in formal logic, not just OR and XOR! Over on Rowan's post I got into debates with a prescriptive grammarian who seems to believe almost obsessively that English "or" is equivalent to formal logic's XOR, and that got me thinking about the various ways we use the word "or" and the logical operators those uses most closely resemble...

• Logical XOR
  aerynne gave a good example of this: "Do you want to go to a movie or stay home?"
  To a typical native speaker of English, the meaning is clearly that you have a choice of two alternatives - you could go to a movie, or you could stay home instead. Which one of those two alternatives do you prefer?
  
  ( It isn't exactly a logical operator... )


• Logical AND
  "You can take a bus or a train to get from Boston to New York"
  
  This is a short form of "You can take a bus to get from Boston to New York or you can take a train to get from Boston to New York", and if I make this statement, what I intend to communicate is that both are true. A typical native speaker of English would understand that intent without spending any thought on the matter. Here, "or" really is acting almost exactly as a formal logical operator - in this case, AND
  
  ( Detailed dissection, discussion, and rebuttal of silly claims... )


• Logical OR
  "Mind if I put on some music?"
  "Sure go ahead, if you have some jazz or blues. Otherwise, I'd rather you didn't."
  
  Logically, if ( [you have some jazz] OR [you have some blues] ) then put some on. If you have one and not the other, that's okay. If you have both, that's okay too. I don't care. Heck, if you happen to have a compilation CD of alternating jazz and blues tracks, that fits just as well as an all-jazz CD or an all-blues CD. The value if the if clause is false only if you have neither.


• Logical -> (implication)
  "Go to the store on your way back, or I'll have to take the car tomorrow."
  "Look both ways or you might get hit by a car."
  
  You could make the case for these being XOR. In fact, as with the first example, these aren't exactly any sort of formal logical operator, and might resemble more than one. But when someone makes a statement of this form, what they mean to communicate are the consequences of the negation of the first part of the statement. IF you don't go to the store THEN I'll have to take the car. The logical operator "or" most closely resembles here is implication.

In English, "or" can be used to connect phrases, clauses, and words in many different ways. Some of these resemble logical operators - many different logical operators. Usually, the meaning is clear to a typical native speaker, from context and semantics. The word "or" by itself is not sufficient to convey its meaning. That doesn't make it any different from many other English words with varying meanings.

Rowan posted about it because sometimes, "or" is ambiguous. Most of the time, though, it's not. And trying to "solve" the ambiguity by tying "or" down to a specifical formal logic operator doesn't work - all it does is make you misinterpret some otherwise clear statements and questions, and make you label others as "bad grammar" when they are not.

[ Edit: ( Or works on three levels: grammatic, logical, and semantic. )

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