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Let's take a compound propositions

Either it is below freezing or it is snowing.

Now if,
$p$: it is below freezing
$q$: it is snowing

Question Link: https://gateoverflow.in/42720/kenneth-rosen-edition-6th-exercise-1-1-question-7-page-no-17

Will it be $p \vee q$ or $p \oplus q$? There are some instances where semantics are required. For example

Either you are ill or appearing for exam. In this both cases can't be true, because if you are ill you can't appear for example and you must be in one state.

References

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    $\begingroup$ +1 to Nathaniel's answer, however, I will say that in Mathematics and Computer Science inclusive or is usually assumed unless otherwise stated (at least in my experience). For instance, in the formulation of a question, one may state: "$x$ is true if and only if either $y$ or $z$, but not both." This of course means exclusive or. $\endgroup$ Commented Mar 12 at 17:54
  • $\begingroup$ I believe the same thing; it's just how we treat it in English that causes bias, and I make mistakes all the time. $\endgroup$
    – tbhaxor
    Commented Mar 12 at 18:14
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    $\begingroup$ It means either. (Sorry, couldn't resist.) $\endgroup$ Commented Mar 12 at 18:25
  • $\begingroup$ @EmilJeřábek It means either, but does it mean both? XD $\endgroup$ Commented Mar 12 at 18:54
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    $\begingroup$ @codeing_monkey it means both either and both, pick either... $\endgroup$ Commented Mar 13 at 10:12

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Don't try to force formal definitions over common english language.

"Either… or" means whatever you want it to mean. There is no convention. Just be precise when you want a formal definition.

Also what is the problem with the answers given in the post you linked?

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  • $\begingroup$ What do you mean by being precise? $\endgroup$
    – tbhaxor
    Commented Mar 12 at 18:13
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    $\begingroup$ After reading this, I understood your point. It makes sense. In some cases, semantics and pragmatics do matter. $\endgroup$
    – tbhaxor
    Commented Mar 12 at 19:12
  • $\begingroup$ It means whatever the reader thinks it means, so make sure they think the right thing! $\endgroup$
    – mudri
    Commented Mar 15 at 9:35
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Natural language is indeed ambiguous and depends a lot on semantics and specific context [ref].

In English, Either/Or usually means Exclusive Or because we tend to use in the situation where two options are presented and exactly one should be True (let's say selected) for entire statement to be True.

But in mathematics, however, Either/Or includes intersection case as well i.e inclusive or. This is because most of the statements in maths work out even when both of the statements are individually True.

But again, when you are dealing with natural languages in the logic, make sure to consider the context and how two statements would interact each other.

For example, if you are ill then you won't appear for example and if you are appearing for exam, then you aren't ill. Here, in this case, exclusive or ($\oplus$) would work better.

Here is the question related to ambiguity between inclusive and exclusive or, which is solved by thinking rationally and in context with how we will deal the situation in real world. https://gateoverflow.in/42729/kenneth-rosen-edition-6th-exercise-1-1-question-15-page-no-18

Also Wikipedia Article says,

Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B".

References

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  • $\begingroup$ I disagree with this answer. In my experience, "either... or" itself, in the absence of more precision, typically means inclusive or; however it is often used in contexts where there is additional information that make the intersection case unlikely or impossible; it is this additional information, and not "either... or" itself, that allow the listener to exclude the intersection case. $\endgroup$
    – Stef
    Commented Mar 13 at 10:14
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In English, no distinction is made between inclusive and exclusive OR.

That is, there is no word or specific idiom that makes these distinctions.

The purpose of the word "or" in English is not as a logical operator, but to connect a series of alternatives together in a sentence.

When the system of logical operators was invented in the 19th century by George Boole, he simply used existing English words to describe the operators in his system. But these terms take on a much more specific definition in his system, than the definition they have in ordinary English.

In ordinary English, it is often possible to resolve by context or common sense, whether it is possible for multiple alternatives to occur jointly and (if so) how the statement should apply in such a case. For example, "wanted dead or alive" (exclusive), or "come hell or high water" (inclusive).

But it's a mistake to think English speakers are constantly performing this resolution - often the distinction is simply not an important element of what is being said.

When using somewhat technical language, as in law, you might find devices like "either A, or B, or both" and "either A or B, but not both".

So it's possible to use English to express exactly the same meaning as the logical operators. But it takes more words and more complicated sentences to do so, and it is done by explicitly addressing the joint cases and stating whether or not they are included.

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