logic - Do comparisons make up a special case for understanding either/or as a logical operator?

This question was prompted by a friend who is a fluent speaker of English but not a native one.

Consider the following cases:

• I either eat the apple or the orange. I eat: apple XOR orange.


• I eat either the apple or the orange. I eat: apple XOR orange.

No difference.

Now consider:

• The banana either costs more than the apple or the orange. banana > apple XOR banana > orange
  
  
  
  
  • That is: The banana costs more than the apple or the orange, but not both.
  
  



• The banana costs more than either the apple or the orange. banana > apple AND banana > orange
  
  
  
  
  • That is: The banana costs more than both the apple and the orange.
  
  

My friend asked me, as a native speaker, to first confirm that this was the correct logical translation and then to explain just what the hell is going on here. What rule governs the distinction in the second pair of propositions?

Usually in these cases, I just explain it away as some form of idiomatic expression: "Sometimes either/or means exclusive-or and sometimes it means and. It's just the way it works."

However, I did notice one thing here. This distinction arises in the propositions involving a comparison between subject and objects: less than or more than.

Am I on to something? Anybody have a more coherent explanation?

Answer

Consider your sentence:

1. The banana costs more than either the apple or the orange.

You have noticed that there is a reading of this sentence where it means:

2. The banana costs more than both the apple and the orange.

This sentence, in turn, means:

3. The banana both costs more than the apple and costs more than the orange.

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It is recognized in Oxford Dictionary (as well as in this related question) that "either" is sometimes used to mean "both," as evinced by this definition:

[DETERMINER]

Each of two

'the road was straight, with fields on either side'

Now, this definition holds that "either" is a determiner. But in the kind of example you point to, it is clearly a conjunction ("either... or").

But your conjunction example can easily be transformed into a determiner-involving variant:

4. The banana costs more than either of them.

This should convince you that the sense of "either" as a determiner meaning "both" might get extended to a conjunction.

So although Oxford doesn't mention it, we can form a corresponding definition for "either... or" as "both... and".

Here is another example of "either... or" being used to mean "both... and":

5. Fido likes either ice cream or bacon before bed.

It seems to me that this kind of use is licensed only in contexts where there is a connotation of choice, but I can't formulate a more precise rule. Maybe someone else can.