Ambiguity of Some Logical Operators

Subsection 7.2.10 Ambiguity of Some Logical Operators - OR

And, finally, we must deal with the fact that some English words that appear to refer to logical operators are themselves ambiguous.

We’ve already seen that the English word “or” can be used to indicate both:

the logical operator ∨ (inclusive or) – Recall that p ∨ q is true whenever p or q or both is/are true.
the logical operator XOR (exclusive or) – Recall that p XOR q is true whenever p or q but not both is true.

Sometimes, context makes it clear what meaning is intended. Sometimes, however, there is ambiguity.

In these examples, context tells us that “or” means inclusive or:

If it rained or the sprinklers went off, the sidewalk will be wet. (The sidewalks will be wet even if both rain and sprinklers.)

You’ll love the concert if you’re into percussion or you love everything Japanese. (You’ll like the concert even if both you’re into percussion and you love everything Japanese.)

If you or your partner works at ZZZ Corp, you are eligible for their insurance. (You can get the insurance even if both of you work for ZZZ.)

In these examples, context tells us that “or” means exclusive or:

Lightening or Black Thunder will win the race tomorrow. (They can’t both win.)

If you have the winning ticket, you get the car or the trip. (You can’t have both.)

I will buy my Calculus textbook from the University Co-op or Amazon. (Students don’t buy multiple Calculus textbooks.)

But in these examples, there is ambiguity that could lead different people to different interpretations (and thus ways of encoding the sentence in logic):

You can have cake or pie for dessert. (Maybe you have to choose, but maybe you can have both.)

Every house in the neighborhood has a play room or a home theater. (Maybe there was space for only one of these, but maybe it’s an upscale neighborhood and some houses have both.)