Are the Logical Statements (A ∨ B) and ¬(A ∨ B) Equivalent?

[ForceBoy]

<Moderator's note: Moved from a technical forum and thus no template.>

I am given two statements:
Either Alice or Bob is in the room.
Either Alice or Bob isn't in the room.

I would represent them as follows:
(A ∨ B)
¬(A ∨ B)

My question is whether the two statements are equivalent. By the first statement, if Bob is in the room, Alice isn't. By the second, if Alice isn't in the room, Bob is. These are converses of each other. Can one represent them with the same formula since they mean the same thing?

 

[andrewkirk]

This is a good example of the ambiguity of natural language. One of the reasons for the invention of formal logical symbolism is to remove that ambiguity. The first statement could mean
A v B (inclusive or)
or
(A v B) & ¬(A & B) (exclusive or).

To find out which was intended, we would need to question the speaker further.

 

[ForceBoy]

So with only the given, one can't make a formula to begin with, right?

 

[andrewkirk]

We can make a formula, but we cannot be confident that that is what the speaker meant. We can however make two different formulas and state fairly confidently that she must have meant one of those two.

 

[pbuk]

Either Alice or Bob is in the room.
...
(A ∨ B)

This makes sense, but as andrewkirk points out (A v B) & ¬(A & B) also fits the words (although it does infer additional information).

Either Alice or Bob isn't in the room.
...
¬(A ∨ B)

This does not make sense. ¬(A ∨ B) is equivalent to ¬A & ¬B which fits the words "Neither Alice nor Bob is in the room". Did you mean to write ¬A ∨ ¬B? This is not equivalent to either (A v B) or (A v B) & ¬(A & B), however again interpreting "or" as "exclusive or" you could write (¬A v ¬B) & ¬(¬A & ¬B) which IS equivalent to (A v B) & ¬(A & B).

I don't think you should get too hung up on this. When an examiner writes "Either Alice or Bob is in the room" he means (A ∨ B). If he meant to use the exclusive or he would write "Either Alice or Bob is in the room but not both".

 

[pbuk]

When an examiner writes "Either Alice or Bob is in the room" he means (A ∨ B). If he meant to use the exclusive or he would write "Either Alice or Bob is in the room but not both".

Perhaps I should have said "When an examiner in the Western education system writes "Either Alice or Bob is in the room" he means (A ∨ B)". I have seen questions in countries where English is not the first language but is used for teaching Maths which are deliberately designed to catch the student out by exploiting ambiguities in the language. I don't know why this is done.

 

[Mark44]

When an examiner writes "Either Alice or Bob is in the room" he means (A ∨ B).

I disagree. My reading of "Either Alice or Bob is in the room" is that one of them is in the room, but not both. That sense of disjunctive-or-ness is conveyed by the addition of "either".

The expression (A ∨ B) is true if one or both of Alice and Bob is in the room, and is false only if neither is in the room. The symbol ∨, is used to indicate conjunctive-or.

The way to represent "Either Alice or Bob is in the room" is by the use of exclusive-or. One could write the expression as A ⊕ B, although many different symbols are used for this purpose, including XOR , EOR, EXOR, ⊻, ⊕, and others (see https://en.wikipedia.org/wiki/Exclusive_or).

Programming languages clearly make the distinction between inclusive-or and exclusive-or. For example, C, C++, and Java use || for inclusive-or and ^ for exclusive-or.

 

[Stephen Tashi]

I disagree. My reading of "Either Alice or Bob is in the room" is that one of them is in the room, but not both. That sense of disjunctive-or-ness is conveyed by the addition of "either".

In common speech, people often use "either" to imply a disjunctive version of "or", but they also use "or" alone in a disjunctive sense. However, I've never seen a textbook on logic where students are suppose to interpret "either... or..." differently than they interpret "... or...".

 

[ForceBoy]

Thank you all very much. I have to admit I thought I had marked this thread as solved but just now see that I hadn't. I also come back after learning more and I understand the problem and solution much better. I would like to thank all of you for your insight.