Alternative way of writing "Either Alice or Bob is not in the room."

[sonadoramante]

Homework Statement

Analyze the logical form of the following statements.
"Either Alice or Bob is not in the room."

Relevant Equations

Let P stand for Alice is in the room.
Let Q stand for Bob is in the room.

(P ^ ~Q) v (~P ^ Q)
Either Alice is in the room and Bob isn't or Alice isn't in the room and Bob is.

Would this be acceptable?

Let P stand for Alice is in the room.
Let Q stand for Bob is in the room.

(P ^ ~Q) v (~P ^ Q)
Either Alice is in the room and Bob isn't or Alice isn't in the room and Bob is.

Would this be acceptable?

 

[sonadoramante]

Alternatively, I came up with ~P v ~ Q.
I compared the truth tables of ~P v ~ Q with that of (P ^ ~Q) v (~P ^ Q) and they aren't logically equivalent.

 

[hutchphd]

You can write down the truth table. What should it look like?

 

[sonadoramante]

Well! I drew truth table for both formulas above however, they were not logically equivalent which tells me that most likely ~P v ~ Q might be the answer.

 

[hutchphd]

Show them please.
Which one corresponds to the initial statement?

 

[sonadoramante]

Okay!

P Q ~P v ~ Q (P ^ ~Q) v (~P ^ Q)
F F T F
F T T T
T F T T
T T F F

First column = P
Second column = Q
Third column = ~P v ~ Q
Fourth column = (P ^ ~Q) v (~P ^ Q)

 

[hutchphd]

So which of the final two columns describes the world (in the question?). In particular what if (P=F and Q=F)...which is a correct answer?
(You understand that there are many possible correct expressions of increasing complexity.)

 

[sonadoramante]

Got it! ~P v ~ Q.

 

[PeroK]

Homework Statement:: Analyze the logical form of the following statements.
"Either Alice or Bob is not in the room."
Relevant Equations:: Let P stand for Alice is in the room.
Let Q stand for Bob is in the room.

(P ^ ~Q) v (~P ^ Q)
Either Alice is in the room and Bob isn't or Alice isn't in the room and Bob is.

Would this be acceptable?

Let P stand for Alice is in the room.
Let Q stand for Bob is in the room.

(P ^ ~Q) v (~P ^ Q)
Either Alice is in the room and Bob isn't or Alice isn't in the room and Bob is.

Would this be acceptable?

I'm not sure I would agree with this. If we say that Alice is not in the room, then that says nothing about where Bob is. That seems to me a general point. Let's take an example:

Call the police if a red flare is fired or the emergency whistle is blown.

If you were writing the software for that emergency system, then would you interpret that requirement as only calling the police if either a red flare is fired or an emergency whistle is blown but not both?

In other words, if someone fires a flare and someone else blows the emergency whistle, then you do nothing because the requirement was one or other (but not both?).

You need to check this with your course professor or tutor, but I'd say in mathematics and computer science "OR" means either or both; and, XOR, means either but not both.

 

[Mark44]

If we say that Alice is not in the room, then that says nothing about where Bob is.

This is just like the other thread, the one about the truthfulness of John and Bill. For this problem, the presence of "Either" in the statement "Either Alice or Bob is not in the room." implies to me that exactly one of them is not in the room

You need to check this with your course professor or tutor, but I'd say in mathematics and computer science "OR" means either or both; and, XOR, means either but not both.

Agreed.

 

[sonadoramante]

This is just like the other thread, the one about the truthfulness of John and Bill. For this problem, the presence of "Either" in the statement "Either Alice or Bob is not in the room." implies to me that exactly one of them is not in the room

Agreed.

Yes! We use inclusive (OR).

 

[Mark44]

Yes! We use inclusive (OR).

What I described is the exclusive OR (XOR).
From Wikipedia (https://en.wikipedia.org/wiki/Exclusive_or)

In English, the construct "either ... or" is usually used to indicate exclusive or and "or" generally used for inclusive.

Based on the presence of "either," the sentence "Either Alice or Bob is not in the room." implies "Alice is not in the room" XOR "Bob is not in the room".

 

[PeroK]

What I described is the exclusive OR (XOR).
From Wikipedia (https://en.wikipedia.org/wiki/Exclusive_or)

Based on the presence of "either," the sentence "Either Alice or Bob is not in the room." implies "Alice is not in the room" XOR "Bob is not in the room".

I'm not sure about this. I don't think normal English usage adequately distinguishes between OR and XOR.

A menu with a choice of starters is interpreted as XOR: "soup OR salad".

But, something like: "either he's overworking OR he's having personal problems" is interpreted as possibly both.

 

[Mark44]

I don't think normal English usage adequately distinguishes between OR and XOR.

Agreed. For example, when someone asks of a new parent, "Is it a girl or a boy?" an unhelpful response is "Yes."

But, something like: "either he's overworking OR he's having personal problems" is interpreted as possibly both.

I guess one could interpret this as you say, but OTOH, the following would normally be interpreted in the XOR sense.
"Either you start coming to work on time or you won't be working here any longer." The intent is clearly exclusive or. I read "either/or" constructs as generally being exclusive or.

 

[pbuk]

I am getting a strong sense of deja vu, perhaps we should have a FAQ.

1. "Either A or B is true" is ambiguous.
2. It can be made unambiguous by completing as either:
   1. "Either A or B is true but not both"
   2. "Either A or B is true or they are both true"
3. However it is better to adopt a more formal language from the outset:
   1. "Exactly one of A and B is true"
   2. "At least one of A and B is true"

If you are asked a question by a teacher or examiner in the ambiguous form in (1) you should follow the examples given in class.

 

[sonadoramante]

@pbuk
It was one of the practice problems (without the solutions provided) from "how to prove it" for our discrete mathematics class.

 

[Stephen Tashi]

I am getting a strong sense of deja vu, perhaps we should have a FAQ.

1. "Either A or B is true" is ambiguous.

We can say it's ambiguous in common speech, but in formal mathematical writing "Either A or B" is True in the case where both A and B are True. A FAQ should be clear about this.

Another case where formal mathematical writing differs from common speech is in phrases like "X has 2 prime divisors". In formal mathematical writing, that statement does not exclude the possibility that X has 3 prime divisors.

 

[vela]

In formal mathematical writing "Either A or B" is True in the case where both A and B are True. A FAQ should be clear about this.

I don't agree with this. When "either" is used, I've always interpreted it as exclusive or.

 

[PeroK]

I don't agree with this. When "either" is used, I've always interpreted it as exclusive or.

The Department of Mathematics at the University of Toronto wouldn't agree with you there:

https://www.math.toronto.edu/preparing-for-calculus/3_logic/we_1_statements.html

The example is:

If $x \in \mathbb R$, then either $x \le 0$ or $x \ge 0$. That must be valid mathematically.

 

[Stephen Tashi]

I don't agree with this. When "either" is used, I've always interpreted it as exclusive or.

We should look at examples of formal mathematical writing.

I've never interpreted "either ... or ..." (by itself) to be mutually exclusive in formal mathematical texts. I expect to see "either ...or... and not both..." if the author intends the "or" to be mutually exclusive.

In the texts on formal logic that I have studied, exercises where students are asked to translate common language sentences into logical expressions use the inclusive interpretation of "either ...or ..." unless there are other phrases that modify that meaning. The logic texts take the legalistic approach. For example, we might know that a person cannot both stay at home all day and also drive around town. However a student is expected to translate the sentence "Either Bob stays at home all day or Bob drives around town", to something like $H \lor D$. Logic texts emphasize not making unstated assumptions - even if those assumptions are unstated known facts.

 

[Mark44]

don't agree with this. When "either" is used, I've always interpreted it as exclusive or.

And so have I, which I've stated in this and a related thread by the OP.

The example is If $x \in \mathbb R$, then either $x \le 0$ or $x \ge 0$. That must be valid mathematically.

In the document you linked to, they also say (underline added):

In every day language we use the phrase "either A or B" to mean that one of the two options holds, but not both. For example, when most people say something like ``You can have either a hot dog or hamburger," they usually aren't offering you both. The use of ``either/or" in everyday English is usually divisive, and meant to imply there are only two options: A or B, but not both A and B. (The use of "or" in this way is sometimes referred to as "exclusive or.")

The example you quoted seems sloppy to me, given that they included "either" in their description.
I would have written it as If $x \in \mathbb R$, then either $x \lt 0$ or $x \ge 0$. These are clearly disjoint sets.

And it's not only in every-day language that we distinguish "either/or" from plain old "or." Many computer programming languages include separate operators for OR (both logical and bitwise) and exclusive OR.

However a student is expected to translate the sentence "Either Bob stays at home all day or Bob drives around town", to something like $H \lor D$. Logic texts emphasize not making unstated assumptions - even if those assumptions are unstated known facts.

What if those assumptions are unstated impossible facts? Bob can't both stay at home all day and drive around town all day. Maybe someone should bring logic texts and mathematics texts into the mid-20th century by distinguishing between inclusive OR and exclusive OR (which is variously symbolized as XOR, EXOR, EOR, ^, and $\oplus$, among several others). See https://en.wikipedia.org/wiki/Exclusive_or.

 

[PeroK]

The example you quoted seems sloppy to me, given that they included "either" in their description.
I would have written it as If $x \in \mathbb R$, then either $x \lt 0$ or $x \ge 0$. These are clearly disjoint sets.

We'll have to agree to disagree on this. I'll leave it to you to contact Toronto University and set them straight!

 

[scottdave]

I'm not sure I would agree with this. If we say that Alice is not in the room, then that says nothing about where Bob is. That seems to me a general point. Let's take an example:

Call the police if a red flare is fired or the emergency whistle is blown.

If you were writing the software for that emergency system, then would you interpret that requirement as only calling the police if either a red flare is fired or an emergency whistle is blown but not both?

In other words, if someone fires a flare and someone else blows the emergency whistle, then you do nothing because the requirement was one or other (but not both?).

You need to check this with your course professor or tutor, but I'd say in mathematics and computer science "OR" means either or both; and, XOR, means either but not both.

This is good. I agree that I don't think the statement is implying XOR.
I wonder what would be a good example using NOTs, though?

 

[PeroK]

I wonder what would be a good example using NOTs, though?

I'm not sure I understand that question.

 

[scottdave]

I'm not sure I understand that question.

Well there is the example of "if a siren or a flare". Both of those are events.
But the OP had Bob or Alice Not in room.

I guess you just call the event as one of them not in the room. If both events happen (neither in the room) then the statement is still true.

 

[PeroK]

Well there is the example of "if a siren or a flare". Both of those are events.
But the OP had Bob or Alice Not in room.

I guess you just call the event as one of them not in the room. If both events happen (neither in the room) then the statement is still true.

The requirement, as stated, might reflect the fact that Alice and Bob are exes, who under no circumstances should be allowed in the same room together.

One thing I was going to say to the OP before this whole digression is that it's often worth simplifying the requirement first before translating it into formal language. In this case, there is one case we want to avoid which is Alice and Bob both in the room. The statement is, therefore:
$$\neg (P \land Q)$$
Which is, of course, equivalent to:
$$\neg P \lor \neg Q$$