#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

# is that if A, then not B, and A or B, but not both the same thing?

Member
50 karma

is that if A, then not B, and A or B, but not both the same thing?

Show Related Discussions

• #### A ---> B --SOME--> C; therefore A ----> C, is that a valid argument?A ---> B --SOME--> C; therefore A ----> C, is that a valid argument? I know A ---Some--> C is an invalid argument, but I am unsure about the not so…

• Legacy Inactive ⭐
4141 karma

Hi @"cynthia.wu82" ! Those two statements are not the same thing. If A, then not B is represented as A----->/B ( B----->/A) and A or B, but not both is /A----->B ( /B----->A). This conversation could also help explain why https://7sage.com/discussion/#/discussion/10964/biconditionals

• Legacy Member
301 karma

@"cynthia.wu82" Yes, they are the same thing. @"nessa.k13.0" correct me if I am wrong, but I believe you're confusing an exclusive or for an inclusive one. The diagram you provided in your response to this thread is a representation of an inclusive or. The "A or B, but both both" is an exclusive or, which is diagrammed as the following: A --> /B (B--> /A).

• Alum Member
edited May 2017 392 karma

Edit: Just saw the but not both. They are not the same thing.

Consider these four possibilities in the biconditional: (A) (B) (/A) (/B)
Biconditional (A<->/B CP: /A<->B) Remember these are logically equivlent
If A is chosen, B cannot be
If B is chosen, A cannot be
If we don't choose (/A) then, B must be chosen
If we don't choose (/B) then A must be chosen

Consider those same four possibilities in if A not B (A->/B CP: B->/A)
If A is chosen B cannot be
If B is chosen A cannot be
If A is not chosen (/A), now what? Nothing in the sufficient conditions triggers B to be chosen, so it is free to be chosen or not chosen
If B is not chosen (/B), now what? Again, nothing in the sufficient conditions triggers A to be chosen, so it is free to be chosen or not to be Chosen.

• Alum Member Sage 7Sage Tutor
10710 karma

@Euthyphro said:
Edit: Just saw the but not both. They are not the same thing.

Consider these four possibilities in the biconditional: (A) (B) (/A) (/B)
Biconditional (A<->/B CP: /A<->B) Remember these are logically equivlent
If A is chosen, B cannot be
If B is chosen, A cannot be
If we don't choose (/A) then, B must be chosen
If we don't choose (/B) then A must be chosen

Consider those same four possibilities in if A not B (A->/B CP: B->/A)
If A is chosen B cannot be
If B is chosen A cannot be
If A is not chosen (/A), now what? Nothing in the sufficient conditions triggers B to be chosen, so it is free to be chosen or not chosen
If B is not chosen (/B), now what? Again, nothing in the sufficient conditions triggers A to be chosen, so it is free to be chosen or not to be Chosen.

You got this!

• Legacy Member
edited May 2017 301 karma

@Euthyphro your post is correct if you assume that one of the elements in the "not both" MUST be selected. The initial thread did not state that. It said "A or B, but not both." Your biconditional would be the case if it would have said, "One of A or B is selected, but not both." The way it is stated in the initial post leaves the option of it being an exclusive or, which is the same thing as "if A, then /B." I am assuming that the initial post meant "One of A or B is selected, but not both," which renders your explanation accurate. I was just being a little too technical. I hope that's okay.

• Alum Member
edited May 2017 382 karma

@TheLSAT

Regarding the "A or B, but not both" statement,

I don't think it is necessary to state "one of" is chosen as we have the OR relationship already.

We have an OR relationship which means that we have to choose at least one item. We also have a not both relationship, which means we cannot choose both items. This results in only one valid outcome: 1 item is selected always. We cannot have both out per the OR rule and we cannot have both in per the not both rule, so our only possible world is at most AND at least 1, that is, only one is selected.

A or B

A B
A B
A B
A B

The last possibility (A B) is not allowed. Why? Well if we diagram the OR statement, we get the following:

/A --> B and the contrapositive of /B-->A

In the case where both A and B are out, we get a contradiction because one of the items being out is sufficient to guarantee the inclusion of the remaining item.

not both A and B
A B
A B
A B
A B

The last possibility A B is not allowed. Why? Well if we diagram the NOT BOTH statement, we get the following:

A-->/B and the contrapositive of B-->/A

In the case where both A and B are in, we get a contradiction because one of the items being in is sufficient to guarantee the exclusion of the remaining item.

Now, combine the above statements to get the "A or B, but not both" statement. You'll find that we have two acceptable outcomes only:

A B
A B

The result is a biconditional (A<-->/B and the contrapositive /A<-->B)

Have a look at the Wolfram Alpha truth tables for these statements as well

not both A and B https://www.wolframalpha.com/input/?i=~[A&B]

A or B but not both https://www.wolframalpha.com/input/?i=A+or+B+but+not+both

• Legacy Member
edited May 2017 301 karma

@dcdcdcdcdc Yes, stating that one is selected is necessary to eliminate the "/A/B" option.
"You can have milk or juice, but not both" is not the same thing as "One of milk or juice is selected, but not both." The former leaves the possibility of not having either, and the latter eliminates that possibility, making the diagram a biconditional.

• Legacy Member
301 karma

And btw, if you painstakingly look at the logic games section of the LSAT, you'll notice that LSAC, more often than not, states that "one of A or B is selected, but not both" in this manner to eliminate the confusion of the exclusive or, where the option of selecting neither is possible. And at times, they might not write it that way, which leaves the option of neither being selected open, but for some reason they rarely, if ever, test that inference.

• Legacy Inactive ⭐
edited May 2017 4141 karma

Hi @TheLSAT !

@TheLSAT said: @"nessa.k13.0" correct me if I am wrong, but I believe you're confusing an exclusive or for an inclusive one. The diagram you provided in your response to this thread is a representation of an inclusive or. The "A or B, but both both" is an exclusive or, which is diagrammed as the following: A --> /B (B--> /A).

I'm referencing the "A or B, not both" the "exclusive or" not the "inclusive or". I'm saying "If A, then not B" and "A or B, but not both" as separate conditions, are not the same thing.

• Alum Member
edited May 2017 382 karma

@TheLSAT ""You can have milk or juice, but not both" is not the same thing as "One of milk or juice is selected, but not both.""

The definition of OR, both inclusive and exclusive, requires at least one item, selected, in, satisfied, etc. Simply stating A or B, requires by its very definition at least one of A or B.

I think maybe the use of "can have" in your example is introducing the idea of being possibly avoided, but in the original statement "A or B, but not both" there is not a possibility or probability.

Your implication that "you can have milk or juice" also allows you to have none of them doesn't mesh with the use of both exclusive and inclusive OR. In this case, OR seems to be operating more as a comma that has no effect on the statement. Instead, the statement merely reverts to a not both relationship.

I'm interpreting you example as akin to "It might be the case that A implies B" as opposed to "A implies B," which, I agree, are very different things. In sum, I agree that the example you provide would indeed be different from the one I presented earlier. However, I think the "can" part introduced might be more suitable for grappling with a LR question (similar to those MSS/MBT types that use phrases such as "may" or "probably").

• Legacy Inactive ⭐
edited May 2017 4141 karma

Yeah I agree @dcdcdcdcdc ! "Or" in English to logic can be pretty ambiguous when we aren't given specifics. Terms like "can" and "may" introduce the idea of neither being an option. In LR if we aren't instructed that one must be chosen, we have to account for the possibility of "or" outcomes resulting in neither option. Thanks for linking the lessons!

• Legacy Member
edited May 2017 301 karma

@dcdcdcdcdc The definition of "or" in the exclusive arena does not mean at least one because at least one denotes that you could have more than one. In the exclusive or arena, the definition of the "or" is at most one. However, this is beyond the point. What I was saying is LSAC does not tend to use the terminology you indicated. They tend to be more specific as in "One of A or B is selected, but not both." This is because the latter is more specific in nature than "A or B, but not both." Anyways, I don't want for us to keep commenting on this post haha. If you'd like to discuss it in more detail, then just PM me.

• Member
edited May 2017 427 karma

@TheLSAT said:
@dcdcdcdcdc The definition of "or" in the exclusive arena does not mean at least one because at least one denotes that you could have more than one. In the exclusive or arena, the definition of the "or" is at most one. However, this is beyond the point. What I was saying is LSAC does not tend to use the terminology you indicated. They tend to be more specific as in "One of A or B is selected, but not both." This is because the latter is more specific in nature than "A or B, but not both." Anyways, I don't want for us to keep commenting on this post haha. If you'd like to discuss it in more detail, then just PM me.

Exclusive OR means "exactly one" not "at most one". "Or, but not both" means one of the options must be true, and the other option is not true. "At most one" suggests zero options being true is a possibility. If we had "not both" by itself (without the "or"), then that meanst "at most one".

• Alum Inactive Sage Inactive ⭐
4181 karma

I think what @TheLSAT is trying to communicate is that, for the purpose of the LSAT (due to the specific language of LSAC), "exclusive or" will always lead to one or the other, when technically (and this part is outside of the LSAT), it may not.

Even if this is true, it's not mentioned in the curriculum, because it isn't relevant to the LSAT. But he's saying that when you simply state, "A or B, but not both" (The LSAT does in fact only rarely do this, at most), you're attaching an assumption (that "must" is a necessary part of that statement). So when he/she is stating "at most one," this meaning is outside the confines of the LSAT. Now, I'm not a logician, so I don't know whether this is the case, but it's an interesting point, and the curriculum simply doesn't necessarily debunk this.

• Member
edited May 2017 427 karma

@danielznelson said:
I think what @TheLSAT is trying to communicate is that, for the purpose of the LSAT (due to the specific language of LSAC), "exclusive or" will always lead to one or the other, when technically (and this part is outside of the LSAT), it may not.

Even if this is true, it's not mentioned in the curriculum, because it isn't relevant to the LSAT. But he's saying that when you simply state, "A or B, but not both" (The LSAT does in fact only rarely do this, at most), you're attaching an assumption (that "must" is a necessary part of that statement). So when he/she is stating "at most one," this meaning is outside the confines of the LSAT. Now, I'm not a logician, so I don't know whether this is the case, but it's an interesting point, and the curriculum simply doesn't necessarily debunk this.

How would it not exclude 0 as an option." Isn't "A" shorthand for the proposition "A is true"?

So "A or B, but not both" = "A is true or B is true, but not both". There isn't an assumed "must", it's part of the meaning of the symbol, "A or B".

TheLSAT's example introduced the word "can", but it doesn't change anything about the logic of "or, but not both". "You can have milk or juice, but not both" = It must be true that we "can" have one, and it must be true that we cannot have the other one -- the possibility where we cannot have any is excluded. This doesn't mean that we "will" have milk or juice, but it must be true that we "can" have one. The can vs. will issue seems separate from the "or, but not both" issue.

• Legacy Member
edited May 2017 301 karma

@anonclsstudent Mhmm okay

• Legacy Member
edited May 2017 301 karma

@anonclsstudent you're right only if the language is definitive. This is beyond the point because I never denied the truth of the proposition when it is definitive.

• Alum Member
edited May 2017 382 karma

@anonclsstudent gets where I am coming from.
@TheLSAT
If you look at the truth tables for "A or B" and for "A or B, but not both" the possibility of having NONE of A or B is not permissible. That's the whole point. Your discussion of exclusive versus inclusive has no bearing on the issue because that difference only sets the upper bound at 1 or 2, but you suggest that 0 is an acceptable lower bound. That goes against the meaning of or in any sense.

Moreover, stating "at least one" only speaks of the lower bound and makes no claim about the upper bound. "At least one" is consistent with both exclusive and inclusive or. By your statements and reasoning so far you accept the following to be true of an inclusive or:

A or B

A is true and B is true = true
A is true and B is false = true
A is false and B is true = true
A is false and B is false = true

• Alum Member
edited May 2017 1997 karma

@dcdcdcdcdc said:
@anonclsstudent gets where I am coming from.
@TheLSAT
That goes against the meaning of or in any sense.
.......
A is true and B is true = true

I may be nitpicking here, but I think if you let "or" include the possibility of both, then it would be fair to assume the possibility of neither as well.

Ultimately, this matter largely boils down to context. You can't make a generalized rule for how to handle this, because it greatly depends on the given question.

If you're driving and you come to a fork in the road, you can either turn left or right. You can't do neither, and you can't do both.

If you're offered cake or pie for dessert, you could take both, or you could take neither (although it may be implied that you can only have one dessert if only one dessert option came with your 3 course meal).

So, use common sense. In LG's, at least in the newer PT's, the verbiage is added to the rules of the game to remove any of these uncertainties.

• Alum Member
382 karma

I gladly give the last word to @Mellow_Z and I think they have summarized it nicely. Much agreed that the or is interpreted often by context and, though I haven't looked to closely, I agree the games tend to eliminate any ambiguity on this issue.

• Member
50 karma

please correct me if I am wrong.

A or B is /A-->B and /B-->A
there are three possibilities:
A, /B
B,/A
A & B are all there
the forbidden situation is /A,/B

but, the A or B, but not both,
A<-->/B or B<-->/A
there are two possiblities
A, /B
B, /A
the forbidden situation is A, B are both there
and /A,/B, A and B are both not there.

• Alum Member
164 karma

To keep it simple, the biconditional requires A or B to be in, but not both. The not both rule could have both out. You can think of it in terms of max out