Hey, I'm a bit confused about the classification here and would love some clarification!
I understand that a biconditional statement means 'If A then B, and if B then A'—so we can write it like:
A → B = ~A or B
B → A = ~B or A
A ↔ B = (~A or B) and (~B or A)
But I’m having trouble with how 7Sage classifies 'A or B, but not both' as a biconditional indicator. To me, it looks like 'A or B, but not both' is actually more like an exclusive-or (XOR), where it's one or the other, but not both.
For example, 'A or B, but not both' translates to:
(A or B) and ~(A and B) = (A or B) and (~A or ~B)
And that doesn't seem to match up with the biconditional logic, which is why I’m a little confused.
Am I missing something here? Or is there a specific context I’m not considering in how 7Sage is approaching this? I’d really appreciate any insight on this!
This clicked (sort of) for me by thinking of it as an exclusion of all other possible causes. In the case of the example sentence, the ONLY path that Luke has to become a Jedi is via being trained by Yoda, and being trained by Yoda is CERTAIN to make him a Jedi.
Therefore, you can derive information in both directions. If Yoda DOES train him then he will become a Jedi. If he DOES become a Jedi Yoda must have trained him.
This is different from typical logical relations where you are limited in what you can know. For example:
if A -> B, the contrapositive is /B -> /A
If we are given information about /A, we do not know what happens in this case because there is no arrow flowing from it to another thing (to egregiously oversimplify the actual mechanism). But if instead we had:
A <-> B and its contrapositive /A <-> /B
We now know what /A causes, it must mean cause /B, whereas in the first case you would not be able to generate any reliable conclusions.
If two events are inextricably linked, and are the sole causes of one another, the "if and only if" wording applies, and you have this certainty that if one event happens, the other MUST as well (no matter which you start from). This logic also applies after negation.
In my opinion the best way to explain/understand a bi-conditional claim is how it is taught in Math. It is taught as an 'iff' claim aka. an if and only if claim.
All an iff claim means is that each element implies the other:
Iff A then B, means: A --> B, and B --> A.
It is helpful to differentiate an iff claim from a regular conditional because of LSAT questions as so:
an animal is a zebra if and only if it has white and black stripes. (our biconditional claim in an argument on the LSAT)
Jerry is an animal that has black and white stripes. (an assertion made as part of the argument)
What inference can you make? (LR question on LSAT)
That Jerry is a zebra. (Answer derived by understanding that it is an iff claim, so both elements imply each other)
Is it the case for anyone else that you understand how to work out these scenarios in English, but the Lawgic translation then makes it confusing to understand?
For those confused about uni-conditionals vs. bi-conditionals, consider these:
Uni-conditional example: "You will receive a medal if you finish the race in first-place."
We can logically create this diagram: finish race in first-place → receive a medal
Steve received a medal. Can we logically conclude that he therefore finished in first-place? No! The rule only tells us what happens if people finish first-place. Perhaps people who finished second and third also received a medal. Maybe every race participant received a medal! Who knows?
To identify if two conditions are bi-conditional (both necessary and sufficient), look for language and logical cues that signal mutual dependence. Here’s a guide on how to identify a bi-conditional:
Language Cues: Certain phrases in English imply a bi-conditional relationship. These include:
"If and only if" (often abbreviated as "iff" in logic).
"If but only if."
"Then and only then."
"If... then... but not otherwise."
"Or... but not both."
When you see these phrases, they usually indicate that both conditions must be true together or false together.
Logical Testing of Both Directions: For a bi-conditional to be true, both of the following need to hold:
Sufficiency: Check if one condition guarantees the other.
If
A
A is sufficient for
B
B, then
A
→
B
A→B (if
A
A happens, then
B
B must happen).
Necessity: Check if the condition is also required.
If
B
B is necessary for
A
A, then
B
→
A
B→A (if
B
B does not happen, then
A
A cannot happen).
If both
A
→
B
A→B and
B
→
A
B→A hold, then you have a bi-conditional, written as
A
↔
B
A↔B.
Counterexamples: To confirm the conditions are truly mutually dependent, test for cases where one might happen without the other:
If you find that one condition can occur without the other, then they are not mutually dependent, meaning they are not bi-conditional.
Example Practice:
Uni-directional (Not Bi-conditional): "If it rains, the ground gets wet." This is only sufficient (raining is enough to make the ground wet), but it is not necessary (the ground can get wet by other means, like a sprinkler).
Bi-directional (Bi-conditional): "The light is on if and only if the switch is up." Here, the light being on and the switch being up are mutually dependent; they are true together or false together.
By using these strategies—checking language cues, testing for both directions of implication, and looking for counterexamples—you can determine if two conditions are bi-conditional.
What are some questions from previous tests that use biconditionals? Seeing this in LR-question context would be helpful for me to make sure I understand this concept.
Why is “or…but not both” also a biconditional indicator? I thought it would fall under the group 4 idea of negating the necessary condition (aka the “not both” indicator)?
If there were to be visual diagrams of bi- conditions, like the circles in earlier lessons, what will it look like? Or is that kind of visual diagrams not applicable to this anymore?
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75 comments
man, is law school worth this? i feel like just when I begin to get it, i don't get it.
What would a "Or... but not both" biconditional look like? Would we have to negate one of the given conditions for the Yoda example?
how long took you to come here at this lesson? for me 1 month
Sounds like you just keep the sufficient?
Why can't you just drop the sufficient part of the bi-conditional?
Hey, I'm a bit confused about the classification here and would love some clarification!
I understand that a biconditional statement means 'If A then B, and if B then A'—so we can write it like:
A → B = ~A or B
B → A = ~B or A
A ↔ B = (~A or B) and (~B or A)
But I’m having trouble with how 7Sage classifies 'A or B, but not both' as a biconditional indicator. To me, it looks like 'A or B, but not both' is actually more like an exclusive-or (XOR), where it's one or the other, but not both.
For example, 'A or B, but not both' translates to:
(A or B) and ~(A and B) = (A or B) and (~A or ~B)
And that doesn't seem to match up with the biconditional logic, which is why I’m a little confused.
Am I missing something here? Or is there a specific context I’m not considering in how 7Sage is approaching this? I’d really appreciate any insight on this!
Luke will become a Jedi if Yoda trains him but not otherwise.
Yoda trains him --> Luke becomes a Jedi.
/Luke becomes a Jedi --> /Yoda trains him.
Is this the correct translation?
This clicked (sort of) for me by thinking of it as an exclusion of all other possible causes. In the case of the example sentence, the ONLY path that Luke has to become a Jedi is via being trained by Yoda, and being trained by Yoda is CERTAIN to make him a Jedi.
Therefore, you can derive information in both directions. If Yoda DOES train him then he will become a Jedi. If he DOES become a Jedi Yoda must have trained him.
This is different from typical logical relations where you are limited in what you can know. For example:
if A -> B, the contrapositive is /B -> /A
If we are given information about /A, we do not know what happens in this case because there is no arrow flowing from it to another thing (to egregiously oversimplify the actual mechanism). But if instead we had:
A <-> B and its contrapositive /A <-> /B
We now know what /A causes, it must mean cause /B, whereas in the first case you would not be able to generate any reliable conclusions.
If two events are inextricably linked, and are the sole causes of one another, the "if and only if" wording applies, and you have this certainty that if one event happens, the other MUST as well (no matter which you start from). This logic also applies after negation.
imma bout to crash out
In my opinion the best way to explain/understand a bi-conditional claim is how it is taught in Math. It is taught as an 'iff' claim aka. an if and only if claim.
All an iff claim means is that each element implies the other:
Iff A then B, means: A --> B, and B --> A.
It is helpful to differentiate an iff claim from a regular conditional because of LSAT questions as so:
an animal is a zebra if and only if it has white and black stripes. (our biconditional claim in an argument on the LSAT)
Jerry is an animal that has black and white stripes. (an assertion made as part of the argument)
What inference can you make? (LR question on LSAT)
That Jerry is a zebra. (Answer derived by understanding that it is an iff claim, so both elements imply each other)
Starting to feel a little excessive
my brain is fried, everything after the kick it up lesson is just mad confusing
I don't quite understand why "or....but not both" would an indicator for bi-conditionals
Is it the case for anyone else that you understand how to work out these scenarios in English, but the Lawgic translation then makes it confusing to understand?
when was BUT described as a conjunction?
For those confused about uni-conditionals vs. bi-conditionals, consider these:
Uni-conditional example: "You will receive a medal if you finish the race in first-place."
We can logically create this diagram: finish race in first-place → receive a medal
Steve received a medal. Can we logically conclude that he therefore finished in first-place? No! The rule only tells us what happens if people finish first-place. Perhaps people who finished second and third also received a medal. Maybe every race participant received a medal! Who knows?
----------------------------------------------------------------
However, consider this bi-conditional example:
"You will receive a medal if and only if you finish the race in first-place."
The arrows now point both ways: finish in first place ↔ receive a medal
Christopher received a medal. Can we therefore logically conclude that he finished in first place? Yes!
(likewise, if we were just told that Christopher "finished in first-place," we could also logically conclude that he therefore received a medal).
This is how I understand but not otherwise:
Basically stating not the sentence right before it.
For example
If Yoda trains him, Luke will become a Jedi
Otherwise
If Yoda does not train him, Luke will not become a Jedi
To identify if two conditions are bi-conditional (both necessary and sufficient), look for language and logical cues that signal mutual dependence. Here’s a guide on how to identify a bi-conditional:
Language Cues: Certain phrases in English imply a bi-conditional relationship. These include:
"If and only if" (often abbreviated as "iff" in logic).
"If but only if."
"Then and only then."
"If... then... but not otherwise."
"Or... but not both."
When you see these phrases, they usually indicate that both conditions must be true together or false together.
Logical Testing of Both Directions: For a bi-conditional to be true, both of the following need to hold:
Sufficiency: Check if one condition guarantees the other.
If
A
A is sufficient for
B
B, then
A
→
B
A→B (if
A
A happens, then
B
B must happen).
Necessity: Check if the condition is also required.
If
B
B is necessary for
A
A, then
B
→
A
B→A (if
B
B does not happen, then
A
A cannot happen).
If both
A
→
B
A→B and
B
→
A
B→A hold, then you have a bi-conditional, written as
A
↔
B
A↔B.
Counterexamples: To confirm the conditions are truly mutually dependent, test for cases where one might happen without the other:
If you find that one condition can occur without the other, then they are not mutually dependent, meaning they are not bi-conditional.
Example Practice:
Uni-directional (Not Bi-conditional): "If it rains, the ground gets wet." This is only sufficient (raining is enough to make the ground wet), but it is not necessary (the ground can get wet by other means, like a sprinkler).
Bi-directional (Bi-conditional): "The light is on if and only if the switch is up." Here, the light being on and the switch being up are mutually dependent; they are true together or false together.
By using these strategies—checking language cues, testing for both directions of implication, and looking for counterexamples—you can determine if two conditions are bi-conditional.
I am overwhelmed.
Is this equivalent to a subset-superset diagram where the two circles overlap exactly?
That's how I visualized it in my head.
#help Is "if and only if" equivalent to "if but only if?"
What are some questions from previous tests that use biconditionals? Seeing this in LR-question context would be helpful for me to make sure I understand this concept.
#help
Why is “or…but not both” also a biconditional indicator? I thought it would fall under the group 4 idea of negating the necessary condition (aka the “not both” indicator)?
I really wish I had watched Star Wars.
If there were to be visual diagrams of bi- conditions, like the circles in earlier lessons, what will it look like? Or is that kind of visual diagrams not applicable to this anymore?