Aren't we suppose to assume that the disjunction is inclusive, that is, an "or" is always an inclusive-or if not stated otherwise? So, when we interpret a disjunction, don't we also have to consider the possibility of the inherent "and" unless we are considering an exclusive-or? For example:
If I am adopted, then either Bill or Mary is adopted (but could be both). I don’t think that this is inclusive or? If both are adopted, doesn't it become the same as interpreting it as an "and"? If that is the case how would the negation of this be any different from the negation of using an "and"?
@Ikaarin Remember that it is not about which specific case happens, it's about which cases satisfy the condition. On 7Sage airlines the inflight refreshment must include pretzels or cola. If they offer pretzels and cola that's OK (because it's part of inclusive or). But it's also OK if they offer pretzels and skip the cola, or cola without pretzels. That's the difference between the OR case and the AND case.
In your example if both Bill and Mary are adopted that would satisfy both the AND case and the [inclusive] OR case. But that doesn't tell you about the conditional statement.
Back to our imaginary airline, if you're aboard and you receive pretzels and cola that doesn't tell you that the policy is that they must serve pretzels AND cola, or pretzels OR cola, or even if they have a refreshment policy at all.
@ToweringTextbooks I think I understand what you're getting at. I find this to be very similar to the "some can include all" rule too. The sentence "Some apples are red" will allow for both the outcomes of:
a. Only one apple being red
b. All apples being red
Having all apples as red does not change "some apples" to "all apples" it just means that "all apples" satisfies the "some apples" condition.
@LeCastille Imagine we had N and O right? So if we know that EITHER O or N weren't adopted we can infer that M wasn't adopted! This is what the contrapositive is saying.
Since we know that they both need to be adopted together then we know that is the condition for M to be adopted as well.
I wanted to share an example I used to illustrate the concept:
M -> N and O
I imagined M being "meal", and N being "noodles" and "O" onions. It is necessary to have noodles and onions to make a tasty pasta meal.
Clearly, if you're out of noodles, you can't make pasta, and if you're out of onions, it might turn out rather flavourless (not a meal). This logic translates pretty cleanly into:
If "and" is on the necessary side for the example if M is adopted then O and N are adopted, and we draw it with two arrows pointing from M does it still mean that if M is adopted then both O and N are adopted? Or since the can be shown as two different arrows pointing from M that means at if M is adopted then at least one of them is also adopted?
@Kadoran If M is adopted then both O and N are adopted, as we have satisfied the sufficient condition so the necessary conditions must be true. We could also show this as 1 arrow pointing from M and leading to "O and N are adopted", making this a singular condition and not 2 separate ones!
Personally I needed to see all 4 variations of conjunctions and disjunctions in their contrapostive form (using Morgan's law of course) before it become clear. Hopefully this helps others.
IF there is a thing called De Morgan's Law, THEN it would help me to learn the contrapositive of conjunctions AND the contrapositive of disjunctions.
Contrapositive:
IF a thing could not help me learn the contrapositive of conjunctions OR the contrapositive of disjunctions, THEN this thing is not called De Morgan's Law.
is the rule still valid if the conjunction or disjunction were swapped in the conditional argument? In the example, the conjunction is in the necessary. Would the law be applied differently if it were in the sufficient
Hi! For logical equivalence questions, is an argument's contrapositive considered logically equivalent to the argument? For example, If Adam likes apples and also likes dessert, he will like apple pie. A and D --> AP . If a question is asking which of the following sentences follows the same logic structure and this is an answer choice, would it be correct?: If Samantha does not like car shows and she also does not like rain, she will definitely not go to the car show this Saturday. /C and /R ---> /SC
I am struggling to find the value in these types of lesson knowing that we will just not have time to do all of those s* on the exam... I understand that you are only supposed to use these techniques on really confusing questions, but we are spending so much time on these concepts
#feedback #help it would be really helpful if we could step back and set general rules rather than solely run through than examples (their english translations can overcomplicate things) for some of these. i wrote a simple A, B, C lawgic example (without any english translation), but want to make sure it's correct.
negating conjunctions:
A and B ⟶ C
/C ⟶ /A or /B
negating disjunctions:
A⟶ B or C
/B and /C ⟶ /A
can you let me know if this is right, or if there are any errors here?
#help #feedback "if N or O is not adopted, then M cannot be adopted" What is this statement's contrapositive in English? I.e., what is M → N and O's application in English? Does it mean that if M was adopted then both N and O were adopted? Are we reading the "or" in the first statement as "and" or inclusive "or"?
#feedback I am confused about what the contrapositive which was solved via Lawgic means in English, it would be very helpful to include a translation, if possible. Thank you!
The M → N AND O application in English means that "If M was adopted, then N and O must be adopted."
The contrapositive using De Morgan's would then be /(N AND O) → /M,
further breaking down into /N OR /O → /M. In English, "If N was not adopted or O was not adopted, then we know that M was not adopted."
Knowing that /N occurred would be enough to know that /M occurred. The same would be true if we only knew that /O occurred because a disjunction in the sufficient condition means they can work independently of one another.
The way I understand it is "or" on the LSAT is always inclusive. "And" means both, while "or" can mean one or the other, or both.
Your translation is correct, M -> N and O means "If M is adopted, then both N and O are adopted." An easier way to say this is that M requires both N and O.
This is logically equivalent to to the original statement, /N or /O -> /M, "If either N or O (or both) are not adopted, then M is not adopted." Because M requires both N and O, if either one is not adopted, then M will not be adopted.
Let me know if I am incorrect in doing so but I kind of separated the process and made it longer to help myself better understand. Meaning in my head I thought of De Morgan's law as being the "negate" part of the contrapositive process "flip and negate". Flipping the conditions is the first step, and then to negate using De Morgan's law is the second step. I still ended up with the same answers. Not sure if this was obvious to everyone else but going about the process in this way helped me understand better.
Negate a Conjunction
Initial conditional statement:
M→N and O
↳flip the two conditions:
(N and O) → M
↳conjunction (and) swapped for the disjunction (or):
(N or O)→M
↳ Negate each the conjuncts:
/N or /O→/M
Negate a Disjunction
Initial conditional statement:
/N or /O→/M
↳flip the two conditions:
/M→/N or /O
↳ disjunction (or) swapped for the conjunction (and):
It does not work, and here is my long-winded explanation of why! Based on your comment, I am assuming you used that statement as a contrapositive of a statement like "If you have C, then you have A and B". C → A and B
Your version of the contrapositive says "If you don't have A and you don't have B, then you don't have C." /A and /B → /C
But what if you have A but no B? Or what if you don't have A but you do have B? This statement does not answer those questions and instead prompts you to make unwarranted assumptions in a way the original statement does not, and therefore it is not logically equivalent to the original statement.
The correct contrapositive would be "If you don't have A or you don't have B, then you don't have C." /A or /B → /C
This way, you capture what happens if you have neither A nor B, if you have just A but no B, and if you have just B and no A. I think the key here is remembering that the "or" is inclusive, it can do more than "and" because it can insinuate one, or the other, or both, while "and" just means both. That is why you can't keep "and" or "or" the same when creating the contrapositive statement, they have different meanings.
Since I could've been wrong in guessing what the original statement was that you provided the contrapositive too, I will also provide an example working backwards from your version of the contrapositive. "If you don't have A and you don't have B, then you don't have C." /A and /B → /C.
This cannot be flipped to C → A and B because in plain english that means "If you have C, then you have both A and B." Well, we don't know if we need both A and B to have C in the original statement. We just know that if you do not have both A and B, then you do not have C. But that could still mean that you can have C with just A or just B.
The correct contrapositive would be C → A or B, "If you have C, then you have A or you have B (or you have both).
I know this was posted a little while ago and you may have figured it out, but I wanted to test my own brain a bit and maybe help anyone else that has the same question :)
So it says that the contrapositive of /N or /O > /M is M > O and N.
When I think of this with a real example it doesn't make sense to me.
So for example: if you don't eat well or you don't exercise then you can't be in shape. The contrapositive according the the rule would be if you are in shape then you must exercise and eat well. Using my common sense though I would think of it as If you are in shape then you must exercise or eat well.
"If you don't eat well or you don't exercise then you can't be in shape" would mean that both eating well and exercising are requirements for being in shape. There isn't a possibility of being in shape if you fail either or both of these requirements. So if you're in shape it must be that you fulfill the requirements of being in shape, which include both eating well and exercising. The "or" in the first statement is an inclusive or.
Where the confusion might come from re. the first statement is that you don't need to fail both of the requirements, just at least one of them. If the statement was like "if you neither eat well nor exercise, then you can't be in shape" then the contrapositive would be your "if you're in shape then you must exercise or eat well (at least one of them)"
Its probably an exclusive vs inclusive "or" thing which I'm not too clear on yet but I think will be coming up soon.
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98 comments
this is genuinely giving me PTSD from discrete math
So if it was initially M- N or O it would negate to /O and /N - /M??
it looks like I didn't study enough Mandarin or Calc for the LSAT!
Can someone help me with this pls?? #help
Aren't we suppose to assume that the disjunction is inclusive, that is, an "or" is always an inclusive-or if not stated otherwise? So, when we interpret a disjunction, don't we also have to consider the possibility of the inherent "and" unless we are considering an exclusive-or? For example:
If I am adopted, then either Bill or Mary is adopted (but could be both). I don’t think that this is inclusive or? If both are adopted, doesn't it become the same as interpreting it as an "and"? If that is the case how would the negation of this be any different from the negation of using an "and"?
@Ikaarin Remember that it is not about which specific case happens, it's about which cases satisfy the condition. On 7Sage airlines the inflight refreshment must include pretzels or cola. If they offer pretzels and cola that's OK (because it's part of inclusive or). But it's also OK if they offer pretzels and skip the cola, or cola without pretzels. That's the difference between the OR case and the AND case.
In your example if both Bill and Mary are adopted that would satisfy both the AND case and the [inclusive] OR case. But that doesn't tell you about the conditional statement.
Back to our imaginary airline, if you're aboard and you receive pretzels and cola that doesn't tell you that the policy is that they must serve pretzels AND cola, or pretzels OR cola, or even if they have a refreshment policy at all.
@ToweringTextbooks I think I understand what you're getting at. I find this to be very similar to the "some can include all" rule too. The sentence "Some apples are red" will allow for both the outcomes of:
a. Only one apple being red
b. All apples being red
Having all apples as red does not change "some apples" to "all apples" it just means that "all apples" satisfies the "some apples" condition.
Makes sense! I think people struggling with this, need to understand that memorization shouldn't be the goal, but this should feel intuitive!
Of course, in a contrapositive, if either of the previously NECESSARY conditions fail, the sufficient cant happen, because both were required!
Just think about it logically. Don't make it more complicated than it has to be.
ok no sensible explanation given to me
@LeCastille Imagine we had N and O right? So if we know that EITHER O or N weren't adopted we can infer that M wasn't adopted! This is what the contrapositive is saying.
Since we know that they both need to be adopted together then we know that is the condition for M to be adopted as well.
So it's just "put that thang down flip it and reverse it" again?
In the last example, M > /(/N or /O)
it was not flipped
I believe the correct answer should be:
N and O > /M
Please explain and correct me if I'm wrong.
@LeCastille the example started off
M > N and O
he did the contrapositive which was /N or /O > M
Remember that because it started off with "and" you switch to "or"
he did that because of the de morgan law
@marlenevelazquez Let's say this is the first statement: M > /(/N or /O)
How do we get the contrapositive?
Why is it not: N and O > /M
If Cain runs he will go to the store and the movies .
Runs → store and movies
The negation
/ store or / movies → /run
If Cain did not go to the store or the movies then he did not run.
for the example provided in the lecture video, could I also say:
/N -> /O and /M
or
/O -> /N and /M
I understand the concept but I was curious as to whether these would be considered incorrect?
Is it necessary to do the conjunction step? Or can we just put it into lawgic and move on.
I wanted to share an example I used to illustrate the concept:
M -> N and O
I imagined M being "meal", and N being "noodles" and "O" onions. It is necessary to have noodles and onions to make a tasty pasta meal.
Clearly, if you're out of noodles, you can't make pasta, and if you're out of onions, it might turn out rather flavourless (not a meal). This logic translates pretty cleanly into:
/N or /O -> /M
Let me know if this helped anyone else!
If "and" is on the necessary side for the example if M is adopted then O and N are adopted, and we draw it with two arrows pointing from M does it still mean that if M is adopted then both O and N are adopted? Or since the can be shown as two different arrows pointing from M that means at if M is adopted then at least one of them is also adopted?
@Kadoran If M is adopted then both O and N are adopted, as we have satisfied the sufficient condition so the necessary conditions must be true. We could also show this as 1 arrow pointing from M and leading to "O and N are adopted", making this a singular condition and not 2 separate ones!
Conjunction/Disjunction vs Contrapostive
A AND B → C
/C → /A OR /B (Contrapostive Morgan's Law)
A OR B → C
/C → /A AND /B (Contrapostive Morgan's Law)
A → B AND C
/B OR /C → /A (Contrapostive Morgan's Law)
A → B OR C
/B AND /C → /A (Contrapostive Morgan's Law)
Personally I needed to see all 4 variations of conjunctions and disjunctions in their contrapostive form (using Morgan's law of course) before it become clear. Hopefully this helps others.
mah boy augustus
IF there is a thing called De Morgan's Law, THEN it would help me to learn the contrapositive of conjunctions AND the contrapositive of disjunctions.
Contrapositive:
IF a thing could not help me learn the contrapositive of conjunctions OR the contrapositive of disjunctions, THEN this thing is not called De Morgan's Law.
is the rule still valid if the conjunction or disjunction were swapped in the conditional argument? In the example, the conjunction is in the necessary. Would the law be applied differently if it were in the sufficient
Hi! For logical equivalence questions, is an argument's contrapositive considered logically equivalent to the argument? For example, If Adam likes apples and also likes dessert, he will like apple pie. A and D --> AP . If a question is asking which of the following sentences follows the same logic structure and this is an answer choice, would it be correct?: If Samantha does not like car shows and she also does not like rain, she will definitely not go to the car show this Saturday. /C and /R ---> /SC
I am struggling to find the value in these types of lesson knowing that we will just not have time to do all of those s* on the exam... I understand that you are only supposed to use these techniques on really confusing questions, but we are spending so much time on these concepts
Agreed! Does anyone have any advice on whether we should be watching and taking notes on every lesson or have another strategy?
#feedback #help it would be really helpful if we could step back and set general rules rather than solely run through than examples (their english translations can overcomplicate things) for some of these. i wrote a simple A, B, C lawgic example (without any english translation), but want to make sure it's correct.
negating conjunctions:
A and B ⟶ C
/C ⟶ /A or /B
negating disjunctions:
A⟶ B or C
/B and /C ⟶ /A
can you let me know if this is right, or if there are any errors here?
Good job! Those are right.
#help #feedback "if N or O is not adopted, then M cannot be adopted" What is this statement's contrapositive in English? I.e., what is M → N and O's application in English? Does it mean that if M was adopted then both N and O were adopted? Are we reading the "or" in the first statement as "and" or inclusive "or"?
#feedback I am confused about what the contrapositive which was solved via Lawgic means in English, it would be very helpful to include a translation, if possible. Thank you!
The M → N AND O application in English means that "If M was adopted, then N and O must be adopted."
The contrapositive using De Morgan's would then be /(N AND O) → /M,
further breaking down into /N OR /O → /M. In English, "If N was not adopted or O was not adopted, then we know that M was not adopted."
Knowing that /N occurred would be enough to know that /M occurred. The same would be true if we only knew that /O occurred because a disjunction in the sufficient condition means they can work independently of one another.
I hope this helps!
The way I understand it is "or" on the LSAT is always inclusive. "And" means both, while "or" can mean one or the other, or both.
Your translation is correct, M -> N and O means "If M is adopted, then both N and O are adopted." An easier way to say this is that M requires both N and O.
This is logically equivalent to to the original statement, /N or /O -> /M, "If either N or O (or both) are not adopted, then M is not adopted." Because M requires both N and O, if either one is not adopted, then M will not be adopted.
Super helpful, thank you so much!!
Makes sense to me now, thank you!! :)
is the phrase /A or /B --> C the same as /(A or B) -->C???
same question for "and" function: would /A and /B --> C be the same as /(A and B) --> C
Let me know if I am incorrect in doing so but I kind of separated the process and made it longer to help myself better understand. Meaning in my head I thought of De Morgan's law as being the "negate" part of the contrapositive process "flip and negate". Flipping the conditions is the first step, and then to negate using De Morgan's law is the second step. I still ended up with the same answers. Not sure if this was obvious to everyone else but going about the process in this way helped me understand better.
Negate a Conjunction
Initial conditional statement:
M→N and O
↳flip the two conditions:
(N and O) → M
↳conjunction (and) swapped for the disjunction (or):
(N or O)→M
↳ Negate each the conjuncts:
/N or /O→/M
Negate a Disjunction
Initial conditional statement:
/N or /O→/M
↳flip the two conditions:
/M→/N or /O
↳ disjunction (or) swapped for the conjunction (and):
/M→/N and /O
↳ Negate each the disjuncts:
M→N and O
Why even switch AND to OR? Doesn't the contrapositive also work if you say that "If you don't have A AND you don't have B, you can't have C"?
It does not work, and here is my long-winded explanation of why! Based on your comment, I am assuming you used that statement as a contrapositive of a statement like "If you have C, then you have A and B". C → A and B
Your version of the contrapositive says "If you don't have A and you don't have B, then you don't have C." /A and /B → /C
But what if you have A but no B? Or what if you don't have A but you do have B? This statement does not answer those questions and instead prompts you to make unwarranted assumptions in a way the original statement does not, and therefore it is not logically equivalent to the original statement.
The correct contrapositive would be "If you don't have A or you don't have B, then you don't have C." /A or /B → /C
This way, you capture what happens if you have neither A nor B, if you have just A but no B, and if you have just B and no A. I think the key here is remembering that the "or" is inclusive, it can do more than "and" because it can insinuate one, or the other, or both, while "and" just means both. That is why you can't keep "and" or "or" the same when creating the contrapositive statement, they have different meanings.
Since I could've been wrong in guessing what the original statement was that you provided the contrapositive too, I will also provide an example working backwards from your version of the contrapositive. "If you don't have A and you don't have B, then you don't have C." /A and /B → /C.
This cannot be flipped to C → A and B because in plain english that means "If you have C, then you have both A and B." Well, we don't know if we need both A and B to have C in the original statement. We just know that if you do not have both A and B, then you do not have C. But that could still mean that you can have C with just A or just B.
The correct contrapositive would be C → A or B, "If you have C, then you have A or you have B (or you have both).
I know this was posted a little while ago and you may have figured it out, but I wanted to test my own brain a bit and maybe help anyone else that has the same question :)
So it says that the contrapositive of /N or /O > /M is M > O and N.
When I think of this with a real example it doesn't make sense to me.
So for example: if you don't eat well or you don't exercise then you can't be in shape. The contrapositive according the the rule would be if you are in shape then you must exercise and eat well. Using my common sense though I would think of it as If you are in shape then you must exercise or eat well.
Where am I going wrong here?
"If you don't eat well or you don't exercise then you can't be in shape" would mean that both eating well and exercising are requirements for being in shape. There isn't a possibility of being in shape if you fail either or both of these requirements. So if you're in shape it must be that you fulfill the requirements of being in shape, which include both eating well and exercising. The "or" in the first statement is an inclusive or.
Where the confusion might come from re. the first statement is that you don't need to fail both of the requirements, just at least one of them. If the statement was like "if you neither eat well nor exercise, then you can't be in shape" then the contrapositive would be your "if you're in shape then you must exercise or eat well (at least one of them)"
Its probably an exclusive vs inclusive "or" thing which I'm not too clear on yet but I think will be coming up soon.