So for negating a conjunction, you have to switch out the "and" for an "or" and negate the conjuncts. but in the example, the conjuncts are already negated from the original contrapositive rule, so wouldn't they now be un-negated if they're negated again? I'm confused
@duaafaquih Yes, it's negating the negation. You can keep negating a negation like a cycle. It's the same thing as un-negating, then negating again, etc.
Mark and Susan don’t have a car. Their friend Anne does. Mark and Susan want to go to the beach. If mark and Susan go to the beach, then Anne must go to the beach.
M and S -> A
If Anne doesn’t go to the beach, then mark or Susan can’t go to the beach (because they don’t have a car).
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"?
@Isra 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
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106 comments
I've been waiting for my goat DeMorgan -someone who took logic
How do you know when to use De Morgan's Laws? How would the question or AC's imply that you should do so? Or just when should I use it?
When are we supposed to have time for this In 2 minutes?
contrapositive - de morgan's law
and -> or
or -> and
slap a negationn on all
I decided to pursue law because I'm not that good at math. Now I feel like I'm back at square one........
So for negating a conjunction, you have to switch out the "and" for an "or" and negate the conjuncts. but in the example, the conjuncts are already negated from the original contrapositive rule, so wouldn't they now be un-negated if they're negated again? I'm confused
@duaafaquih Yes, it's negating the negation. You can keep negating a negation like a cycle. It's the same thing as un-negating, then negating again, etc.
Mark and Susan don’t have a car. Their friend Anne does. Mark and Susan want to go to the beach. If mark and Susan go to the beach, then Anne must go to the beach.
M and S -> A
If Anne doesn’t go to the beach, then mark or Susan can’t go to the beach (because they don’t have a car).
/A -> /M or /S
Hope this helps someone
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"?
@Isra 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