Since we have been working on some logical reasoning

Question

Since we have been working on some logical reasoning questions I'm going to try to collect the principles I'm following here for everybody's reference. Please feel free to discuss and add more.

If X then Y This is the equivalent of: If non Y then non X. Example: If it rains, then I will take an umbrella with me. I don't have a umbrella with me. That must mean it is not raining. This is NOT equivalent to: If Y then X, or If Y then non X, or if non Y then X. In fact, if we know "If X then Y" and Y occurred, X may or may not happen. Example. If it rains, then I will definitely take an umbrella with me. I have a umbrella with me today. Is it raining? It may or may not be raining. I said if it rains I will take an umbralla with me. But I could also take an umbralla with me just for the sake of it, even if it doesn't rain. By the same token, if it is not raining, do I have an umbralla with me? I may or may not have. Using symbals: X->YnonX These two below are the same thing: nonX->YX X->nonY non X Y if and only if X This is the equivalent of: If X then Y, AND if Y then X. Also, if non X then non Y. If non Y then non X. Example: I will take an umbralla with me if and only if it rains. If it rains, then I have the umbralla with me. If I have the umbralla, then it must be raining. If I don't have the umbralla, then it mustn't be raining. If it isn't raining, then I don't have the umbralla with me. Y unless X This is the equivalent of: If non X then Y. Also, if non Y then X. Example: I will take an umbralla with me unless it is sunny. If it is not sunny, I will take an umbralla with me. If I don't have an umbralla with me, it must mean that it is sunny. However, if it is sunny, I may or may not take an umbralla with me. If I have my umbralla with me, it may or may not be sunny.

rnk_m · Answer

Take this statement the understanding will be lot easier:

If it rains then it is cloudy.(X ---> Y)

Clearly "it is cloudy" is the necessary but not sufficient for rain.

If it is cloudy then it does not mean it will rain.(Y does not -----> X)

But if it's not cloudy then it won't rain.(~Y -----> ~X)

cybera · Answer

woow..wooww

what a POST...

Thanks a BILLION

HongHu · Answer

Paul has made a very good example to show the relationship between some and all.
 

  What we know:  
Some A are not G. And all B are G.

  What can we infer from here?  
Very limited. If it is an A then it may or may not be G. If it is B then it is definitely G. If it is a G then it may or may not be A, and it may or may not be B.  If it is not G then it must not be a B. 

  What can we NOT infer from here?  

Some A are G. -- That's very plausible, but very WRONG! We know that some A are not G, but it is possible that all A are not G, we just don't know. For example: some snakes do not have feet. From this statement, can we conclude that some snakes do have feet? NO.

By the same token, we can't say some G are A. We don't know if some animals with feet are snakes.

Some G are not A. -- Again this is wrong. Although some A are not G, but perhaps G only include the As that is G. Say some integers are odd numbers. Obviously it would be wrong to say some odd numbers are not integers. 

All G are B. -- This is pretty obvious. All odd numbers are integers but not all integers are odd.

  Now lets look at the statement one by one.  

A) Some Alphas are not Betas
We know that some A are not G. And if it is not G then it must not be B. Therefore some A are not B. 
Correct 

B) No Gammas are Alphas
We konw some A are not G. It is possible that some A are G, and it is also possible that no A is G. We just don't know. If no A is G then no G is A, but if some A are G then some G must be A. Some integers are not odd, but it is incorrect to say that no odd numbers are integers.
In other words this is NOT always true. 

C) All Gammas are Betas
We know that all B are G, but we don't know if all G are B. All odd numbers are integers but not all integers are odd.

D) All Alphas are Gammas
Obviously wrong because we know some A are not G.

E) Some Alphas are Gammas
We don't know about it. It may or may not be true. Perhaps some A are G. Perhaps no A is G. All we know is some A are not G. 
Say A=integers, G=odd numbers. Some integers are not odd numbers, but some integers are odd numbers.
Now let's say A=even numbers, G=odd numbers. Some even numbers are not odd numbers. But it will be wrong to say some even numbers are odd numbers.
The key here is when we make a statement about "some", we are not saying anything about the rest. The rest may or may not be different from the "some" that we have made a statement about.

HongHu · Answer

More discussions on if ... then ... 

 Question: 
If i say X happens only because of Y is that the same as saying X if and only if Y

 Answer: 
No. X happens only because Y happened, in other words, if Y does not happen then X will not happen. That means X->Y. If X happened, we know for sure that Y has happened. But we don't know if Y happens whether X will happen. So it is only half of the "if and only if" condition. Y is necessary for X to happen, but we don't know if it is sufficient.

 Question: 
I Agree that X happens only because of Y means X--->Y, that takes care of the "only if condition". Now in this example:
 
It gets "chilly in summer" because of the "unexpected cold front" 

Does it not mean 
unexpected cold front ----> chilly in summer 

All Iâ€™m doing after this point is adding an "only " clause to it

 Answer: 
Yes the word "only" makes all the difference. 

If you say X happens because of Y, then it means Y->X. It gets cold because the cold front moves in. If cold front, then it gets cold. Other things may lead to X too. When we see X it doesn't mean Y must be there. Winter makes it cold too, even without a cold front moving in. So when you feel that it is getting cold, you can't say that there must be a cold front moving in. 

If you say X happens "only" because of Y, then it means X->Y. If you say that it only gets cold when there is a cold front, then whenever it gets cold, you know that there is a cold front. What you don't know from that statement though, is whether everytime when the cold front moves in it gets cold.

"if" and "only if" is the formal usage. "because" is simply common language, so you need to carefully read the sentence and translate it into the logistic languages. I do not think there is an equivalent for "if and only if" using "because", but there will be many other common languages that would express the same thing.

 Question: 
One more time. X is the only cause of Y. Can you throw more light on this. What does it mean and what does it not mean?

 Answer: 
Only X causes Y, no others causes Y. When we see Y we know X must be there. Y->X. 

I don't think X causes Y every time though, so we can't say X->Y. 

eg. Lacking food is the only cause of starving to death. If somebody is starved we know he must be in lack of food. But if somebody lacks food he might not be starved to death.

HongHu · Answer

Futuristic provided me with another good example for logic type CRs.

The first step I did is to symbolize the stem:
 Not all T are F. All L are T. Therefore not all L are F.

The correct conclusion is that it may be true that not all L are F, but it is equally true that all L are F. If we know that all T are L, but not all T are F, then we know for sure not all L are F.

It may be helpful to think the other way. Some Ts are not F. All L are T. But all Ls may be the Ts that are F, or they may be the Ts are not F, we don't know.

So now we know what the error is, then we can proceed.

All M are C. Not all O are C. Therefore not all O are M. 
Compare this with the stem. 
All L are T. Not all T are F. Therefore not all L are F. 
Note L corresponds to M and T corresponds to C. The second condition is not the same.

All H are M. Not all H are F. Therefore not all H are F. 
Again you see the flow is different. Both first and second condition starts with H, while in stem it flows like this: L->T->F.

Not all F are W, all G are F. Therefore not all G are W. 
Stem: 
Not all T are F. All L are T. Therefore not all L are F. 
Exactly the same flow.

Not all P are W. Some nonW are U. Therefore all nonW are U. 
Not same, don't you think?

Some C are nonS. All C are I. Therefore some nonS can be I.
Again not the same logic.

For this type of questions it is very helpful to symbolize it, then you can ignore the contents and simply compare the logic structure. For this question you don't even need to understand the logic and find the error. Simply translation and comparation would be sufficient. However, it is often very helpful if you could understand why a logic is fause for logical questions.

HongHu · Answer

Another good example where logic can be applied:

Fact: BD are similar to RD in appearance but RD can burn hydrogen and BD cannot. 
Fact:Stars have lithium when first formed. 
Fact: All stars but the coolest BD can destroy lithium completely. 
Conclusion: If there is no lithium, it is not a coolest BD.

Ask for assumption. 
Looking at the facts the first one doesn't do much except to tell us BD is cool. Two and three are important. The logic here is that all NonC(oolest BD) can do D(estroy lithium). Therefore if D it is nonC. Obviously the logic is wrong in saying if A is B then B is A. We know that all nonC can D, and some C cannot D, but perhaps some C can D also? Now look at the options.

Aha, exactly what we are looking for. Obviously it is assuming that none of the C can Destroy lithium, so that we can reach the conclusion. (If all A is B, and none B is nonA, then B is A.)

You then look at the rest of the options, and make sure there isn't something that is equally plausible. If so then you need to reread everything again.

We already know that hydrogen is irrelevent here. 
 
Obviously helium is irrelevent also. 
 
This one is a little curious, but since we already formed our answer, we know this is not it. It could be the answer if the question stem is different. Say if the conclusion is that "It must be a coolest BD if there are lithium in the star." Then the right answer may very well be this one, since you would be assuming all D process should have been completed when you found the star, otherwise there may be some lithrium left undestroyed when you found a star even if it's not the coolest BD. 
 
Again we know this is irrelevant. It can be rather quick to go through the options if you already know what you are looking for.

job_man · Answer

wow my head hurts after reviewing this material

I obviously have a while to go before going to battle with the GMAT

HongHu · Answer

Heh. It's perfectly normal. Logical CRs are perhaps one the hardest type questions. Just keep practising using symbols and you'll find the feel of it.

Artemov · Answer

Is there a way out for solving these syllogism problems with the help of Venn diagram?
I have heard it is.
Will any of the experts enlighten on this please?

HongHu · Answer

You could, I would think. However I think that may actually make the whole thing more complicated than it needs to be.

jyotsnasarabu · Answer

Positive Negative 

Universal All No 

Particular Some Some........not 



1) Some A's are not B's means atleast 1 A is not a B 

2) All A's are B's means All B's are A's 

3) A positive + A positive = always is a positive conclusion 

4) A positive + a negative= Always leads to a negative conclusion 

5) a negative + a Negtive = leads to no conclusion 

6) A universal + a universal statement always gives the followin conclusions- either universal/a particular/a no conclision 

7) a universal Conclusion+particular conclusion =always gives a particular conclusion 

8) a particular + a particular = no conclusion 

9) All A's are B's 
All B's are C's 
the conclision is exists = All C's are A's and vice versa 

10) Some A's are C's 
Some C's are A's = a conclusion cannot be derived................... 


All A's are B's means that A->B is distributed But B-> A is not distributed 

Some A's are B's means A->B is not ditributed nor is B->A distributed 

No A's are B's means A->B is not true and B->A is not true . So here both 
A and B are Distributed 


11) All A's are B's 
All C's are B's 
No conclusion since the middle term B is not distributed in both 

For a conclision to exist the middle term needs to be distributed atleast once................................. 

12) All A's are B's 
All A's are c's 
Conclusion is Some B's are C's or vice versa 

14) All A's are B's 
Some A's are C's 
Some b's are C's and vice versa 

13) All A's are B's 
Some B's are C's 
No conclusion since B is not distributed...................... 

15) No A's are B's 
Some A's are not C's 
No conclusion since both premises are negative 

16) many A's are b's 
All b's are not C's 
conclusion is some a's are not c's and vice versa 

17)All a's are b's 
No b's are c's 
conclusion :No a's are c's and vice versa..................... 

18) All a's are b's 
All b's are c's 
Conclision exists as b is distributed once............All a's are c's and vice versa. Some a's are c's and vice versa is also right....... 

19) All a's are b's 
All c's are b's 
consluaion: No conclusion since the term b is not distributed 

20) All a's are b's 
All a's are c's 
conclusion: Some b's are c's and vice versa 

21) All a's are b's 
Some a's are c's 
conclusion : Some b's are c's 

22) All a's are b's 
Some b's are c's 
Conslusion: No conclusion since b is not distributed at all........... 

23) Many a's are b's 
No b's are c's 
Conclusion: Some a's are nto c's 

24)No a's are b's 
some a's are not c's 
conlusion: No concluaion sonce both premises are negative 

All+All = All /some 
All+Some = Some 
All+No= No/Some not 
No+Some = Some not 

Hope the above mentioned funda's help 

sincerely, 
jyotsna

vineetgupta · Answer

Thanks HongHu for the awesome post.
And guys,these things are actually tested on the GMAT in one way or the other.
So, make sure u have a good hold of these concepts.

wilso · Answer

I have personally found using variable and mapping arguments to be very effective in understanding CR questions. Can anyone point me in the direction of some books or other sources that will help me get a better understanding of this concept? 

thanks

surbab · Answer

It is really very helpful honghu. 

thanks a lot!!!

bmwhype2 · Answer

small mistake on this one. https://www.gmatclub.com/phpbb/viewtopic.php?t=49756

empty_spaces · Answer

all apples are fruits.

are all fruits apples?

bmwhype2 · Answer

very helpful starting point for Logic / Critical Reasoning https://regentsprep.org/Regents/Math/math-a.cfm

dmetla · Answer

Thanks this was useful information

rudransh · Answer

Nice post, information provided here indeed couldn't find in MGMAT CR book

gurpreetsingh · Answer

do we get such questions on gmat? I think this concept is a part of logical reasoning not part of critical reasoning.

Since we have been working on some logical reasoning

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Since we have been working on some logical reasoning

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