Logical reasoning

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{{#x:box| We'll put here some valuable info on logical reasoning taken from the Verbal forum. }}

Contents

Conditional statements

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 umbrella with me. But I could also take an umbrella 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 umbrella with me? I may or may not have.

Using symbols:

X->Y<nonY>nonX

These two below are the same thing:

nonX->Y<nonY>X

X->nonY <Y>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 umbrella with me if and only if it rains. If it rains, then I have the umbrella with me. If I have the umbrella, then it must be raining. If I don't have the umbrella, then it mustn't be raining. If it isn't raining, then I don't have the umbrella 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 umbrella with me unless it is sunny. If it is not sunny, I will take an umbrella with me. If I don't have an umbrella with me, it must mean that it is sunny. However, if it is sunny, I may or may not take an umbrella with me. If I have my umbrella with me, it may or may not be sunny.

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 I'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.

Conditions

Necessary

If A is a necessary condition of B, that means A must happen for B to happen. In other words, if B happened, A must be true. If A is not true, then B can't happen.

In summary: If B then A. If non A then non B.

Example: I will take my umbrella with me only if it rains.

Raining is a necessary condition for my taking the umbrella with me.

If it is not raining, you can be sure that I don't have my umbrella with me.

Sufficient

If A is a sufficient condition of B, that means if A happens B must happen. In other words, if B did not happen, A must be false.

In summary: If A then B. If non B then non A.

Example: I will take my umbrella with me if it rains.

Raining is a sufficient condition for me to take the umbrella with me.

If it is not raining, you are not sure whether I have my umbrella with me. But if I don't have my umbrella with me, you can be sure that it is not raining.

Necessary and sufficient

If A is a necessary and sufficient condition for B, that means if A happen B must happen, and if A does not happen, B does not happen. In other words, A=B. It is equivalent with B if and only if A.

In summary: If A then B. If B then A. If non B then non A. If non A then non B.

Example: I will take my umbrella if and only if it rains. If it is raining, you can be sure I have the umbrella. If it is not raining, I don't have the umbrella. If I have my umbrella with me, you can be sure that it is raining. If I don't have it with me, it mustn't be raining.

Sometimes, always, never

We have to be careful about sometimes, always, and never.

  • If A = always doing something
  • then non A = not doing something sometimes

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  • If A = never doing something
  • then non A = doing something sometimes

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  • If A = doing something sometimes
  • then non A = never doing something

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  • If A = not doing something sometimes
  • then non A = always doing something

{{#x:vspace|1em}} For example:

  • Birds sing sometimes. A never sings. Therefore A is not a bird.
  • Birds don't sing sometimes. A always sings. Therefore A is not a bird.

Compare to:

  • Birds always sing. A doesn't sing sometimes. Therefore A is not a bird.
  • Birds never sing. A sings sometimes. Therefore A is not a bird.

Compare to:

  • Birds sing sometimes. A sings sometimes. Is A a bird? We don't know. A may be a person who sings sometimes.
  • Birds sing sometimes. A doesn't sing sometimes. Is A not a bird? We don't know. A maybe a bird who sings sometimes and doesn't sing the other times.

Some and all

{{#x:box| Which of the following conclusions can be deduced from the two statements below?

  • Some Alphas are not Gammas
  • All Betas are Gammas

A) Some Alphas are not Betas

B) No Gammas are Alphas

C) All Gammas are Betas

D) All Alphas are Gammas

E) Some Alphas are Gammas }}

  • 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.

Logical fallacies

A logical fallacy is an error of reasoning. It either has an error in the logical structure of deduction (formal fallacy), or is falsely inducted from one or more of its premises (informal fallacy).

Here are some fallacies.

Questionable cause

Circular reasoning

In the fallacy of circular reasoning, you assume to be true what you are supposed to be proving.

This product is the best because it is better than any other products.

A more complicated example:

A: This product is perfect.

B: But there is an article in yesterday's newspaper saying that there is a defect in this product.

A: That article must be wrong. How could a perfect product have a defect?

Post hoc reasoning

Also called false cause reasoning. It argues because A preceded B, therefore A caused B.

All plants that survive in metal soils produce histidine. Histidine must be the reason that they can survive the harmful soil.

My red shirt is lucky. Every time I wear it in a soccer game my team wins.

A related fallacy is called "Gamblers fallacy".

Gamblers fallacy

An argument that assumes that a departure from what occurs on average or in the long term will be corrected in the short term.

In a coin toss experiment, the result for the first five throw is head. Therefore we have a very high probability of getting a tail in the six throw.

Slippery slope

An argument that falsely assumes that one thing must lead to another.

Company A has been growing much faster than company B in the past five years. Soon A will surpass B.

You can never give anyone a break. If you do, they'll walk all over you.

Loaded questions

A question constructed in such a way that agreement or disagreement with one term seems to imply agreement with the second.

Are you going to admit that you're wrong?

Have you enjoyed spoiling the dinner for everybody?

Informal Relevance Fallacy

Argument by authority

An argument that bases the judgment of an assertion on the source of the assertion.

This medicine is effective because my doctor said so.

A related fallacy is sometimes called "Personal attack."

Personal attack

When one dispute an argument based on the source of the argument.

Jerry recommended this milk. But I distrust Jerry. So I won't buy this milk.

Appeal to common belief

A fallacious argument that concludes a proposition to be true because many or all people believe it.

Since 60% of the people polled believed in the Bible, the Bible must be true.

Fifty million Elvis fans can't be wrong.

Doctors recommended this medicine than any other medicine. You should buy it.

Brand X vacuums are the leading brand in America. You should buy Brand X vacuums.

A similar type of fallacy is "Appeal to tradition."

Appeal to tradition

A common logical fallacy in which someone proclaims his or her accuracy by noting that "this is how it's always been done."

This is right because we've always done it this way.

Your invention is a bad idea because it has no historical precedent.

Argument from ignorance

A logical fallacy in which it is claimed that a premise is true only because it has not been proven false, or that a premise is false only because it has not been proven true.

Nobody has seen it. It could not have happened.

It's similar to "Shifting the Burden of Proof."

Shifting the burden of proof

The God exist. Unless you can prove that the God doesn't exist, you have to accept it as the truth.

It's also similar to "Argument from incredulity."

Argument from incredulity

What he said is very hard to believe. It simply cannot be true.

Faulty generalization

Hasty generalization

An argument that bases a generalization on insufficient number of instances.

The weather must be perfect in San Diego all the time. I've been there several times, and the sky was always blue and the temperature ideal.

Three of my patients have been harmed by this vaccine. It must be unsafe.

A reverse type of fallacy is "Sweeping generalization."

Sweeping generalization

A type of fallacy which applies a general rule to a specific case.

Our GMAT class has raised the participants an average of 100 points in their GMAT scores. You had a GMAT score of 600 and have taken our class. You will score 700 in your next attempt for sure.

Biased sample

The conclusion is drawn based on a biased sample that is not representative of the entire population.

I can't believe our restaurant isn't the best in town. All our regular customers love us.

Propositional Fallacies

Alternative disjunct

A or B

A

Therefore not B

Number x belongs to a set that contains numbers that are multiples of 2 or 3. It is a multiple of 2, therefore it must not be a multiple of 3.

Affirming the consequent

If P, then Q.

Q.

Therefore, P.

All humans are mortal. Meow is mortal. She must be a human. (She could be a cat.)

Argument from fallacy

If A then B

Not A

Therefore not B

I close my eyes when I'm sleeping. I'm not sleeping, so my eyes must not be closed.

When I was little I was told that an Angel always pushes a dropped pencil down to the floor. Now I know that angels do not exist. Therefore a pencil will never drop down to the floor.


Examples

{{#x:box| Not all tenured faculty are full professors. Therefore, although every faculty member in the linguistics department has tenure, it must be the case that not all of the faculty members in the linguistics department are full professors.

The flawed pattern of reasoning exhibited by the argument above is most similar to that exhibited by which one of the following?

(A) Although all modern office towers are climate-controlled buildings, not all office buildings are climate-controlled. Therefore, it must be the case that not all office buildings are modern office towers.

(B) All municipal hospital buildings are massive, but not all municipal hospital buildings are forbidding in appearance. Therefore, massive buildings need not present a forbidding appearance.

(C) Although some buildings designed by famous architects are not well proportioned, all government buildings are designed by famous architects. Therefore, some government buildings are not well proportioned.

(D) Not all public buildings are well designed, but some poorly designed public buildings were originally intended for private use. Therefore, the poorly designed public buildings were all originally designed for private use.

(E) Although some cathedrals are not built of stone, every cathedral is impressive. Therefore, buildings can be impressive even though they are not built of stone. }} 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.

{{#x:box| (A) Although all modern office towers are climate-controlled buildings, not all office buildings are climate-controlled. Therefore, it must be the case that not all office buildings are modern office towers. }} 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.

{{#x:box| (B) All municipal hospital buildings are massive, but not all municipal hospital buildings are forbidding in appearance. Therefore, massive buildings need not present a forbidding appearance. }} 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.

{{#x:box| (C) Although some buildings designed by famous architects are not well proportioned, all government buildings are designed by famous architects. Therefore, some government buildings are not well proportioned. }} 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.

{{#x:box| (D) Not all public buildings are well designed, but some poorly designed public buildings were originally intended for private use. Therefore, the poorly designed public buildings were all originally designed for private use. }} Not all P are W. Some nonW are U. Therefore all nonW are U. Not same, don't you think?

{{#x:box| (E) Although some cathedrals are not built of stone, every cathedral is impressive. Therefore, buildings can be impressive even though they are not built of stone. }} 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 false for logical questions.

{{#x:vspace|2em}}

{{#x:box| Brown dwarfs, dim red stars that are too cool to burn hydrogen, are very similar in appearance to red dwarf stars, which are just hot enough to burn hydrogen. Stars, when first formed, contain substantial amounts of the element lithium. All stars but the coolest of the brown dwarfs are hot enough to destroy lithium completely by converting it to helium. Accordingly, any star found that contains no lithium is not one of these coolest brown dwarfs.

The argument depends on assuming which one of the following?

(A) None of the coolest brown dwarfs has ever been hot enough to destroy lithium.

(B) Most stars that are too cool to burn hydrogen are too cool to destroy lithium completely.

(C) Brown dwarfs that are not hot enough to destroy lithium are hot enough to destroy helium.

(D) Most stars, when first formed, contain roughly the same percentage of lithium.

(E) No stars are more similar in appearance to red dwarfs than are brown dwarfs. }}

  • 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.

{{#x:box| (A) None of the coolest brown dwarfs has ever been hot enough to destroy lithium. }}

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.

{{#x:box| (B) Most stars that are too cool to burn hydrogen are too cool to destroy lithium completely. }}

We already know that hydrogen is irrelevent here.

{{#x:box| (C) Brown dwarfs that are not hot enough to destroy lithium are hot enough to destroy helium. }}

Obviously helium is irrelevent also.

{{#x:box| (D) Most stars, when first formed, contain roughly the same percentage of lithium. }}

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.

{{#x:box| (E) No stars are more similar in appearance to red dwarfs than are brown dwarfs. }}

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.

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{{#x:box| The article is largely based on the material provided by User:HongHu. }}

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