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Logic problem

Posted By: TheFallibleFiend

Logic problem - 10/09/03 01:12 PM


My oldest daughter is applying to a local magnate school for science and technology. It's a pretty famous school and I've hired numerous of their students as interns over the years - all of them exceptional people. The county is offering a test preparation course (free to the students in her school). The first session she got the following sheet to introduce the subject of logical reasoning to her and her cohorts, which though not strictly a word post, is pretty interesting.

--------------------------

Session 2: Warm up
Logical Reasoning

All Seniors at The Academy take calculus and physics.
Some seniors take both courses. Rocky, a senior at
The Academy, is not taking physics. Based only on
the given information, which of the following MUST be
true?

I. Rocky is taking calculus.
II. Marsha, who is a senior, is taking calculus.
III. If John is taking calculus, then John is a senior.

a. I only
b. II only
c. I and II only
d. II and III only
e. I, II, and III

--------------------------

k

Posted By: Faldage

Re: Logic problem - 10/09/03 01:14 PM

f. They don't know what and means.

BTW, is that a school where they teach you to be a magnate?

Posted By: AnnaStrophic

Re: Logic problem - 10/09/03 01:31 PM

All Seniors at The Academy take calculus and physics.

Shouldn't that be "calculus or physics? (Is this what you were saying, Faldage?)

Posted By: Faldage

Re: Logic problem - 10/09/03 01:38 PM

They need at least one or in the water.

Posted By: shanks

Re: Logic problem - 10/09/03 03:04 PM

You two communicate by computer?

The problem with the problem is that even 'or' can be ambiguous - does it mean 'one and only one of the two', or does it mean 'and/or' (a construction I find a touch ugly and neologistic, but awfully useful for clarification).

Anyway, I know what my anwer is going to be, but I'm not telling ;-)

cheer

the sunshine warrior

Posted By: TheFallibleFiend

Re: Logic problem - 10/09/03 03:16 PM


In logic problems, one normally assume the inclusive or, unless the problem states otherwise. In this particular case, the author clarifies the response with the second sentence.

k


Posted By: tsuwm

Re: Logic problem - 10/09/03 03:36 PM

> In this particular case, the author clarifies the response with the second sentence.

In this particular case, unless the teaching of logic has changed dramatically since I studied it, the author has obfuscated the question by using "and" in the first sentence.

..and I too was bemused by "magnate school". :)

(still, I agree with shanks that there is one obviously best answer)
-ron o.
Posted By: Faldage

Re: Logic problem - 10/09/03 03:42 PM

In this particular case, the author clarifies the response with the second sentence.

In this particular case the author shows an inability to properly word logic problems. AND means that both arguments must be true to return a true result. OR, as you correctly point out, is interpreted to mean that either one, the other or both arguments being true would return a true result.

The student should not be held responsible for the inadequacies of the test-writer.

"Some seniors take both courses" is not inconsistent with "All Seniors at The Academy take calculus and physics." If it is true that "All Seniors at The Academy take calculus and physics" then it is also true that "Some seniors take both courses."


If it is assumed that the test-writer meant OR in the first statement then the correct answer is a. I only. If the student decides that AND was truly meant and the test is to find out of the student knows that SOME is included in ALL, then either Rocky is NOT a senior at The Academy and b. II only is true or Rocky is taking phyusics and c. I and II only is true. I'll admit that if you dig into this long enough you'll decided that the correct answer is f. IV only is true (where IV. The test-writer is a doofus who shouldn't be writing logic question) but that they're looking for a. I only, but the student shouldn't have to dig into it long enough.


Edit: Obviously pipped
Posted By: TheFallibleFiend

Re: Logic problem - 10/09/03 03:55 PM



I agree that using "and" has obfuscated the problem in the first place. In fact, it's made the problem statement a contradiction. What I meant to say and did not (because I thought it was an underlying assumption of the subthread) is that had the problem maker used the word "OR" in place of "AND" the second sentence would have clarified whether the (putative) OR was intended to be inclusive or exclusive.

As it stands, the second sentence just restates the first sentence.

Let me be clear. I don't think the person is necessarily an idiot and I don't want to hold him or her up for ridicule. My daughter has lost nothing so far, except a little time. My only concern is that this get fixed and that the kids, my daughter and all the others, get taught the correct stuff in time for the real test.

k


Posted By: Faldage

Re: Logic problem - 10/09/03 04:06 PM

I don't suppose we can expect students who are attracted to this magnet school to have taken any classes in formal logic. I still think it could have been worded better.

BTW, "magnate school" scores .325T (325mT) on the googlometer versus "magnet school"'s 120T. However, "magnate school" did not rate a "Did you mean 'magnet school'?"

Posted By: TheFallibleFiend

Re: Logic problem - 10/09/03 05:02 PM


I've heard the term "magnet school" spoken more than I've seen it written. I've only seen it written a few times previously and usually it was "magnate school." I don't know which is correct, but I do notice that when I googled I got a lot more hits with the former than the latter. Also the first one seems more appropriate.

k


Posted By: tsuwm

Re: hit-meters - 10/09/03 05:10 PM

I don't think the returns are statistically significant as of yet, but for the record:
google-meter - 28 hits
googlemeter - 1 hit
googlometer - 2 hits
googleometer® - (the elegant combined form) 4 hits

-joe (you can't beat our meter) hedonometer

Posted By: Faldage

Re: hit-meters - 10/09/03 05:20 PM

IIRC, I coined it and I coined it "googlometer". Accept no substitutes.

Posted By: tsuwm

Re: hit-meters - 10/09/03 06:29 PM

>I coined it and I coined it "googlometer". Accept no substitutes.

you may have coined "googlometer", but otherwise you got a fight on your hands. (btw, none of the usages herein seem to show up via Google.)

EDIT: (oh, and someone out there insists, rather crudely, that he invented the googlometer as of Aug 10, 2001.)
Posted By: AnnaStrophic

Re: Logic problem - 10/09/03 06:47 PM

In no particular logical order:

I gotta get WordWind over to this thread. She teaches at a magnet school.

Aw shucks, Shanks, this is one of the many forms of communication available . Faldage and I are miles away from each other during the work week, and we post here like any other two people might. I asked for clarification. Verstehen, mate?


Posted By: musick

Re: Logic problem - 10/09/03 08:33 PM

e.

Posted By: Alex Williams

flogging a dead horse - 10/09/03 09:08 PM

The question is definitely hampered by its own inconsistent logic. The first sentence states that all seniors take both calculus and physics. The second sentence contradicts the first: Rocky is a senior and yet he is not taking physics. We conclude that the question writer meant calculus OR physics. If that conclusion is correct then (a) is the correct answer because if Rocky, a senior, is not taking calculus, then he must be taking physics. (In symbolic form, C OR P means not C --> P and not P --> C.) Continuing on the conclusion that the writer meant C OR P rather than C AND P , we can't be sure that II is correct. Marsha is a senior who may be taking calculus, but she may have opted to take only physics and we don't know for sure. III is not necessarily correct, because the given premises do not say that ONLY seniors may take calculus. In this example John may be a junior or sophomore who is taking calculus.

If the writer did mean to say that all students do in fact take both calculus and physics, then the statement about Rocky being a senior and not taking physics is false. This ruins the question, but looking at the 3 choices of things that MUST be true... I is difficult to decide since Rocky seems to be lying about either his schedule or his class. I suggest we detain him at Guantanamo Bay. II is correct, because Marsha is a senior and therefore she must be taking both calculus and physics. III is still not necessarily true since the same possibility that John is an underclassman enrolled in physics remains open.

Posted By: maverick

Re: flogging a dead horse - 10/09/03 10:49 PM

> II is correct, because Marsha is a senior and therefore she must be taking both calculus and physics.

Nope. It doesn't state Marsha is a senior at the Academy!

Based only on the given information, which of the following MUST be true?

As for John, he may be at school in Oz working under a completely different timetable for all we have been told.


Posted By: Alex Williams

Re: flogging a dead horse - 10/10/03 01:25 AM

Nope. It doesn't state Marsha is a senior at the Academy!

True. I was assuming that all the students described were at the academy in question. Now that you've pointed it out, I think it would be a cheap trick to phrase a question thusly. There is an implication that the students mentioned are all at the same academy, although I suspect that the person who wrote the question may consider it fair game to trick the examinees thusly.

Posted By: consuelo

Two to beam up, Scotty - 10/10/03 03:11 AM

"Logic is the beginning of wisdom not the end"

- (Spock to Valeris STII)


Posted By: TheFallibleFiend

Re: Logic problem - 10/10/03 01:26 PM


Excellent! If the premises are false, then any conclusion is always TRUE!

k


Posted By: Faldage

Re: Logic problem - 10/10/03 01:36 PM

If the premises are false, then any conclusion is always TRUE!

Attually®, premises can be true or false, conclusions can be valid or invalid and a true conclusion can result validly from false premises.

All lizards have feathers

All robins are lizards

Therefore: All robins have feathers.

The conclusion is both valid and true but the premises are both false.

Posted By: TheFallibleFiend

Re: Logic problem - 10/10/03 06:19 PM


Sorry, got in a rush. If the premises are false, then regardless of whether the conclusions are true or false, the proposition itself evaluates to true.
(Of course you know this, but everyone may not.)

F->T is T

F->F is also T


BTW, I just got an email from the county thanking my daughter for catching the error - which appears to have happened at the printers.

k


Posted By: Faldage

Re: Logic problem - 10/10/03 06:43 PM

F->T is T

F->F is also T


You lost me.

Posted By: TheFallibleFiend

Re: Logic problem - 10/10/03 08:09 PM



There's an explanation at http://pluto.fss.buffalo.edu/classes/psy/segal/416f2001/logic/98logic2.html.

Go to the bottom of the page and check out columns 1,3, and 5. In logic, that "contains" symbol (the sideways U) means "implies" or "if ... then ..."

"If P, then Q" is a proposition that has a value that depends on the values of P and Q. Extracting from that truth table:

P Q P->Q
T T T
T F F
F T T
F F T

The only time P->Q is false is when the premise (antecedent) is true and the conclusion (consequent) is false.

I was kidding a little in my response to Mav - but only a little. I'm not sure that a contradiction can be treated as a false statement. Intuitively it seems that it could, though.

k


Posted By: Faldage

Re: Logic problem - 10/10/03 10:24 PM

I see part of the problem. You're calling "if P then Q" a proposition. I would call it the major premise. And it looks to me like you've got it backwards.

The syllogism would go:

Major Premise: If P then Q

Minor Premise: P

Conclusion: Q

From this it is valid to say:

If not Q then not P

but not:

If not P then not Q

Now. Are you talking about the last Table on the page?

a. I would call the normal AND

b. is the standard inclusive OR

c. looks like P AND NOT Q

d. is the standard exclusive OR (XOR)

The standard colloquial defintion of "or" is d. The conflict between b. and d. is the basis for the joke question, "Did you take the bus or bring your lunch?"

Posted By: TheFallibleFiend

Re: Logic problem - 10/10/03 10:56 PM


And it looks to me like you've got it backwards.


I'm not sure what I have backwards. A proposition is a sentence that is either true or false. In a conditional sentence, the premise (the if part) can be true or false and the conclusion (the then part) can be true or false. That is, each component (P and Q) is a proposition and the statement P->Q (which can be read as "P implies Q" or "if P then Q") is also a proposition. If you call P the minor premise and P->Q the major premise, it's okay with me. I don't specifically recall seeing that nomenclature, but the terminology doesn't change the underlying logic, so I accept it.


Now. Are you talking about the last Table on the page?



a. I would call the normal AND

b. is the standard inclusive OR

c. looks like P AND NOT Q

d. is the standard exclusive OR (XOR)

The standard colloquial defintion of "or" is d. The conflict between b. and d. is the basis for the joke question, "Did you take the bus or bring your lunch?"


a. yes. this is AND.
b. yes. this is OR.
c. no. this is "P implies Q" that sideways U symbol reads (P implies Q, which can also be written P->Q which can also be written as "not P or Q").
d. yes. equivalence is identical to XOR.

Back to c.

Let's look at the table again, only I'm going to add new fields for "NOT Q" and "P and not Q" (C means conditional, Q' means not Q, and Z means "p and not q"

P Q C Q' Z
T T T F F
T F F T T
F T T F F
F F T T F


We note Column C (P implies Q) is not the same as column Z ("P and not Q"). This time I add columns P' and W (not P or Q).

P Q C P' Z
T T T F T
T F F F F
F T T T T
F F T T T


We note that column C is the same as column Z.
You can check out http://www.merriam.uiuc.edu/ps481/lectures/topic6/advancedrules.html
and search for "Material implication" and an little more detailed explanation at http://secure.yournotes.com:81/notes/fall98/cse260/cse260_082898.shtml
(the same thing I've done above, but considerably neater).

k


Posted By: Buffalo Shrdlu

Re: Logic problem - 10/10/03 11:31 PM

nice job with the [ pre's ] and queues...

Posted By: Faldage

Re: Logic problem - 10/11/03 12:35 AM

I don't think we're talking about the same thing. Maybe you're not looking at it backwards, but we're looking at it from different directions. I think we actually agree. I screwed up on that c. I misread it or something. I don't see the lazy U, my browser is showing a rectangle*, but that's no excuse.

Meanwhile, I'm trying to figure how this relates to our original problem. I think we agree that it was messed up and if they want to say it was a screw up at the printer, I guess we'll just have to accept that as long as they got it fixed.

*Pretty much its feeble attempt at reproducing something it doesn't have in its font.

Posted By: AnnaStrophic

Re: Logic problem - 10/11/03 09:07 AM

BTW, I just got an email from the county thanking my daughter for catching the error - which appears to have happened at the printers.

FF, what precisely was the error?

Posted By: Buffalo Shrdlu

Re: Logic problem - 10/11/03 12:38 PM

was the error?

and or, no?

Posted By: Wordwind

Re: Logic problem - 10/12/03 12:37 AM

I've just read the top few posts--and the problem appeared to be poorly worded. However, based one the wording, only one statement appeared to be true. Still, I had the same problem with the wording that Faldage and AnnaS did. And someone way up there suggested and/or instead, which would have clarified the problem. The capitalization in the problem was off, too.

AnnaS: I teach at a magnet school, but it's not for any of the scienes; it's the magnet school for the fine and performing arts where logic ain't necessarily so.

Posted By: dodyskin

Re: Logic problem - 10/12/03 11:23 AM

what's a magnet school?

Posted By: Faldage

Re: Logic problem - 10/12/03 01:40 PM

A magnet school is a school that attracts students that aren't anemic.

Posted By: Buffalo Shrdlu

Re: Logic problem - 10/12/03 04:08 PM

that's a Sominex®, Fald...

Posted By: moss

Contradiction not clarification - 10/13/03 01:15 PM

In this particular case, the author clarifies the response with the second sentence.

With respect, the author does not "clarify" the response with the 2nd sentence, the author contadicts it.

Here are the sentences:

"All Seniors at The Academy take calculus and physics.
Some seniors take both courses."

The phrase "Calculus and physics" is not equivalent to "calculus or physics".

The 2nd sentence here contradicts the 1st, and the reader does not have sufficient information inside the 4 corners of the "logical problem" to decide which sentence should prevail.

It is simply arbitrary to say the 2nd sentence should prevail over the 1st. Why not the 1st over the 2nd?

In this situation, the only logical thing to do is to answer "None of the above" and ask for more information to resolve the apparent contradiction.

Since there is no "None of the above" option, one should 'think outside the box' and provide the correct answer [literally outside the box] in writing.

Of course, this is a very good lesson for real life.

How many times do we assume something from incomplete or inconsistent input, and end up running off in the wrong direction, wasting time or money, our own or someone else's?

Getting the facts straight in the beginning is the real lesson to be learned from this particular "logic problem". That and having the courage and self-confidence to resist the stampede to the 'wrong' answer simply because it is the best of all the 'wrong' answers provided.

Congratulations to the authors of this problem!

[I just hope the computers which are programmed to tabulate test results will recognize the correct answer scribbled outside the box. Otherwise, we will be punishing, and, worse, extirpating the courage and creativity of our most gifted test-takers.]

Posted By: musick

Re: Contradiction not clarification - 10/13/03 06:48 PM

How many times do we assume something from incomplete or inconsistent input, and end up running off in the wrong direction,...

- and -

I just hope the computers which are programmed to tabulate test results will recognize the correct answer scribbled outside the box.

You've proved your point nicely... and I agree with the rest of *it, as well.

Sometimes the devil's advocate just ain't taken seriously enough.

Posted By: TheFallibleFiend

Re: Logic problem - 10/14/03 01:16 PM



d. is the standard exclusive OR (XOR)



You are in error, here. And I was in error when I said it was correct. Equivalence is the negation of exclusive or.



k


Posted By: TheFallibleFiend

Re: Contradiction not clarification - 10/14/03 01:24 PM

"Clarifies" was the wrong choice of word.

The first sentence says that all students take both classes. The second says that some students take both classes. If all students take both classes, then it is also true that some students take both classes. Therefore, the second sentence does not contradict the first sentence, but is contained in it AS IT IS CURRENTLY WORDED.

HOWEVER, if the problem had been worded correctly, " ... students take calculus OR physics" THEN the second sentence WOULD HAVE clarified that the OR was intended to be an inclusive OR (just in case the student had any doubt).

I agree with your point, though. All real learning takes place inside a student's head. When students understand something, they have an obligation to themselves to speak up, to have it explained to them. If they do not like the explanation, they should be able to say, "I have no choice but to acquiesce to your answer, but I don't agree with it."

k


Posted By: TheFallibleFiend

Re: Logic problem - 10/14/03 01:38 PM



FF, what precisely was the error?



The error is, as Faldage pointed out, with the use of AND in the first sentence. It should be an OR. In casual conversation, one can use words as one pleases. The criterion of correctness is whether the recipient of the sentence understands it. But in logic problems, some words have very specific meanings, AND being one of them. If you say "P and Q," that means both.

Now we could argue that the student should know what was meant. It then becomes a problem of linguistic interpretation and not a problem about about logical reasoning.

(Back when I took AI, one of the more challenging exercises we had was to take a list of sentences in common English, convert them to statements in the predicate calculus, then reword them to say what the author really meant and convert THOSE to predicate calculus. I know it sounds trivial, but it took a lot of effort to do this.)

k



Posted By: AnnaStrophic

Re: Logic problem - 10/14/03 03:46 PM

Thanks, FF. That's also what I originally thought. But after my first post there was a lot of back-and-forth on where the error lay, if it did indeed exist, which is why I asked you just to make sure I still understood. So you daughter figured this out!? Man, she should be running that school!

Posted By: TheFallibleFiend

Re: Contradiction not clarification - 10/14/03 05:30 PM



How many times do we assume something from incomplete or inconsistent input, and end up running off in the wrong direction, wasting time or money, our own or someone else's?



Getting the facts straight in the beginning is the real lesson to be learned from this particular "logic problem".


Back in engineering school, almost every class in the core curriculum began with a review of the basic problem solving method. For some classes, there were 8 steps, for others 5. But they were all variations on a theme. They were all variations of the basic four-step method outlined by the hungarian mathematician George Polya in his book "How to Solve it."
(Details at http://makeashorterlink.com/?Q33D12536)

Step 1. Define the problem.
Step 2. Plan a solution.
Step 3. Carry out the plan.
Step 4. Look back (check the answer).
Summary of method outlined in "How to Solve it," 2nd ed., 1957.

(In statics or dynamics courses, there is, among other things, the additional step of drawing a diagram, for example. But this is just a refinement to step 1, above.)

One of the most common and important reasons for projects to fail is the failure of the engineer to commit to step 1. It's a difficult thing to do when the culture says "get things DONE" to waste time sitting back to reflect on what exactly it is that one has to get done.

Dewey has essentially the same idea with the first "step" being the recognition that a problem exists.
From http://makeashorterlink.com/?V32D25536
"Upon examination, each instance reveals, more or less clearly, five logically distinct steps: (i) a felt difficulty; (ii) its location and definition; (iii) suggestion of possible solution; (iv) development by reasoning of the bearings of the suggestion; (v) further observation and experiment leading to its acceptance or rejection; that is, the conclusion of belief or disbelief."
"How we Think," 1910

Stated so clearly, one can only marvel that so many projects fail in this regard. I think there are several reasons: 1) economically, projects are pushed to show results and 2) psychologically, engineers like to show results.

In any case, all of this is just to reiterate that I agree with your point.

k



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