Thursday, February 16, 2006

Revenge on Susan

Susan posted a few problems to solve here.
Just for revenge, I'm going to post a few of my own.

Each of the following sets of propositions can serve as premisses for a valid sorites. For each, find the conclusion. Note: some of these are from Lewis Carroll, and some from Copi and Cohen.

1. a. No one reads the Times unless he is well educated.
    b. No hedgehogs can read.
    c. Those who cannot read are not well educated.

2. a. When I work a logic example without grumbling, you may be sure it is one that I can understand.
    b. These sorites are not arranged in regular order, like the examples I am used to.
    c. No easy example ever makes my head ache.
    d. I can't understand examples that are not arranged in regular order, like the examples I am used to.
    e. I never grumble at an example, unless it gives me a headache.

3. a. Babies are illogical.
    b. Nobody is despised who can manage a crocodile.
    c. Illogical persons are despised.

4. a. No ducks waltz.
    b. No officers ever decline to waltz.
    c. All my poultry are ducks.

5. a. Every one who is sane can do Logic.
    b. No lunatics are fit to serve on a jury.
    c. None of your sons can do Logic.

Five men who were buddies in the last war are having a reunion. They are White, Brown, Peters, Harper, and Nash, who by occupation are printer, writer, barber, neurologist,and heating contractor. By coincidence, they live in the cities of White Plains, Brownsville, Petersburg, Harper's Ferry, and Nashville, but no man lives in the city having a name similar to his, nor does the name of his occupation have the same initial as his name or the name of the city in which he lives.

The barber doesn't live in Petersburg, and Brown is neither a heating contractor nor a printer - nor does he live in Petersburg or Harper's Ferry. Mr. Harper lives in Nashville and is neither barber nor writer. White is not a resident of Brownsville, nor is Nash, who is not a barber nor a heating contractor. With only the information given, determine the name of the city in which Nash resides.

The following is an LSAT-type game.

1. Frank is the same height as Hank.
2. George is taller than Frank.
3. Eric is taller than Adam.
4. Adam is taller than David and Carl.
5. Bob is shorter than Carl.

Which one of the following must be false?
a. George is taller than Hank.
b. Carl is taller than David.
c. Adam is taller than Frank.
d. David is the same height as Carl.
e. Bob is the same height as Eric.

Visit Math Help Boards for friendly, free and expert math help.


At 2/16/2006 02:24:00 PM , Blogger Susan said...

Aahh! You're keeping me from my quilting :).

Logic, no fair. I only had one watered-down course in that, and that was 3 years ago. I should get a grade curve for my limited background.

Warning! Answers ahead!


(1) Since hedgehogs cannot read, they are not well-educated, which means they do not read the Times. Therefore no hedgehogs read the Times

(2) These sorities are not in regular order, so you don't understand. Since you don't understand, you are grumbling. Since you are grumbling, your head aches. Since your head aches, it is not an easy example.

(3) Since babies are illogical they are despised (that's awful!). Since they are despised, babies can not manage a crocodile

(4) Since all my poultry are ducks, they don't waltz. Since no officers decline to waltz, my poultry are not officers

(5) Everyone who is sane can do logic, which means anyone who can not do logic is not sane. None of my sons (hey!) can do logic, therefore they are not sane. Since no lunatics are fit to serve on a jury, my sons are not fit to serve on a jury No one talks about my sons that way!

First paragraph problem:

I know I'm insane, but I actually handwrote out all 125 possibilities to this one and crossed them out one-by-one.

By crossing out choices, I found that Harper had to be from Nashville. This necessitated that Brown was from White Plains and a neurologist. This necessitated that Harper was a printer from Nashville. This means that Nash must be a writer from either Petersburg or Harper's Ferry. Here I noticed that both Nash and White both from either Petersburg or Harper's Ferry; therefore, since Peters is from either Harper's Ferry or Brownsburg, he must be from Brownsburg (and a Heating Contractor). This forces White to be from Harper's Ferry; thus Nash is a writer from Petersburg. White is a barber from Harper's Ferry, btw.

Second paragraph problem:

Translating to inequalities :)
F = H
G > F
E > A
A > D
A > C
B < C

Composing, we have, B < C < A < E, so Bob must be shorter than Eric. This is not possible if Bob is the same height as Eric, therefore (e) must be false.


How did I do?

At 2/16/2006 05:04:00 PM , Blogger Adrian C. Keister said...

(1) Correct.
(2) Comment: if you have a->b, and b->c, you're only allowed to conclude that a->c. You may not conclude c unless you further assume a (or naturally b, but that would be more trivial, and also not a sorites). I don't think you've fallen into that trap, since it does appear that the claim "These sorites are not in regular order" is an atomic sentence, not an implication.
(3) Correct. Please note: these are only temporary assumptions for the sake of argument. ;-)
(4) You could also conclude there are no officers amoung my poultry. Correct.
(5) Correct. Er, no one is talking about your sons at all yet.

First paragraph. That is more or less what I did. I created five tables, one for each person, each with 5 x 5 squares. It's much like Sudoku, if you've ever done those, only this one's 3D. ;-)

Second paragraph. There are three "threads" here:

It might have been more complete to explain why none of the others have to be false.
Correct, but not complete.

Summary: I'd give you maybe a 90 or 95.

In Christ.

At 2/16/2006 07:48:00 PM , Blogger Susan said...

For the buddy reunion problem, I seriously considered constructing a 3-D table with cardboard strips, but decided against it. That would have been way to nerdy ;). I also considered the way you said, but decided not. My mother is addicted to Sudoku. . . Have you ever done cross sums? I've done a few of those and couldn't put them down.

You know, for the second paragraph problem I almost did explain why the others didn't have to be false, and I also almost said something like "If required, I can explain why they don't have to be false. . . " but I didn't. It may have had something to do with the fact that my allotted quilting time for the day was rapidly getting eaten up as I typed. Just a hunch ;).

You know, I went from rapidly solving those problems to 3 straight hours of tutoring high school math :). This morning was spent grading homework and tests. My brain has been well-used today.

Well, that was fun.

At 2/19/2006 09:49:00 PM , Blogger Adrian C. Keister said...

You mean, way too nerdy? *tsk,tsk* ;-)

Never done cross sums. What are they like?

Almost doing isn't the same as doing. "Well, I almost designed the bridge so it didn't fall down. Really, I was 99.99% there." Or another favorite example, "I only burned the house down once..." *chuckles*

Glad your brain was well-used. As it turns out, I had a very good day Saturday. Got some good research done, as well as cleaning my room. Mrs. Wontrop is especially happy about the latter...

In Christ.

At 2/20/2006 08:28:00 AM , Blogger Susan said...

You've never done cross-sums? Ah, they are fun. I've only done a few, in the problem-solving course I took. It's set up like a crossword puzzle, but the only clues are labels for each row or column that tell the total sum for that row or column. You can't reuse any number in a row or column. Very similar to Sudoku.

You mean, way too nerdy?

Aahh! I am the queen of typos, me thinks. I have a very good knowledge of grammar (thanks to Mother Dear), but unfortunately my fingers don't always get the message. The past 6 months or so I have been especially apt to type words of similar pronunciation (even if vastly different in spelling). I usually catch my mistakes when I read back through, but not always. I have dubbed Mother Dear my official editor. I'll dub you junior editor.

Well, I'm happy that you got your room cleaned this weekend. Poor Mrs. Wontrop. What she must put up with. . . She has my pity.

At 3/01/2006 10:35:00 PM , Blogger Adrian C. Keister said...

Reply to Susan.

"Junior Editor" sounds grand.

Hey, your Mrs. Wontrop quote is getting up there on teasing. Not bad!

However, I seem to see quite the tangent developing. And we're both busy. Shall we cordially tie this comment off?

In Christ.

At 3/08/2006 08:24:00 PM , Blogger Susan said...

Okay, so do TIOC rules say that I have to acknowledge your suggestion? Agreed to TIOC.


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