Q23 - Teachers are effective only when

[noah]

This is definitely an inference question worth diagramming! With so many conditional statements, it's important to keep track of what you can infer. Here are translations that could work:

First sentence: Teachers effective --> Students independent learners
Second sentence: ~ Teachers deciding --> ~ Student independent deciding
Third sentence: Students independent learners --> Student independent deciding

Conclusion: Teachers effective --> Teachers deciding

The premises can be linked, using the contrapositive of the second one, to prove the conclusion correct:

Teachers effective --> Students independent learners --> Student independent deciding --> Teachers deciding

For use below, I'll abbreviate: TE --> SIL --> SID --> TD

We're asked what cannot be true for teachers who have enabled students to make decisions for themselves. If we know SID, we also know TD because of the last conditional relationship. (E) cannot be true because if teachers have empowered their class to make decisions ("Student independent deciding") then we know "Teachers deciding," meaning the teachers do have the power to make decision in their own classroom.

(A) could be true: SID and SIL can both occur.
(B) could be true: SID doesn't require TE or ~ TE (thought TE does require SID)
(C) is proved possible for the same reason as (B)
(D) is true: SID requires TE.

[peg_city]

I'm going through Manhattan's logical reasoning book on page 367 and this question pops up. I start diagramming the inferences..

1) Is easy
TE -> SIL

2) Not easy

To me it reads TD -> ~SID, which is obviously wrong. The question before this in the logical reasoning book is Q24 on PT 24 (S2) and it reads "No Mathematical proposition can be proven true by observation" which is diagrammed MP -> ~PTO. How is this inference different from the other one? They both indicate a negation in the first few words. What indicator can I use when faced with this type of sentence again?

3) Not easy again

I diagrammed it as SID -> SIL which is backwards. What word in that sentence should indicate to me which way the inference should be drawn?

4) easy

Thanks very very Much

[noah]

1) Is easy
TE -> SIL

Great.

2) Not easy

To me it reads TD -> ~SID, which is obviously wrong. The question before this in the logical reasoning book is Q24 on PT 24 (S2) and it reads "No Mathematical proposition can be proven true by observation" which is diagrammed MP -> ~PTO. How is this inference different from the other one? They both indicate a negation in the first few words. What indicator can I use when faced with this type of sentence again?

Think about this statement: "Jim can't run until his leg heals." What do you know for sure? If that's tough to say, identify the necessary (required) condition? Here it's the leg healing. So, what is the leg healing necessary for? Jim to run.

JR --> LH or ~ LH --> ~ JR

You may be tempted to say LH --> JR, but we don't know that for sure since something else may come up to stop him from running, even if his leg heals.

The sentence you were working on is basically "Not until TD can they enable SID." So what's the necessary condition? TD! So, it's SID --> TD.

Instead of only applying a formulaic "this word means necessary/sufficient", consider the meaning of the statement and what's necessary. It's good to know the basic tendencies, but don't lose your common sense reading skills. That said, "not until" gives us a sense that what's coming next in the sentence must be necessary.

Now, as for the line in PT24, S2, Q24, notice how we explain it in the explanations - we don't break up that first sentence as you do.
But, your translation if fine. However, I don't see how it relates to the statement about teachers. There's no "not until".

3) Not easy again

I diagrammed it as SID -> SIL which is backwards. What word in that sentence should indicate to me which way the inference should be drawn?

Again, what's the necessary thing in this sentence? Is it suggesting that students making decisions requires them to become independent learners, or that for students to become independent learners, they must make decisions?

If you need a hint, look at what's "essential." Whatever is essential is necessary.

Tell me if that clears it up.

[daniel.g.winter]

Noah, thanks for the help here, but I have a question. I originally got this wrong, and I really want to understand how to get it right. I was using the PowerScore Logical Reasoning Bible method of diagramming conditional reasoning, and I am at a loss as how to diagram the second sentence using their method. The LR Bible states that when "until" is used in a sentence, "whatever term that is modified by until becomes the necessary condition, and the remaining term is negated and becomes the sufficient condition."

Following that method, the second sentence becomes:

~SD --> TPD

But that's not what will help me make the chain of inferences that I need to arrive at. Am I missing something? Does the inclusion of "NOT" before the until change things? I'm thinking that the "not" before until is applied to the students making their own decisions, and thus when that is negated you arrive at:

SD --> TPD

Which is the correct inference, and thus makes the rest of the question pretty straightforward. However, if that is not correct, I want to know so I don't continue making this mistake. I want to be able to attack problems like this with ease. This one gave me a lot of trouble and chewed up valuable time, and I didn't even get it correct.

You used a different method to diagram that second sentence, and then used the contrapositive of that. Care to explain how you did it?

Thanks in advance!

[noah]

I will give this a shot, but I can't help but say that this is why we don't push a dogmatic approach to conditional logic. Understanding the meaning of the sentence and translating it into conditional logic is a much more resilient skill than the one you're working on.

Here are two until statements:

I will cry until you kiss me.

I will not smile until you kiss me.

For the first one, which part seems necessary? Kissing! What is it necessary for? Stopping crying. So, ~ cry --> kiss

Or, you could have thought that crying is necessary. It's necessary that you're crying if...you didn't get a kiss: ~ kiss --> cry.

For the second one, kissing is necessary for what? A smile!

smile --> kiss

or, you can say that if you know for sure (necessary) that you're not smiling if there's been no kiss ~ kiss --> ~ smile.

So, for the first, we took the "hope for change," i.e. kissing, and said that if it doesn't happen (negate it), we will still have crying.

For the second one, our "hope for change," i.e. kissing, negated means we'll definitely still have not smiling.

I'll leave it to you to write out a more formal rule, but I would also encouage you to take a step back from these sentences and think what they mean instead of applying a rule. If you think this approach might suit you better, check out the chapter on conditional logic in our LR book.

I hope that helps!