Conditional Reasoning Diagramming Drill #12

[cindyannconway]

I am having trouble seeing my error when I do drill 12 on page 77 of the LGB. Here is an explanation of what I think I should be doing. Can someone point out where I am going wrong:

Q: The A train cannot arrive until the C train arrives, unless it arrives immediately before the C train.

I see two conditionals in there identified by 'until' and 'unless'.

I start by tackling the first one following the rule "In the case of 'unless', 'except', 'until' and 'without', the term that is modified by 'unless' etc becomes the necessary condition. The remaining term is negated and becomes sufficient.

C train arriving is modified by 'until' so it is the necessary condition, and the negated A becomes the sufficient, therefore
A C

Then I looked at the second conditional 'unless it arrives immediately before C'. Following the same rules as stated above, unless modified AC so I set it as the necessary condition, but I need to negate the sufficient condition so I get:

/(A C) AB

However the answer given on p77 is (A C) AC.

Where is my thinking incorrect.

Thanks!

[Adam Tyson]

Your thinking is actually spot on, cindyannconway, but the confusion is almost certainly coming from the way you are reading the diagrams in plain English, and part of that comes from the difference in meaning between the tools used in your sufficient condition, which uses an arrow:

A C

and ours, which uses a long dash:

A C

For our purposes in this question, A C means "if A has arrived, then C must have also arrived." In other words, there is no way that A could arrive before C does, because when A is here, C must also be here.

But A C means the exact opposite! It means "A arrives before C", so A's arrival does NOT indicate that C has also arrived.

Your final diagram, if I were to read it out loud in plain English, would be "if A's arrival is not sufficient to prove that C has also arrived, then A must arrive immediately before C." Our final diagram would be read as "if A arrives before C, then A arrives immediately before C." These two sentences mean the same thing! You're good! It's just a confusing thing to interpret.

I think this one is easier to deal with if you paraphrase it, to simplify it, rather than follow the more formal process that you went through. In the end, the author is saying that A can't be before C unless it's right before it. That's the genesis of our diagram - no need to to a nested conditional with both "until" and "unless" issues. But the formal approach is still valid, as long as you understand what it means at the end.

Good work!

[cindyannconway]

Thanks Adam! That clears it up for me.

[AJGunning]

The reasoning for the necessary and sufficient organization makes sense to me, but the language around (A C) as opposed to (C A) [which I had diagrammed originally] still confuses me. Any tips on how to see through the confusing wording for this diagram?

[Jeremy Press]

Hi AJ,

Take this in pieces. Everything prior to the "unless" is the general rule we follow, and getting the general rule right is the necessary first step.

First, put this language into a scenario that's more real world to you.

Exercise language: "The A train cannot arrive until the C train arrives."

Real-world example: "You cannot watch the Cubs game until you do your homework." (Sorry, I'm a Chicagoan, couldn't resist!)

So in the real-world example, the GENERAL RULE is that homework has to happen before watching the Cubs game, and in the trains example, the GENERAL RULE is that the C train has to arrive before the A train arrives. (C A)

What's the "unless" doing? It's giving us an exception to the general rule. Consider an addition to my real world example: "You cannot watch the Cubs game until you do your homework, UNLESS you do your homework immediately after you watch the game." What is that "unless" exception for? It's the necessary condition for deviating from the GENERAL RULE. In other words: "If (EVER) you're going to watch the Cubs BEFORE homework, then you MUST do your homework immediately afterwards." How would I diagram that? (Cubs Homework) CH (CH = Cubs immediately before homework).

In our trains example, the "unless" is the exception to the general rule. It's the necessary condition for deviating from the GENERAL RULE that C comes before A. In other words: "If the A train arrives before the C train, then it must arrive immediately before the C train." How would I diagram that? Just like Adam did above. (A C) AC.

Remember, PowerScore's "Unless Equation" is the route to accurately diagramming "unless" statements. Diagram the "unless" portion of the statement as the necessary condition. Then, think about how you would diagram the remaining condition, but instead of diagramming it as it's stated, diagram its logical opposite! That's basically the method I'm tracing for you in more detail above.

I hope this helps!

Jeremy

[sseyedali]

Hi there, I just had a clarification question on this:

I can see that "The A train cannot arrive until the C train arrives, unless it arrives immediately before the C train" symbolizes:

(A---C) ---> AC

And so the contrapositive is ~AC --> (C---A)

However, I'm getting caught up in the language here. If the contrapositive suggests that "If A is not immediately before C, then it is NOT the case that the A train cannot arrive until the C train arrives", why would that imply the symbolism above? I feel like if this were a logical reasoning question there would be a lot of emphasis on the fact that the negation of "The A train cannot..." doesn't imply that, in fact, the C train arrives earlier than the A train--it only implies that it COULD/CAN do so.

I don't mean to be annoying with this, I just want to know if there is any clarification or if I'm reading this incorrectly. Again, I understand how symbolically, the negation of (A---C) in such a game setup would mean that (C---A), but I am struggling with the wording of the sentence, and its entailments.

[Dave Killoran]

However, I'm getting caught up in the language here. If the contrapositive suggests that "If A is not immediately before C, then it is NOT the case that the A train cannot arrive until the C train arrives", why would that imply the symbolism above? I feel like if this were a logical reasoning question there would be a lot of emphasis on the fact that the negation of "The A train cannot..." doesn't imply that, in fact, the C train arrives earlier than the A train--it only implies that it COULD/CAN do so.

I don't mean to be annoying with this, I just want to know if there is any clarification or if I'm reading this incorrectly. Again, I understand how symbolically, the negation of (A---C) in such a game setup would mean that (C---A), but I am struggling with the wording of the sentence, and its entailments.

I'm not sure I'm following your question here, to be honest. Are you wondering why "not (A---C)" has been turned into (C---A)? It seems so at first, but then in your final paragraph you express understanding there. Note that it works that way given the nature of the drill, where no ties are possible. So, there are only two possibilities with A and C in this game: C is before A, or A is before C. In LR this would be different since there typically wouldn't be the "no ties" rule, but in LG we want to get to the operational truth as directly as possible, hence the re-symbolizing there. If it was LR, it would have remained as "not (A---C)."

Hopefully that coves it, but if I've missed the point of your question above, just let us know and we'll try again Thanks!

[sseyedali]

That is helpful, thank you!

[zelal.h]

Hello,

I'm a little confused on one of the conditional reasoning diagramming drills.
It says that "The A train cannot arrive until the C train arrives, unless it arrives immediately before the C train." In the book it says that this translates to "If the A train arrives before the C train, then it must immediately before the C train."

when I read the first part that A can't arrive until C I thought that this would translate to "If the C train arrives before the A train) and would be diagrammed (C--A) because C has to arrive first in order for A to arrive. However, it is the opposite and I'm confused as to why that is.

Any help would be greatly appreciated.
Thank you!

[Stephanie Oswalt]

Hello,

I'm a little confused on one of the conditional reasoning diagramming drills.
It says that "The A train cannot arrive until the C train arrives, unless it arrives immediately before the C train." In the book it says that this translates to "If the A train arrives before the C train, then it must immediately before the C train."

when I read the first part that A can't arrive until C I thought that this would translate to "If the C train arrives before the A train) and would be diagrammed (C--A) because C has to arrive first in order for A to arrive. However, it is the opposite and I'm confused as to why that is.

Any help would be greatly appreciated.
Thank you!

Hi zelal.h!

Thanks for the post! I've moved your question to the thread discussing this topic. Please review the above discussion, and let us know if this helps, or if you still have further questions! Thanks!