Conditional Possibilities vs Certainties #2 and #4

[james a]

Hi,

I am getting nowhere fast with the Conditional Possibilities vs Certianties Drill (p. 2-58 & 2-59 in test prep book). I have no idea how to interpret the diagrams and answer the questions of "must be, could be, cannot be" true. I have read the previous pages over and over, but the interpretation of the diagrams is eluding me.

Example:
If F does not occur, what must be true, could be true, cannot be true?

F
D----> +
G

First of all, do I diagram this as below (along with contrapositive) and work it out from there?

F (with slash through it)
D----->+
G

Secondly, how in the heck do I interpret the results? The answers are in the book but I am still struggling.

[Dave Killoran]

Hi James,

You've asked about one of my favorite drills, and one that gets at the heart of the various options the test makers have when working with conditional reasoning. The issue here is: what happens when multiple pieces of information are combined? We start with a conditional relationship, and then we add a new piece of information to it. What then follows from that combination?

The example item you reference, #2, appears as follows. We are given:


F
D +
G

Added to that statement is:

F

What now results?

Well, the first thing that happens is that since F does not occur, then via the contrapositive of the first statement above (which is enacted by F), we know that D cannot occur (D). So, it must be true that D cannot occur (D)and on the flip side, it cannot be true that D occurs). In an LR question, you could see the combination above, and they might ask you what cannot be true. The correct answer would be then that D occurs.

G is the real wild card here, because when F does not occur, we still do not know anything about G. Thus, G could occur, or it might not occur. In the example question cited above, some wrong answers would revolve around G.

The key here is to think about what occurs when the statements in question are combined. If you can master the ideas behind this drill, it shows that you have a solid grasp of what is possible under conditional statements.

That's a start, and I'm happy to go over other examples in this drill if you'd like. Thanks!

[james a]

Thanks Dave.

I moved to question 4 on page 2-59.

1. What is the 'n's relationship to 'q', when 'q' occurs? In the answers they have 'n' occurs and 'n' does not occur both listed under 'could be true'. Why?

2. The contrapositive is throwing me off a little because it appears to contradict the original diagram as it pertains to something 'occuring' or not.

I just started this stuff and I am having a hard time, so please have some patience with my simple questions.

[Dave Killoran]

Hi James,

No problem, I'm glad to help!

1. When a necessary condition occurs, the sufficient can occur, but it does not have to occur. So, when Q occurs, N can occur, but it does not have to occur. Thus, in that situation, you could have N or N.

2. First, go back into the lesson book and re-read the articles that start on pages 2-6 and 2-39. Those should help a bit with working with these relationships. There's also a concept module about Lesson 2 in the Online student Center that would really help you on this topic.

Keep in mind that conditional statements are just that: conditional. If you have a statement such as A B, it means that if A occurs, then B will occur. So, consider these two facts:

• Premise: A B
  
  Premise: B

What can we conclude from those two premises? Well, the first premise stipulates that if A occurs, then B occurs. But the second premise indicates that B does not occur (note that this isn't conditional; it's stated as a fact). So, if B does not occur, then looking at the first premise we can conclude that A does not occur. And that's our conclusion, which is the result of the contrapositive:

• Conclusion: A

Thus, the statements aren't contradictory, because one of them is conditional, and we can add them together.

Please let me know if that helps. Thanks!

[Adam Tyson]

James, as you go through your conditional reasoning materials, remember that the contrapositive is found by both reversing the order of the terms and also negating them all. In addition, where you have either "and" or "or" joining two terms (whether they are sufficient or necessary in the original diagram), you must change "and" to "or" and change "or" to "and".

As Dave said, the contrapositive is inherent in every conditional relationship, and you should always be thinking about it, mentally diagramming it (if not doing it on paper), and looking for answers that apply it. Wrong answers will reverse but not negate (a mistaken reversal), or they will negate but not reverse (a mistaken negation), or they will try to link terms that are not conditionally linked (like tying together F and G in the example you diagrammed here - F happening has no impact on whether or not G happens, and vice versa).

This formula is often in play on the test: CR + MBT = CP. That means that when you have Conditional Reasoning and a Must Be True question, the answer will often (not always) involve the contrapositive of the original relationship.

Have fun with conditionals! They are prevalent on the test and in everyday life. I often suggest to my students that they practice by listening for indicator words in everyday conversations, on the radio, etc., and mentally diagramming those relationships and the contrapositives. When the reporter on the radio says "If the House passes this budget, the Senate will take it up in the next session", that's conditional and you can practice with it! What does it mean if the Senate doesn't take it up next session? What if they do?

Hope that helps!