#24 - No mathematical proposition can be proven true by

[pacer]

I thought that this question is asking for an assumption that is sufficient.

I read in some books that when the question states

which of the following is an assumption on which the conclusion depends? - this means that the task is to find a necessary assumption

and when the question read on "The conclusion follows logically if which of the following assumptions is true?" - the task if to find an assumption that is sufficient

Based on this, this question is asking for a sufficient assumption

This argument is assuming that "Observation is the only method for determining truth"

Observation -> truth

which is more in line with choice C


How do you know it is asking for something necessary?

[Ron Gore]

Hi Pacer,

You are absolutely correct that this is a Justify the Conclusion (or a sufficient assumption question.) In this case, the missing information that would prove the conclusion is a conditional relationship, which by definition includes both a sufficient and a necessary condition. So don't dismiss the answer choice because it contains a necessary condition indicator. Consider the entire context of the answer choice to see if it matches.

Your prephrase was strong, and this is a very straightforward Justify question for which the prephrase is both mechanical and devastating. Here's how I would diagram the argument:

MP = mathematical proposition
PTO = proven true by observation
KT = known to be true

P: MP PTO

C: MP KT

Your prephrase is that the correct answer choice will link to together the two previously unconnected portions of the argument, in this case PTO and KT, in a way that proves the conclusion KT is valid. In other words, the correct answer choice will say:

PTO KT

Since the prephrased link between the premise and the conclusion is a conditional relationship, it will have both a sufficient and a necessary condition.

Answer choice (D) is incorrect because it states the Mistaken Reversal of our prephrase, and the Mistaken Negation of the relationship as stated in answer choice (E). This answer choice would be diagrammed as:

KT PTO

Answer choice (E) contains the logical equivalent of our prephrase, though it is stated in terms of the contrapositive:

KT PTO

Hope that helps!

Ron

[kristinajohnson@berkeley.edu]

Can we always link the necessary condition in the premise and the necessary condition in the conclusion? I feel like sometimes I'm linking the sufficient conditions and sometimes I'm linking the necessary conditions and sometimes I'm linking sufficient and necessary conditions, but maybe not? Is there a formula like that I can think of instead of the other formulas in the book that I don't find helpful in practice?

"No mathematical proposition can be proven true by observation. It follows that it is impossible to know any mathematical proposition to be true."

Premise: MP THEN NOT PTO

Conclusion: MP THEN NOT KT

Need NOT PTO THEN NOT KT

Answer choice correct THEN conclusion valid: NOT PTO THEN NOT KT correct THEN MP THEN NOT KT valid
How does this help in practice?

Premise + answer choice = conclusion: MP THEN NOT PTO + NOT PTO THEN NOT KT = MP THEN NOT KT
However, (for this problem) a conditional link is being established through the necessary premise and the sufficient answer choice to verify the conclusion. Is that a formula I can use?

In practice (for this problem) it's more like if necessary from premise then necessary from conclusion = answer choice. It would be so nice to be able to have a formula like this! Is that always true? Are there a few formulas that could be applied to every Justify problem?

[kristinajohnson@berkeley.edu]

I need to add to my previous post my initial mistake, and the questions I have from it.

"No mathematical proposition can be proven true by observation. It follows that it is impossible to know any mathematical proposition to be true."

Correct answer choice D, "Knowing a proposition to be true requires proving it true by observation."

I initially attempted to solve this problem with not mp on the necessary side for both the premise and conclusion but the (wrong) answer I came to was a mistaken negation of the correct answer, why? Shouldn't we be able to move things around as long as they are logically the same?

Premise: MP THEN not PTO (if PTO then not MP)

Conclusion: MP THEN not KT (if KT then not MP)

need if PTO then KT (I know this turned out to be wrong, just showing what I did.)

if PTO then KT then not MP (if MP then not KT then not PTO). This is a mistaken reversal of correct answer choice D. My anxiety stems from it all adding up and checking out when compared to what we're given. I guess it's incorrect to link the contrapositive sufficient conditions in the premise and conclusion, respectively. But if I linked the contrapositive sufficient conditions from the conclusion and premise, in that order, I would have had the right answer. Curious. Worrisome.

Thank you very much.

[Jeff Wren]

Hi kristina,

You asked:

"Can we always link the necessary condition in the premise and the necessary condition in the conclusion?"

Your question suggests that you may be misunderstanding how Justify questions work, especially those involving conditional reasoning.

Before we get into conditional reasoning with Justify questions, let's start with a (hopefully) straightforward example of conditional reasoning in a Must Be True question.

Imagine that the stimulus states:
1. If A, then B (diagrammed A ->B)
2. If B, then C (diagrammed B-> C)

The question then asks what Must Be True (i.e. what is the inference that we can make).

We can link these two conditional statements to form a chain, A -> B -> C
We can then drop the middle term B to infer: if A, then C (diagrammed A -> C)
And we would be looking for that answer, or its contrapositive (Not C -> Not A)

With Justify questions, rather than finding an inference that we can draw from the facts, we are going in the other direction and instead looking for the missing premise that would prove the conclusion to be true when added into the argument.

So if the argument states:

Premise: If A, then B (diagrammed A ->B)
Conclusion: If A, then C (diagrammed A -> C)

Then you need to figure out the missing piece of the argument that would prove the conclusion. Since we already have A -> B, the missing piece is B-> C because that would prove our conclusion just like in the Must Be True example above.

The argument in this question follows the exact pattern shown above. One thing that makes the argument tricky is the specific wording of the terms used. Another thing that makes Answer E tricky is that it is in the form of the contrapositive, but you should be prepared for that possibility.

So to answer your earlier question, it is not a question of whether we can always connect the necessary terms. Instead, what you are looking for is: which answer, if it were true and added to the argument, would 100% get you from the premises to the conclusion. In other words, what is the missing conditional statement that needs to be added to your diagram to get you to the conclusion? In some cases, it will be linking the necessary terms. In other cases, it will not. It all depends on what they give you in the premises and how those premises relate to the conclusion.

For example, if the argument had given you:

If B, then C (diagrammed B ->C) as the premise rather than:
If A, then B (diagrammed A ->B)

Then the answer would have been:
If A, then B (diagrammed A ->B)
because now that is the missing piece of the argument.

You should be diagramming each of the conditional premises and the conclusion and look at your diagram to try to figure out what is missing. If needed, you could diagram out each conditional answer and add each one to your diagram (one by one) to figure out which one is the missing piece. Of course, that could be quite time consuming, and with practice, you hopefully will be able to narrow down the answers without having to test each one individually. Ideally, you could prephrase the missing conditional statement before looking at the answers and then simply find the answer that matches your prephrase.

Let's look at one more example.

Imagine you have an argument with four terms, A, B, C, and D.

The conclusion states: If A, then D (diagrammed A -> D)

What you need to do to prove this conclusion is to form a chain from A to D in the premises: if A, then B, then C, then D (diagrammed A -> B -> C -> D)

If it is a Justify question, then one of those links will be missing from the argument and that will be the answer. Perhaps it is the first part (if A, then B) that is missing, perhaps it is the second part (if B, then C), or the third part (if C, then D). It just depends on what premises were included and what part is missing.