Syllogisms! 1

Logic Level 1

All pens are roads.

All roads are houses.

We are given the two statements above. Which of the following conclusions must be true?

(1) : All houses are pens.

(2) : Some houses are pens.

Assume that the set of pens is non-empty.

Neither (1) nor (2) are true (2) is true only None of these choices Both (1) and (2) are true (1) is true only

2 solutions

Pranshu Gaba
Jul 5, 2016

It is a good idea to use set theory and draw Venn diagram in such problems.

From the given statements, we get that $\text{Pens} \subset \text{Roads} \subset \text{Houses}$ . The following is a Venn diagram for the same.

From the Venn diagram, we can see that statement (2) must be true, that is, some houses are pens.

It is not necessary that all houses are pens, since there may exist some houses that are not pens.

This struck me as a very awkwardly worded problem. We're given that all pens are roads and all roads are houses. Given that alone I would agree with statement 2. The thing is, we're only allowed to assume that the set of pens is not empty and we are asked to analyze statements a thru d. Statement a can be true if the set of roads that are not pens and the set of houses that are not roads are both empty.

George Argiriadis - 4 years, 10 months ago

Nice diagram.

In the problem this confusing part should be removed:

--begin quote--

Enter your answer as:

a) if only conclusion 1 follows.

b) if only conclusion 2 follows

c) If either of the conclusions follow.

d) If Neither conclusion follows.

e) If both the conclusions follow.

--end quote--

steve miller - 3 years, 7 months ago

Thanks, it is removed.

Pranshu Gaba - 3 years, 7 months ago

You would need to provide more information, such as "not all roads are pens" to get the answer you have specified here. Let's look at the information you have provided, but first let's call Pens 'A', Roads 'B, and Houses 'C'. With your guidelines, we can say the following: A=B, and B=C. Without having any other information, we can safely arrive at the conclusion that A=C. For one to arrive at the 'correct answer' they have to presuppose that the statement "B does does not always equate to A" is true.

Aidan Moore - 3 years, 5 months ago

The question says "Which of the following conclusions must be true?" I have shown that from the given information, (2) must be true.

We are not given "All roads are pens", so how can we conclude that the sets are equal? There might be some roads that are not pens. We would need more information to conclude if B = A, so (1) may or may not be true. We need that extra information.

Pranshu Gaba - 3 years, 5 months ago

I thought the same .

Maunil Chopra - 2 years, 2 months ago

Akhash Raja Raam
Jun 28, 2016

All pens are roads but all roads are not pens. Using this logic solving this problem is easy!

Syllogisms! 1

2 solutions

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