In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. Some L are R.
II. Some A are R.
Conclusion:
I. All A are L.
II. All R are L.

Question

In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.

Statements:

I. Some L are R.

II. Some A are R.

Conclusion:

I. All A are L.

II. All R are L.

Prepp Team · Accepted Answer

Solving Syllogism Statements and Conclusions

This question asks us to analyze given statements and determine which of the provided conclusions logically follow. This type of problem falls under the category of logical syllogism or deductive reasoning.

We are given two statements:

I. Some L are R.
II. Some A are R.

And two conclusions:

I. All A are L.
II. All R are L.

We need to assume the given statements are true, even if they contradict common knowledge, and then see which conclusion(s) must be true based solely on these statements.

Analyzing the Syllogism Statements

Let's break down what each statement tells us:

Statement I: Some L are R. This means there is at least one element that belongs to both the category L and the category R. It indicates a partial overlap between L and R. It does not say anything about all L or all R.
Statement II: Some A are R. Similarly, this means there is at least one element that belongs to both the category A and the category R. It shows a partial overlap between A and R. It does not provide information about all A or all R.

Both statements involve the term 'R'. 'R' acts as a connector between 'L' and 'A'. However, both connections are only partial ('Some'). There is no statement directly connecting 'L' and 'A', and importantly, there are no 'All' statements provided.

Evaluating the Syllogism Conclusions

Now, let's evaluate each conclusion based on the statements:

Conclusion I: All A are L.

This conclusion claims that every element in category A is also in category L. Do the statements support this? Statement II says 'Some A are R', and Statement I says 'Some L are R'. While A and L both have some relationship with R, this doesn't guarantee any relationship between A and L, let alone a universal inclusion of all A within L. It is possible that the 'Some A that are R' are completely separate from the 'Some L that are R', or they could overlap, but we cannot definitively conclude that ALL A are L. Therefore, Conclusion I does not logically follow.

Conclusion II: All R are L.

This conclusion claims that every element in category R is also in category L. Do the statements support this? Statement I says 'Some L are R'. This only guarantees that there's an overlap; some Rs are L. It does not imply that *all* Rs are L. There could be Rs that are not L (perhaps they are only A, or something else entirely). Statement II says 'Some A are R', which also doesn't provide any information about all R being L. Therefore, Conclusion II does not logically follow.

Summary of Findings

Based on the analysis of the statements, neither of the conclusions can be logically deduced. 'Some' statements provide limited information about partial overlaps and cannot be used to derive universal 'All' conclusions, especially when the connection between the terms in the conclusion is indirect or only partial through a middle term.

Statement Type	Relationship	Information Provided
Some X are Y	Partial inclusion/overlap	At least one X is Y. At least one Y is X. Cannot conclude about all X or all Y.

Since neither Conclusion I nor Conclusion II logically follows from the given statements, the correct answer is that neither conclusion follows.

Syllogism Revision Table

Statement	Analysis	Valid Inferences (Simple)
Some L are R	Partial overlap between L and R.	Some R are L. Existence of at least one element in L and R.
Some A are R	Partial overlap between A and R.	Some R are A. Existence of at least one element in A and R.

Additional Information on Syllogism Logic

Syllogism problems involve deductive reasoning, where you start with general statements (premises) and try to reach a specific conclusion that must be true if the premises are true. Here are some key points about syllogisms with 'Some' statements:

'Some' means 'at least one'. It does not mean 'only some' or 'not all'.
From two 'Some' statements, no universal conclusion ('All' or 'No') can be logically drawn between the terms.
If the middle term (R in this case) is only partially connected to the other two terms (L and A), you cannot establish a definite universal relationship or lack thereof between those outer terms (L and A).
To derive an 'All' conclusion, you typically need at least one 'All' statement among the premises.

Mastering syllogism involves understanding the relationships implied by different types of statements ('All', 'No', 'Some', 'Some Not') and the rules for combining them to form valid conclusions. Visual aids like Venn diagrams can often help in understanding the possible relationships and testing the validity of conclusions.