Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements. Statements: I. All X are A. II. Some R are X. Conclusions: I. Some A are R. II. All R are A.
Only conclusion I follows
Analyzing Statements and Conclusions Logic Problem
This problem requires us to analyze the given statements and determine which of the provided conclusions logically follow from them. We will use principles of logical deduction, often visualized using Venn diagrams, to solve this type of syllogism question.
Understanding the Statements
We are given two statements:
- Statement I: All X are A. This means that the set of all things called 'X' is completely contained within the set of all things called 'A'. If something is an X, it must also be an A.
- Statement II: Some R are X. This means there is at least one thing that is both 'R' and 'X'. The set of 'R' and the set of 'X' have at least one element in common.
Evaluating the Conclusions
Now let's look at each conclusion based on the information from the statements.
Conclusion I: Some A are R.
From Statement II, we know that there is at least one element that belongs to both R and X. Let's call this element 'e'. So, 'e' is an R, and 'e' is an X.
From Statement I, we know that all X are A. Since 'e' is an X, it must also be an A.
Therefore, we have an element 'e' which is both an R and an A. This means that some A are R (or equivalently, some R are A).
Conclusion I logically follows from the given statements.
Conclusion II: All R are A.
From Statement II, we know that *some* R are X. Since all X are A (Statement I), these specific R's that are also X must be A.
However, Statement II only talks about *some* R being X. It does not provide any information about the R's that are *not* X. These other R's might or might not be A.
For example, consider a scenario where:
- X = {p, q}
- A = {p, q, r, s} (All X are A)
- R = {p, t} (Some R are X, specifically 'p')
In this example, Statement I (All X are A) and Statement II (Some R are X) are true. However, Conclusion II (All R are A) is false because element 't' is R but not A.
Since we can construct a valid scenario where the statements are true but Conclusion II is false, Conclusion II does not logically follow from the statements.
Summary of Analysis
Based on our analysis:
- Conclusion I (Some A are R) follows from the statements.
- Conclusion II (All R are A) does not follow from the statements.
Therefore, only conclusion I follows.
Revision Table: Logic Conclusions
| Statement | Relationship | Implication |
|---|---|---|
| I. All X are A. | X <sub>⊂</sub> A (X is a subset of A) | If something is X, it is also A. |
| II. Some R are X. | R ∩ X ≠ ∅ (Intersection of R and X is not empty) | There is at least one element common to R and X. |
Combining these, if there is an element in R and X (from Statement II), and everything in X is also in A (from Statement I), then that element must be in R and A. Hence, Some R are A, which means Some A are R.
Additional Information on Syllogisms
Syllogisms are a form of logical reasoning where a conclusion is drawn from two given or assumed propositions (premises). The structure typically involves a major premise, a minor premise, and a conclusion. In this problem:
- Major Premise (rewritten): Some X are R (from Statement II, just swapping order)
- Minor Premise: All X are A (Statement I)
- Conclusion I: Some A are R
- Conclusion II: All R are A
This specific syllogism structure is a valid form when Conclusion I is drawn (Some R are X, All X are A $\implies$ Some R are A). However, it does not support the conclusion "All R are A". Understanding the different types of categorical propositions (All, No, Some, Some Not) and their relationships is crucial for solving these logic problems.
Important Questions from Syllogism
The statements below are followed by conclusions labeled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All twenty are thirty.
All thirty are forty.
All forty are sixty.
All sixty are seventy.
Conclusions:
I. Some forty are thirty.
II. Some seventy are sixty.
III. No thirty is twenty.The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All Strong are animals.
Some animals are Tigers.
All Tigers are Sharp.
Conclusions:
I. Some Strong are Sharp.
II. No Strong is Sharp.The statements below are followed by conclusions labelled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some women are weak.
Some weaks are female.
All female are iron.
All iron are gold.Conclusions:
I. Some weaks are iron.
II. Some gold are weaks.
III. Some women are female.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some teaspoons are glasses.
All teddies are teaspoons.
Conclusions:
I. Some teddies are glasses.
II. Some glasses are teddies.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally Established facts, decide which conclusion(s) logically and definitely follow(s) from the Information, Given in the Statements.
Statements:
All bangles are rings.
Some rings are toys.
Some toys are dolls.
Conclusions:
I. Some rings are bangles.
II. Some dolls are rings.
