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 A are T.
II. Some T are H.
Conclusions:
I. No T is A.
II. No H is T.
III. No A is H.

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 A are T.

II. Some T are H.

Conclusions:

I. No T is A.

II. No H is T.

III. No A is H.

Prepp Team · Accepted Answer

Analyzing Syllogism Statements and Conclusions

This question asks us to analyze a set of statements and determine which of the given conclusions logically follows from them. We must assume the statements are true, even if they seem contrary to common knowledge.

The given statements are:

I. Some A are T.
II. Some T are H.

The given conclusions are:

I. No T is A.
II. No H is T.
III. No A is H.

Understanding the Syllogism Statements

Let's break down what each statement tells us:

Statement I: Some A are T. This is a particular affirmative statement. It means there is at least one entity that belongs to both category A and category T. This implies an overlap between A and T. If some A are T, it also logically follows that some T are A.
Statement II: Some T are H. This is also a particular affirmative statement. It means there is at least one entity that belongs to both category T and category H. This implies an overlap between T and H. If some T are H, it also logically follows that some H are T.

Evaluating Syllogism Conclusions

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

Conclusion I: No T is A. This is a universal negative statement. It claims there is absolutely no overlap between T and A. However, Statement I explicitly says "Some A are T," which means there *is* some overlap (at least one element in common) between A and T. Since Statement I is assumed true, Conclusion I directly contradicts it. Therefore, Conclusion I does not logically follow from the statements.
Conclusion II: No H is T. This is a universal negative statement. It claims there is absolutely no overlap between H and T. However, Statement II explicitly says "Some T are H," which means there *is* some overlap (at least one element in common) between T and H. Since Statement II is assumed true, Conclusion II directly contradicts it. Therefore, Conclusion II does not logically follow from the statements.
Conclusion III: No A is H. This is a universal negative statement. It claims there is absolutely no overlap between A and H. The statements given are "Some A are T" and "Some T are H". Both are 'Some' statements (particular affirmative). When we have two 'Some' statements involving a middle term (T in this case, linking A and H), we cannot logically derive a definite conclusion, especially not a universal negative conclusion like "No A is H". It is possible that A and H overlap, and it is also possible they do not, based *only* on these two statements. For example, if A= {1,2}, T = {2,3}, H = {3,4}, then Some A are T ({2}), Some T are H ({3}), and No A is H. But if A = {1,2}, T = {2,3}, H = {2,3,4}, then Some A are T ({2}), Some T are H ({2,3}), and Some A are H ({2}). Since we can construct scenarios where A and H overlap, the conclusion "No A is H" does not necessarily follow. Therefore, Conclusion III does not logically follow from the statements.

Summary of Conclusion Analysis

Based on our analysis of the statements and how they relate to each conclusion:

Conclusion I (No T is A) is contradicted by Statement I (Some A are T).
Conclusion II (No H is T) is contradicted by Statement II (Some T are H).
Conclusion III (No A is H) cannot be logically derived from two particular affirmative statements.

Since none of the individual conclusions logically follow from the given statements, the correct assessment is that neither conclusion follows.

Syllogism Revision Table | Key Concepts

Statement Type	Representation	Conversion	Example
Universal Affirmative (All X are Y)	All of X is included in Y.	Some Y are X (if X exists).	All dogs are mammals.
Universal Negative (No X is Y)	No part of X is included in Y.	No Y is X.	No fish are birds.
Particular Affirmative (Some X are Y)	At least one part of X is included in Y.	Some Y are X.	Some students are athletes.
Particular Negative (Some X are not Y)	At least one part of X is not included in Y.	Conversion not generally valid.	Some fruits are not sweet.

Syllogism Additional Information | Rules & Types

Syllogism is a form of logical argument where a conclusion is derived from two premises (statements). Understanding the types of statements and basic rules is crucial.

Types of Statements: Statements are classified as Universal Affirmative (A), Universal Negative (E), Particular Affirmative (I), and Particular Negative (O).
Validity: A conclusion logically follows if it is impossible for the conclusion to be false when the statements are true.
Rules for 'Some' Statements: Two 'Some' statements (I-I combination) with a common middle term cannot lead to a universal conclusion (All or No) or a definite particular conclusion about the two extreme terms unless combined with other statement types following specific rules (which is not the case here).
Contradiction: A statement and its contradictory cannot both be true. For example, "Some A are T" and "No A is T" are contradictories (or "Some T are A" and "No T is A"). If one is true, the other must be false.

In this specific problem, the first two conclusions directly contradict the implications of the given statements, and the third conclusion cannot be validly drawn from the combination of statement types provided.