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. No R is a G. II. All G’s are H. Conclusions: I. No H is a R. II. No G is a R.
Syllogism Solution: Analyzing Statements and Conclusions
This problem requires us to analyze the logical relationship between different categories based on given statements and determine which conclusions logically follow.
Understanding the Statements
We are given two statements:
- Statement I: No R is a G. This means there is absolutely no overlap between the category 'R' and the category 'G'. They are completely separate sets.
- Statement II: All G’s are H. This means the entire category 'G' is contained within the category 'H'. Everything that is a G is also necessarily an H.
We must accept these statements as true, regardless of our common knowledge.
Evaluating the Conclusions
We need to check if the following conclusions are necessarily true based only on the given statements.
Conclusion I: No H is a R.
Let's consider the relationship between the categories based on the statements. We know G is entirely inside H (Statement II), and R is entirely separate from G (Statement I). However, Statement II only defines the relationship of G to H. It does not restrict the part of H that is not G.
It is possible that some parts of H (specifically, the parts that are not G) could overlap with R. For example, if H represents 'animals', G represents 'dogs', and R represents 'cats', then Statement I ('No cats are dogs') and Statement II ('All dogs are animals') are true. However, the conclusion 'No animals are cats' (No H is R) is false because cats are animals.
Therefore, Conclusion I does not logically follow from the given statements.
Conclusion II: No G is a R.
Statement I explicitly states, "No R is a G". This establishes a mutual exclusion between categories R and G. If no member of R is a member of G, then it is logically equivalent and necessary that no member of G can be a member of R. The relationship of being disjoint is symmetrical.
Therefore, Conclusion II logically follows directly from Statement I.
Determining Which Conclusion Follows
Based on our analysis:
- Conclusion I (No H is a R) does not follow.
- Conclusion II (No G is a R) follows.
Thus, only conclusion II logically follows from the given statements.
Syllogism Analysis Revision Table
| Statement/Conclusion | Type | Relationship Implied | Logically Follows? | Reasoning |
|---|---|---|---|---|
| Statement I: No R is G | Universal Negative (E) | R and G are disjoint sets | N/A | Given premise |
| Statement II: All G are H | Universal Affirmative (A) | G is a subset of H | N/A | Given premise |
| Conclusion I: No H is R | Universal Negative (E) | H and R are disjoint sets | No | Statement II doesn't restrict parts of H not in G from overlapping with R. |
| Conclusion II: No G is R | Universal Negative (E) | G and R are disjoint sets | Yes | Directly equivalent to Statement I ("No R is a G"). |
Additional Information on Syllogisms and Logic
Syllogisms are fundamental in deductive reasoning. They consist of premises (statements) and a conclusion. The goal is to determine if the conclusion is necessarily true if the premises are assumed to be true.
Categorical syllogisms involve statements that relate categories (or sets). The four standard types of categorical statements, often denoted by vowels, are:
- A: All S are P (Universal Affirmative)
- E: No S is P (Universal Negative)
- I: Some S are P (Particular Affirmative)
- O: Some S are not P (Particular Negative)
Visual tools like Venn diagrams can be very helpful in solving these types of problems. You draw circles representing each category and shade or mark regions based on the statements to see what regions must be empty or must contain members. If the conclusion's relationship is guaranteed by the diagram based on the statements, it follows logically.
Statement I ("No R is a G") means the overlap area between R and G is empty. Statement II ("All G's are H") means the area of G outside H is empty. When you combine these, the area where G and R overlap is empty, and all of G is within H. The critical insight for Conclusion I is that there's still an area within H (the part not covered by G) whose relationship with R is not specified by the statements, allowing for potential overlap.
Important Questions from Conventional Syllogism
- Consider the following Statements and Conclusions :
Statements : Mohan is a good sportsman.
Sportsmen are healthy.
Conclusions : I. All healthy persons are sportsmen.
II. Mohan is healthy.
Which one of the following is correct? Premises : All metals are grey in colour. Some metals are heavy.
Conclusion: I. All heavy metals are grey in colour.
II. All light metals are not grey in colour.Premises : All film stars are playback singers. All film directors are film stars.
Conclusion: I. All film directors are playback singers.
II. Some film stars are film directors.Premises : All apples are golden in colour. No golden coloured things are cheap.
Conclusion: I. All apples are cheap.
II. Golden coloured apples are not cheap.Read the given statement and conclusions carefully. Assuming that the information given in the statement 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 statement.
Statement:
All reds are black.
Conclusions:
1. Some blacks are red.
2. Some reds are black.
