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: 1) All pupils are crows. 2) All crows are cats. Conclusion I. All cats are crows. II. Some crows are pupils.
Only conclusion II follows.
Logic Statements and Conclusions Explained
This question asks us to analyze two statements and determine which of the given conclusions logically follow from them. We must assume the statements are true, even if they contradict general knowledge.
The two statements are:
- All pupils are crows.
- All crows are cats.
Let's analyze each statement and conclusion.
Analyzing Statement 1: All pupils are crows
This statement means that the group of 'pupils' is completely contained within the group of 'crows'. If someone is a pupil, they must also be a crow, according to this statement.
Analyzing Statement 2: All crows are cats
This statement means that the group of 'crows' is completely contained within the group of 'cats'. If something is a crow, it must also be a cat, according to this statement.
Analyzing Conclusion I: All cats are crows
This conclusion suggests that the group of 'cats' is completely contained within the group of 'crows'. Let's see if this follows from the statements.
From Statement 2, we know 'All crows are cats'. This establishes a relationship where the 'crows' group is a subset of the 'cats' group. However, this does not automatically mean that the 'cats' group is a subset of the 'crows' group. There could be other things that are 'cats' but are not 'crows'.
Think of an example: All dogs are animals. This statement is true. Does it mean all animals are dogs? No, because there are other animals like cats, birds, etc. Similarly, 'All crows are cats' does not imply 'All cats are crows'.
Therefore, Conclusion I does not logically follow from the given statements.
Analyzing Conclusion II: Some crows are pupils
This conclusion suggests that there is at least one individual who is both a 'crow' and a 'pupil'. Let's look back at the statements.
Statement 1 says, 'All pupils are crows'. If every single pupil is a crow, then it naturally follows that there must be some crows that are pupils (specifically, all the pupils are crows). Unless the group of 'pupils' is entirely empty (which is not usually assumed in these types of problems), if there are pupils, and every pupil is a crow, then some members of the 'crows' group are definitely 'pupils'.
Consider this: If a basket contains only red apples, then it is true that 'All apples in the basket are red'. It is also true that 'Some red things in the basket are apples'. Since every apple is red, the group of apples is part of the group of red things in the basket. This means some of the red things are those apples.
Therefore, Conclusion II logically follows from Statement 1.
Summary of Logic Conclusions
Based on our analysis:
- Conclusion I (All cats are crows) does not follow from the statements.
- Conclusion II (Some crows are pupils) logically follows from Statement 1.
| Statement/Conclusion | Relationship | Follows Logically? | Reasoning |
|---|---|---|---|
| Statement 1 | All pupils are crows | Given | Pupils are a subset of Crows |
| Statement 2 | All crows are cats | Given | Crows are a subset of Cats |
| Conclusion I | All cats are crows | No | Reverse of "All Crows are Cats" does not necessarily hold. Cats could include things that are not crows. |
| Conclusion II | Some crows are pupils | Yes | If "All Pupils are Crows", then the set of Pupils is within the set of Crows. Therefore, some members of the Crows set must be Pupils. |
Thus, only conclusion II logically follows from the given statements.
Revision Table: Logic Statements and Conclusions
| Concept | Description | Key Point |
|---|---|---|
| Categorical Syllogism | A logical argument where conclusions are drawn from two premise statements. | Involves quantifiers like All, No, Some. |
| Universal Affirmative (All A are B) | Statement meaning the entire set A is included in set B. | Does NOT imply "All B are A". IMPLIES "Some B are A" (assuming A is not empty). |
| Existential Import | The implication that universal statements ("All A are B") mean that set A is not empty, allowing for "Some A are B" or "Some B are A" inferences. | Often assumed in these types of logic problems. |
Additional Information: Logic and Reasoning
This type of question is based on deductive reasoning, specifically involving categorical statements. The key is to understand the relationship between categories (like Pupils, Crows, Cats) established by the words 'All' and 'Some'.
- 'All A are B' means every single member of group A is also a member of group B. This implies that group A is a sub-group of group B.
- 'Some A are B' means there is at least one member that belongs to both group A and group B. This implies the intersection of group A and group B is not empty.
It's important not to use outside knowledge about pupils, crows, or cats. The conclusions must follow strictly and only from the information given in the statements. The statements create a specific, perhaps fictional, world governed by the stated rules.
Drawing Venn diagrams can often help visualize these relationships. If you draw a circle for 'Pupils' inside a circle for 'Crows', and then the 'Crows' circle inside a circle for 'Cats', you can clearly see that 'All pupils are cats' follows, 'All cats are crows' does not necessarily follow, and 'Some crows are pupils' does follow.
Important Questions from Syllogism
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:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
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:
All directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
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 follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
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:
Some cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
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:
All employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
