Two statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements : Some animals are mammals. No mammal is herbivore. Conclusions: I. No herbivore is mammal. II. Some mammals are animals. I. No animal is herbivore.
Only conclusions I and II follow.
Analyzing Statements and Conclusions
This question requires us to analyze the given statements and determine which of the conclusions logically follow from them, irrespective of commonly known facts.
Given Statements:
- Statement 1: Some animals are mammals.
- Statement 2: No mammal is herbivore.
Given Conclusions:
- No herbivore is mammal.
- Some mammals are animals.
- No animal is herbivore.
Step-by-Step Analysis of Statements and Conclusions
Let's break down each statement and conclusion to see the relationships between the categories: Animals, Mammals, and Herbivores.
Analyzing Statement 1: Some animals are mammals.
This statement indicates a partial overlap between the set of 'Animals' and the set of 'Mammals'. It means there is at least one animal that is a mammal, and possibly some animals that are not mammals, and possibly some mammals that are not animals (though in biological terms, all mammals are animals, logic problems often treat categories distinctly based *only* on the statements given). For the purpose of this logic problem, we only know about the overlap specified.
Analyzing Statement 2: No mammal is herbivore.
This statement establishes a complete separation between the set of 'Mammals' and the set of 'Herbivores'. There is absolutely no overlap between these two groups. If something is a mammal, it cannot be a herbivore, and if something is a herbivore, it cannot be a mammal.
Evaluating Conclusion I: No herbivore is mammal.
This conclusion is the converse of Statement 2 ("No mammal is herbivore"). If it is true that "No mammal is herbivore", then it logically and necessarily follows that "No herbivore is mammal". These two statements express the same relationship of complete separation between the two categories, just phrased differently.
Conclusion I logically follows from Statement 2.
Evaluating Conclusion II: Some mammals are animals.
Statement 1 says "Some animals are mammals". If there is an overlap such that some members of group A are also members of group B, it logically follows that some members of group B are also members of group A. For example, if "Some students are intelligent", it means there are intelligent people who are students, so "Some intelligent people are students" is also true.
Conclusion II logically follows from Statement 1.
Evaluating Conclusion III: No animal is herbivore.
We know "Some animals are mammals" and "No mammal is herbivore". This means that the portion of animals that are mammals are definitely not herbivores. However, Statement 1 only talks about *some* animals being mammals. It doesn't give us any information about the animals that are *not* mammals. It is possible that some animals that are not mammals *are* herbivores. We cannot conclude that *no* animal at all is a herbivore based only on the given statements. The statements do not provide enough information to establish a complete separation between 'Animals' and 'Herbivores'.
Conclusion III does not logically follow from the statements.
Summary of Conclusions
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. No herbivore is mammal. | Yes | This is the converse of Statement 2 ("No mammal is herbivore"). |
| II. Some mammals are animals. | Yes | This is the converse of Statement 1 ("Some animals are mammals"). |
| III. No animal is herbivore. | No | Statements only link mammals and herbivores. Animals not stated to be mammals could be herbivores. |
Based on the analysis, only conclusions I and II logically follow from the given statements.
Revision Table: Key Concepts in Logic Problems
| Concept | Explanation |
|---|---|
| Statements | Given propositions assumed to be true for the purpose of the problem. |
| Conclusions | Inferences drawn from the statements. Must be logically derivable. |
| Logic Follows | A conclusion logically follows if it must be true whenever the statements are true. |
| Converse | Swapping the subject and predicate of a statement. For "No A is B", the converse "No B is A" is logically equivalent. For "Some A are B", the converse "Some B are A" is logically equivalent. |
Additional Information: Understanding Syllogisms and Venn Diagrams
This type of logic problem is related to syllogisms, which are a form of deductive reasoning where a conclusion is drawn from two or more statements (premises). Analyzing these problems can often be visualized using Venn diagrams.
- A Venn diagram for "Some A are B" would show two overlapping circles.
- A Venn diagram for "No A is B" would show two separate circles.
In this problem:
- Statement 1 ("Some animals are mammals") implies overlapping circles for Animals and Mammals.
- Statement 2 ("No mammal is herbivore") implies separate circles for Mammals and Herbivores.
When you combine these, the Mammals circle is separate from the Herbivores circle, and the Animals circle partially overlaps the Mammals circle. The part of the Animals circle that is *outside* the Mammals circle (animals that are not mammals) could potentially overlap with the Herbivores circle or be separate. Since we don't have information about the relationship between non-mammal animals and herbivores, we cannot conclude that "No animal is herbivore". However, the separation between Mammals and Herbivores is definite, and the overlap between Animals and Mammals is definite and implies the converse.
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.
