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. Statement I: Some fish are frogs. Statement II: Some frogs are whales. Conclusion I: Some fish are whales. Conclusion II: No fish is a whale.
Either I or II follows
Understanding the Logical Reasoning Problem
This question presents two statements and asks us to determine which of the given conclusions logically follows from them, assuming the statements are true, regardless of whether they align with real-world facts.
Analyzing the Statements
- Statement I: Some fish are frogs. This establishes a relationship between the set of fish and the set of frogs, indicating that there is at least one entity that belongs to both sets.
- Statement II: Some frogs are whales. This establishes a relationship between the set of frogs and the set of whales, indicating that there is at least one entity that belongs to both sets.
Both statements use the quantifier "Some," which in logic means "at least one" and possibly more, but not necessarily all.
Analyzing the Conclusions
- Conclusion I: Some fish are whales. This suggests the possibility of an overlap between the set of fish and the set of whales.
- Conclusion II: No fish is a whale. This suggests there is no overlap between the set of fish and the set of whales; the two sets are entirely separate.
Evaluating Logical Conclusions from Statements
Let's consider how the statements relate the sets of Fish, Frogs, and Whales:
- We know Fish and Frogs share some members (from Statement I).
- We know Frogs and Whales share some members (from Statement II).
The question is, does this guarantee any relationship between Fish and Whales? Not necessarily. The "some frogs" that are fish could be different individuals from the "some frogs" that are whales. For example:
- Imagine a group of frogs. Some of these frogs could be fish (as stated). Some other frogs (or possibly the same ones) could be whales (as stated).
- Possibility 1: The group of frogs that are fish overlaps with the group of frogs that are whales. In this case, there would be some entities that are fish, are frogs, AND are whales. This scenario supports Conclusion I ("Some fish are whales").
- Possibility 2: The group of frogs that are fish is completely separate from the group of frogs that are whales. In this case, the statements are still true, but there is no required overlap between fish and whales. It is possible that no fish are whales. This scenario supports Conclusion II ("No fish is a whale").
Since the statements allow for both the possibility that some fish are whales and the possibility that no fish are whales, neither Conclusion I nor Conclusion II logically follows on its own as a certain consequence of the statements.
Understanding the Relationship Between the Conclusions
However, let's look at the conclusions themselves. "Some fish are whales" and "No fish is a whale" are contradictory statements. If one is true, the other must be false, and vice-versa. There is no middle ground; for any given entity that is a fish, it either is a whale or it is not a whale. Therefore, either there is at least one fish that is a whale (Conclusion I is true), or there are no fish that are whales (Conclusion II is true). One of these two statements *must* be true.
Drawing the Final Logical Inference
Because the statements do not force one conclusion to be true over the other, but the two conclusions presented are contradictories covering all possibilities, the correct logical inference is that Either Conclusion I or Conclusion II follows. We cannot know which one is true based on the premises, but we know one of them must be.
Revision Table: Syllogism Components
| Component | Role | Example from Problem |
|---|---|---|
| Statements | Premises assumed to be true. | Some fish are frogs. Some frogs are whales. |
| Conclusions | Propositions derived from statements. | Some fish are whales. No fish is a whale. |
| Logical Follows | The conclusion must be true if the statements are true. | Determining if conclusions about fish and whales follow from the given statements. |
Additional Information: Syllogistic Logic Basics
This problem is an example of categorical syllogisms, which deal with relationships between categories (like Fish, Frogs, Whales) using quantifiers like "Some," "All," and "No."
- Statements using "Some...are..." are called Particular Affirmative propositions.
- Statements using "No...is..." are called Universal Negative propositions.
When both premises in a syllogism are "Some" statements, it is often impossible to draw a definite conclusion about the relationship between the first and third terms. However, as seen here, if the options include contradictory conclusions, the "Either/Or" option is the correct logical outcome because the truth must lie with one of the contradictories, even if the premises don't specify which.
Important Questions from Conventional 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:
No bank is an office.
All offices are stalls.
Conclusions:
I. No bank is a stall.
II. No stall is a bank.
III. Some stalls are offices.
IV. All the stalls are offices
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 flowers are beautiful.
Vaidehi is beautiful.
Conclusions:
I. Vaidehi is a flower.
II. Some beautiful are flowers.
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:
1. All rugs are blankets.
2. All blankets are pillows.
3. Some blankets are frames.
Conclusions:
I. All pillows are rugs.
II. Some pillows are rugs.
III. All rugs are frames
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 fingers are toes.
Some toes are rings.
Some rings are hands.
Conclusions:
I. Some hands are toes.
II. Some rings are fingers.
III. Some hands are fingers.
V. Some fingers are rings.
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 polygons are angles.
All angles are diagonals.
All cones are cubes.
All cubes are decagons.
No diagonal is a cube.
Conclusions:
I. Some diagonals are polygons.
II. All diagonals are decagons.
III. No polygon is a cone.
IV. Some cubes are angles.
