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 snakes are crocodiles. All crocodiles are turtles All birds are crocodiles. Conclusions : I. All snake are turtles. II. All birds are turtles. III. Some snakes are birds.
Only conclusion I and II follow
Analyzing Logic Statements and Conclusions
This problem requires us to analyze a set of statements and determine which of the given conclusions logically follow based on the information provided in the statements. In logic problems like this, we must assume the statements are true, even if they contradict real-world knowledge. We need to find conclusions that are necessarily true if the statements are true.
Understanding the Given Statements
We have three statements:
- Statement 1: All snakes are crocodiles.
- Statement 2: All crocodiles are turtles.
- Statement 3: All birds are crocodiles.
We can visualize these relationships. Statement 1 tells us that the set of snakes is entirely contained within the set of crocodiles. Statement 2 tells us the set of crocodiles is entirely contained within the set of turtles. Statement 3 tells us the set of birds is entirely contained within the set of crocodiles.
Analyzing the Conclusions Based on Statements
Now let's examine each conclusion:
Conclusion I: All snakes are turtles.
Let's combine Statement 1 and Statement 2:
- All snakes are crocodiles.
- All crocodiles are turtles.
If every snake is a crocodile, and every crocodile is a turtle, then it logically follows that every snake must also be a turtle. The set of snakes is a subset of crocodiles, and the set of crocodiles is a subset of turtles. Therefore, the set of snakes is a subset of turtles.
Conclusion I logically follows from the statements.
Conclusion II: All birds are turtles.
Let's combine Statement 3 and Statement 2:
- All birds are crocodiles.
- All crocodiles are turtles.
If every bird is a crocodile, and every crocodile is a turtle, then it logically follows that every bird must also be a turtle. The set of birds is a subset of crocodiles, and the set of crocodiles is a subset of turtles. Therefore, the set of birds is a subset of turtles.
Conclusion II logically follows from the statements.
Conclusion III: Some snakes are birds.
Let's look at the statements involving snakes and birds:
- All snakes are crocodiles.
- All birds are crocodiles.
Both snakes and birds are subsets of crocodiles. However, the statements do not provide any information about the relationship between snakes and birds themselves. The set of snakes could be entirely separate from the set of birds within the set of crocodiles, or they could overlap. Based *only* on the given statements, we cannot conclude that there is definitely an overlap (i.e., some snakes are birds). While it's possible, it's not a logical necessity derived from the statements.
Conclusion III does not logically follow from the statements.
Summary of Conclusions Analysis
Let's summarize our findings in a table:
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. All snakes are turtles. | Yes | From Statements 1 & 2: Snakes < Crocodiles < Turtles implies Snakes < Turtles. |
| II. All birds are turtles. | Yes | From Statements 3 & 2: Birds < Crocodiles < Turtles implies Birds < Turtles. |
| III. Some snakes are birds. | No | Statements place both within Crocodiles but give no relation between them. |
Based on the analysis, only Conclusion I and Conclusion II logically follow from the given statements.
Revision Table: Checking Logic Statements
| Step | Statements Used | Inference Made | Checks |
|---|---|---|---|
| 1 | All snakes are crocodiles. (S < C) | Set of S is inside Set of C. | Basic understanding of 'All' relation. |
| 2 | All crocodiles are turtles. (C < T) | Set of C is inside Set of T. | Basic understanding of 'All' relation. |
| 3 | Statements 1 & 2 combined. | S < C and C < T implies S < T. (Conclusion I) | Transitive property of 'All' relation. Valid. |
| 4 | All birds are crocodiles. (B < C) | Set of B is inside Set of C. | Basic understanding of 'All' relation. |
| 5 | Statements 4 & 2 combined. | B < C and C < T implies B < T. (Conclusion II) | Transitive property of 'All' relation. Valid. |
| 6 | Statements 1 & 4 combined. | S < C and B < C. | Both S and B are subsets of C. No direct relation stated between S and B. Cannot infer S and B overlap or are separate. |
| 7 | Conclusion III: Some snakes are birds. (Some S are B) | Does not follow from S < C and B < C. | 'All A are C' and 'All B are C' does not necessitate 'Some A are B'. Invalid inference. |
Additional Information on Syllogisms and Logic
This problem is an example of a categorical syllogism, a form of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. The statements are premises, and the conclusions are deductions.
Key terms in categorical syllogisms often include quantifiers like "All," "No," and "Some," which describe the extent to which the subject term is related to the predicate term.
- All A are B: Every member of set A is also a member of set B. (A is a subset of B).
- No A are B: No member of set A is a member of set B. (Sets A and B are disjoint).
- Some A are B: At least one member of set A is also a member of set B. (Sets A and B have at least one element in common; their intersection is not empty).
- Some A are not B: At least one member of set A is not a member of set B. (Set A is not a subset of set B).
Understanding how these quantifiers relate to each other and how relationships between sets (like being a subset) work is crucial for solving these logic problems. The principle used in Conclusion I and II (If All A are B and All B are C, then All A are C) is a fundamental rule in logic.
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.
