In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements. Statements: I. Some red are black. II. No white is black. Conclusion: I. No black is white. II. No white is red. III. No black is red.
Only conclusion I follows
Understanding Syllogism Statements and Conclusions
This question requires us to analyze given statements and determine which of the provided conclusions logically follow. We must assume the statements are true, even if they contradict common knowledge.
Given Statements:
- Statement I: Some red are black.
- Statement II: No white is black.
Given Conclusions:
- Conclusion I: No black is white.
- Conclusion II: No white is red.
- Conclusion III: No black is red.
Analyzing the Syllogism Logic
We can analyze these statements and conclusions using logical deduction or by visualizing them, for example, with Venn diagrams (though we will explain the logic directly here).
Let's break down each statement:
- Statement I: Some red are black. This means there is at least one element that belongs to both the 'red' category and the 'black' category. It does *not* mean all red are black, nor does it mean all black are red. There might be red things that are not black, and black things that are not red.
- Statement II: No white is black. This means there is absolutely no overlap between the 'white' category and the 'black' category. They are entirely separate sets. If something is white, it cannot be black, and if something is black, it cannot be white.
Evaluating Each Conclusion
Let's evaluate each conclusion based on the truth of the statements:
Conclusion I: No black is white.
Statement II says "No white is black". This establishes a complete separation between 'white' and 'black'. If no element of 'white' is in 'black', it logically follows that no element of 'black' can be in 'white'. This is simply the contrapositive or converse restatement of Statement II, and it is logically equivalent. Therefore, this conclusion follows directly from Statement II.
Conclusion II: No white is red.
We know "Some red are black" (Statement I) and "No white is black" (Statement II). Statement II tells us white things are separate from black things. Statement I tells us some red things are black. This means the portion of red things that are black *cannot* be white (because nothing black is white). However, Statement I only talks about *some* red things being black. It doesn't give us any information about the red things that are *not* black. White could potentially overlap with this part of the red category. We cannot conclude that *no* white is red based on the given statements. It is possible that some white are red, as long as those red things are not among the ones that are also black.
Conclusion III: No black is red.
Statement I says "Some red are black". This directly implies that there is at least one element that is both red and black. If some red are black, it means there are black things that are red. Conclusion III, "No black is red", directly contradicts Statement I. Therefore, this conclusion does not follow and is false based on the statements.
Summary of Conclusion Analysis
| Conclusion | Analysis based on Statements | Logically Follows? |
|---|---|---|
| I. No black is white. | Directly follows from Statement II ("No white is black"). | Yes |
| II. No white is red. | Statements do not provide information about the relationship between 'white' and the part of 'red' that is not 'black'. Cannot be concluded. | No |
| III. No black is red. | Directly contradicts Statement I ("Some red are black"). | No |
Based on the analysis, only Conclusion I logically follows from the given statements.
Final Answer on Syllogism Conclusions
Only conclusion I logically follows from the given statements.
Revision Table: Key Syllogism Concepts
| Syllogism Term | Explanation | Example Type |
|---|---|---|
| Statement | A premise given as true for the purpose of the argument. | All A are B, Some A are B, No A is B, Some A are not B |
| Conclusion | A proposition that is asserted to follow logically from the statements. | Must be derived strictly from the statements provided. |
| Logical Deduction | Reasoning from general statements to reach a specific conclusion. | If all humans are mortal, and Socrates is human, then Socrates is mortal. |
| Converse | Swapping the subject and predicate of a statement. (e.g., "No A is B" converse is "No B is A"; "Some A are B" converse is "Some B are A") | Converse of "No white is black" is "No black is white". Converse of "Some red are black" is "Some black are red". |
Additional Information: Analyzing Syllogisms
Syllogism questions test your ability to understand and apply logical rules, not your knowledge of the real world. The statements are your only source of truth.
- 'Some' means at least one, and possibly all.
- 'No' means zero overlap.
- 'All' means complete inclusion.
When analyzing conclusions, check if the conclusion *must* be true if the statements are true. If there is *any* possibility, however remote in a diagram or mental model, that the conclusion could be false while the statements are true, then the conclusion does not logically follow.
For "Some A are B", the converse "Some B are A" is always true. For "No A is B", the converse "No B is A" is always true. However, for "All A are B", the converse "All B are A" is NOT necessarily true.
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.
