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: I. No blue is white. II. No black is blue. Conclusions: I. All whites are blacks. II. No blue is black. III. Some whites are not blues.
Only conclusions II and III follow
Understanding Syllogism Statements and Conclusions
Syllogism questions test your ability to draw logical conclusions from given statements, regardless of whether those statements are factually correct in the real world. We must assume the statements are true and see which conclusions necessarily follow from them.
Analyzing the Syllogism Problem
We are given two statements and three conclusions. Our task is to determine which of the conclusions logically follow from the given statements.
The Statements:
- I. No blue is white.
- II. No black is blue.
The Conclusions:
- I. All whites are blacks.
- II. No blue is black.
- III. Some whites are not blues.
Analyzing the Statements in Detail
Let's break down what each statement means in terms of sets or categories:
- Statement I: No blue is white.
This means there is absolutely no overlap between the set of 'blue' things and the set of 'white' things. The intersection of Blue and White is empty. In set notation, Blue $\cap$ White = $\emptyset$. This also implies No white is blue.
- Statement II: No black is blue.
This means there is no overlap between the set of 'black' things and the set of 'blue' things. The intersection of Black and Blue is empty. In set notation, Black $\cap$ Blue = $\emptyset$. This also implies No blue is black.
Evaluating Each Conclusion Based on the Statements
Now, let's examine each conclusion one by one to see if it is logically supported by the statements.
Conclusion I: All whites are blacks.
Does this follow from "No blue is white" and "No black is blue"?
- Statement I tells us Blue and White are separate.
- Statement II tells us Black and Blue are separate.
The statements establish a relationship between Blue and White, and between Black and Blue. However, they provide no direct or indirect information about the relationship between White and Black. White and Black could be completely separate, they could overlap, or one could be entirely contained within the other (like Conclusion I suggests). Since we can't definitively say based *only* on the statements that all whites must be blacks, this conclusion does not follow.
Conclusion II: No blue is black.
Does this follow from "No blue is white" and "No black is blue"?
- Statement II explicitly states "No black is blue".
The statement "No black is blue" is logically equivalent to "No blue is black". Since Statement II directly provides this information, Conclusion II logically follows from Statement II.
Conclusion III: Some whites are not blues.
Does this follow from "No blue is white" and "No black is blue"?
- Statement I states "No blue is white". This is equivalent to "No white is blue".
If "No white is blue", it means there are absolutely no white things that are also blue. This implies that any white thing you consider cannot be blue. Assuming there is at least one 'white' thing (standard assumption in such problems unless specified otherwise), then that white thing is not blue. Therefore, "Some whites are not blues" must be true. Conclusion III logically follows from Statement I.
Summary of Conclusion Analysis
| Conclusion | Analysis | Follows? |
|---|---|---|
| I. All whites are blacks. | Statements relate Blue to White and Blue to Black, but not directly White to Black. Cannot be concluded. | No |
| II. No blue is black. | Directly follows from Statement II (No black is blue, which is equivalent). | Yes |
| III. Some whites are not blues. | Follows from Statement I (No blue is white, which is equivalent to No white is blue. If no white is blue, then some whites must not be blue, assuming whites exist). | Yes |
Based on the analysis, only Conclusion II and Conclusion III logically follow from the given statements.
Final Answer
The conclusions that logically follow are Conclusion II and Conclusion III.
Revision Table: Syllogism Rules
| Statement Type | Interpretation | Example |
|---|---|---|
| All A are B | Every element of set A is in set B (A ⊆ B) | All cats are animals |
| No A is B | No element of set A is in set B (A ∩ B = ∅) | No dog is a cat |
| Some A are B | At least one element of set A is also in set B (A ∩ B ≠ ∅) | Some students are intelligent |
| Some A are not B | At least one element of set A is not in set B (A \ B ≠ ∅) | Some birds cannot fly |
Additional Information on Syllogism Logic
Syllogisms are a form of deductive reasoning. This means that if the premises (statements) are true, the conclusion must also be true. We are not using inductive reasoning (making generalizations from observations) or abductive reasoning (finding the most likely explanation).
Key principles applied here:
- Equivalence: "No A is B" is equivalent to "No B is A".
- Contrapositive (partial): While not a direct contrapositive, the truth of "No A is B" (meaning A and B are separate) directly implies that any A that exists cannot be a B, thus "Some A are not B" (assuming A exists).
- Lack of Information: If the statements do not connect two categories, we cannot assume a relationship between them. For example, statements about A-B and B-C do not automatically give a specific relationship between A and C (unless the structure is one that links them, like "All A are B" and "All B are C" implies "All A are C"). In our problem, White is linked to Blue, and Black is linked to Blue, but White and Black are not linked to each other through Blue in a way that forces "All whites are blacks".
Practicing with Venn diagrams can often help visualize these relationships, although formal logic rules are the most reliable method for complex syllogisms.
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.
