Two statements are given followed by two conclusions numbered I and II. 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: All black is white. Some white is grey. Conclusions: I. Some grey is black. II. Some white is black.
Only conclusion II follows
Understanding the Syllogism Problem
This question asks us to analyze two given statements and determine which of the provided conclusions logically follows from them. We must assume the statements are true, even if they contradict common knowledge. This type of problem falls under the category of logical deduction, specifically syllogisms.
Analyzing the Given Statements
We have two statements:
- Statement 1: All black is white.
- Statement 2: Some white is grey.
These are categorical propositions. Statement 1 is a Universal Affirmative (A type), stating that the entire class of 'black' is included in the class of 'white'. Statement 2 is a Particular Affirmative (I type), stating that there is at least one member common to the classes 'white' and 'grey'.
Evaluating the Conclusions
We need to check the validity of two conclusions based *only* on the given statements:
- Conclusion I: Some grey is black.
- Conclusion II: Some white is black.
Checking Conclusion I: Some grey is black
This conclusion attempts to establish a relationship between 'grey' and 'black'. Let's consider the statements:
- All black is white. (Black <span>⊆</span> White)
- Some white is grey. (Intersection of White and Grey is non-empty)
From "All black is white", we know that the region for 'black' is entirely contained within 'white'. From "Some white is grey", we know that there is some overlap between 'white' and 'grey'. However, the statements do not tell us if this overlap between 'white' and 'grey' is located within the region that is also 'black'. The part of 'white' that is 'grey' could be the part of 'white' that is *not* 'black'.
For example, imagine White represents a large circle. Black is a small circle completely inside the White circle. Grey is another circle that overlaps with the large White circle. This overlap could happen entirely outside the small Black circle. Therefore, "Some grey is black" does not necessarily follow from the statements.
Checking Conclusion II: Some white is black
This conclusion attempts to establish a relationship between 'white' and 'black'. Let's look at the first statement:
- Statement 1: All black is white.
This is a Universal Affirmative proposition ("All S is P"). A valid immediate inference from "All S is P" is its conversion by limitation, which is "Some P is S".
Applying this conversion rule to "All black is white", we get:
- Some white is black.
This is exactly Conclusion II. Since Conclusion II can be directly and validly inferred from Statement 1, it logically follows from the given statements. Statement 2 ("Some white is grey") is irrelevant to the validity of Conclusion II.
Determining Which Conclusion(s) Logically Follow
Based on our analysis:
- Conclusion I (Some grey is black) does not logically follow from the statements.
- Conclusion II (Some white is black) logically follows from Statement 1.
Therefore, only Conclusion II follows.
Revision Table: Key Syllogism Conversions
| Proposition Type | Form | Example | Valid Conversion | Converted Form |
|---|---|---|---|---|
| A (Universal Affirmative) | All S is P | All dogs are mammals. | Conversion by Limitation | Some P is S (Some mammals are dogs.) |
| E (Universal Negative) | No S is P | No dogs are cats. | Simple Conversion | No P is S (No cats are dogs.) |
| I (Particular Affirmative) | Some S is P | Some dogs are brown. | Simple Conversion | Some P is S (Some brown things are dogs.) |
| O (Particular Negative) | Some S is not P | Some dogs are not brown. | No Valid Conversion | - |
In our problem, Statement 1 "All black is white" is an 'A' type proposition. Its valid conversion is "Some white is black", which is Conclusion II.
Additional Information: Logical Reasoning & Syllogisms
Syllogisms are a form of deductive reasoning where a conclusion is drawn from two given or assumed propositions (premises). The structure typically involves a major premise, a minor premise, and a conclusion.
In this problem, we are dealing with immediate inferences as well, which are conclusions drawn directly from a single premise through operations like conversion. Conversion involves switching the subject and predicate terms of a proposition while maintaining its truth value (if the conversion is valid).
Understanding the standard forms of categorical propositions (A, E, I, O) and their valid conversions is crucial for solving such logic problems. Visual aids like Venn diagrams can also be helpful in checking the validity of conclusions by representing the relationships between the sets described in the statements.
Similar Questions
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:
All teachers are engineers.
All clerks are teachers.
Conclusions:
I. Some engineers are clerks.
II. Some teachers are clerks.
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Two Statements are given followed by Two conclusions numbered I and II. 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.
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All paintings are calligraphies.
All sketching are calligraphies .
Conclusions:
I. At least some calligraphies are paintings.
II. No sketching is a painting.
Two statements are given followed by two conclusions numbered I and II. 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 vehicles are helicopters.
All helicopters are cars.
Conclusions:
I. All vehicles are cars.
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Given below are two statements and two conclusions. Take the statements to be true even if they are at variance with commonly known facts, and decide whether the conclusion(s) follow(s) the given statements.
Statements:
All eyes are noses.
Some noses are lips.
Conclusions:
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Two statements are given followed by conclusions numbered 1 and 2. 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.
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Two statements are given followed by two conclusions numbered I and II. 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.
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All keys are locks.
All locks are handles.
Conclusions:
I. Some locks are keys.
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Three 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.
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All switches are fans.
No heater is a fan.
Some wires are heaters.
Conclusions:
I. Some wires are fans.
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Three statements are given followed by two conclusions numbered I and II. 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:
All H are G.
All V are H.
Some V are I.
Conclusions:
I. At least some G are I.
II. All I being H is a possibility.
Three 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:
All apples are pineapples.
Some apples are mangoes.
No mango is a grape.
Conclusions:
I. Some pineapples are not grapes.
II. No pineapple is a grape.
III. Some apples are not grapes.
Three 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:
No river is a sea.
Some rivers are lakes.
All rivers are ponds.
Conclusions:
I. Some lakes are seas.
II. Some ponds are seas.
III. Some lakes are ponds.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
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. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
Given below are two statements, Consider these statements to be true even if they seem factuality absurd, Read the conclusions and then decide which of the given conclusions logically follow(s) from the given statements?
Statements :
1. All vases are flowers.
2. No flowers is a plant.
Conclusions :
1. No vases is a plant.
2. Some plant are vases
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. All A are S.
II. No D is A.
Conclusions:
I. Some S are A.
II. All S are D.
III. No A is D.
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 land are hard.
II. No stone is land.
Conclusions:
I. Some hard are land.
II. Some stone are hard.
