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 pencils are colours. All colours are paints. Some paints are drawings. Conclusions: I. Some pencils are drawings. II. Some paints are colours. III. Some colours are pencils
Only conclusions II and III follow
Solving Syllogism Conclusions from Statements
This question asks us to analyze three statements and determine which of the given conclusions logically follow from them. We must assume the statements are true, even if they contradict common knowledge.
Analyzing the Statements
Let's represent the statements to understand the relationships:
- Statement 1: All pencils are colours. (P → C)
- Statement 2: All colours are paints. (C → Pt)
- Statement 3: Some paints are drawings. (Pt ∩ Dr $\neq \emptyset$)
From the first two statements, we can deduce a direct relationship between pencils and paints:
- Since All Pencils are Colours, and All Colours are Paints, it logically follows that All Pencils are Paints. (P → Pt)
The third statement introduces a relationship between paints and drawings, indicating an overlap. However, this overlap might be only in the part of paints that are not colours, or it might include parts of colours or even pencils. We cannot be certain about the relationship between pencils/colours and drawings based solely on these statements.
Evaluating the Conclusions
Now let's examine each conclusion based on our analysis of the statements:
Conclusion I: Some pencils are drawings.
We know All Pencils are Paints, and Some Paints are Drawings. The set of Paints overlaps with Drawings. The set of Pencils is entirely contained within the set of Paints. However, we do not know if the portion of Paints that overlaps with Drawings includes any of the Pencils. It is possible that the overlap between Paints and Drawings is only in the area of Paints outside the Colours (and therefore outside the Pencils). Thus, this conclusion does not necessarily follow.
Conclusion II: Some paints are colours.
Statement 2 says "All colours are paints". If all members of one group (Colours) are part of another group (Paints), it logically implies that some members of the second group (Paints) must be from the first group (Colours). This is a standard inference in syllogism. If 'All A are B', then 'Some B are A' is always true.
Applying this rule, from "All colours are paints", we can conclude "Some paints are colours". This conclusion logically follows from Statement 2.
Conclusion III: Some colours are pencils.
Statement 1 says "All pencils are colours". Similarly, if all members of one group (Pencils) are part of another group (Colours), it implies that some members of the second group (Colours) must be from the first group (Pencils). If 'All A are B', then 'Some B are A' is always true.
Applying this rule, from "All pencils are colours", we can conclude "Some colours are pencils". This conclusion logically follows from Statement 1.
Summary of Conclusions
- Conclusion I: Some pencils are drawings - Does NOT follow.
- Conclusion II: Some paints are colours - FOLLOWS.
- Conclusion III: Some colours are pencils - FOLLOWS.
Therefore, only conclusions II and III logically follow from the given statements.
Revision Table: Syllogism Statements and Conclusions
| Statement/Conclusion | Relation | Logical Follows? |
|---|---|---|
| Statement 1: All Pencils are Colours | P $\subseteq$ C | Given |
| Statement 2: All Colours are Paints | C $\subseteq$ Pt | Given |
| Statement 3: Some Paints are Drawings | Pt $\cap$ Dr $\neq \emptyset$ | Given |
| Conclusion I: Some pencils are drawings | P $\cap$ Dr $\neq \emptyset$? | No |
| Conclusion II: Some paints are colours | Pt $\cap$ C $\neq \emptyset$? | Yes (from All C are Pt) |
| Conclusion III: Some colours are pencils | C $\cap$ P $\neq \emptyset$? | Yes (from All P are C) |
Additional Information on Syllogism Logic
Syllogism is a form of logical reasoning where a conclusion is derived from two or more premises (statements). The standard form involves a major premise, a minor premise, and a conclusion. In this case, we have three statements acting as premises.
Key logical deductions often used:
- If "All A are B", then "Some A are B". (Direct inference)
- If "All A are B", then "Some B are A". (Conversion by Limitation, valid)
- If "Some A are B", then "Some B are A". (Simple Conversion, valid)
- "No A are B" is equivalent to "No B are A". (Simple Conversion, valid)
- "Some A are not B" cannot be simply converted.
Venn diagrams are a common tool to visualize syllogisms. Circles represent categories, and their overlap (or lack thereof) represents the relationships described by the statements. By drawing the diagrams for the statements, we can visually check if a conclusion is always true in all possible valid diagrams.
In this specific problem, drawing the diagram for "All Pencils are Colours" inside "All Colours are Paints" and then adding "Some Paints are Drawings" shows that the overlap between Paints and Drawings doesn't guarantee any overlap with the smaller Pencils circle or the Colours circle inside the Paints circle.
Similar Questions
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Statements:
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All scientists are doctors.
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Statements:
All reds are pinks.
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All pinks are oranges.
Conclusions:
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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.
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Statements:
I. All M are A.
II. No A is R.
Conclusion:
I. No M is R.
II. Some A are M.
III. Some R are A.
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 c are f.
No f is t.
All p are c.
Conclusions:
I. Some t are c.
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Important Questions from Conventional Syllogism
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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.
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Statements:
I. All A are S.
II. No D is A.
Conclusions:
I. Some S are A.
II. All S are D.
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Statements:
I. Some land are hard.
II. No stone is land.
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