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 R are M. II. All L are M. Conclusions: I. Some M are not R. II. Some L are R. III. Some M are L.
Only conclusion III follows
Logical Reasoning Solution: Analyzing Syllogism Statements
This question asks us to analyze logical statements and determine which conclusions necessarily follow from them. We are given two statements and three conclusions.
Understanding the Statements
The statements describe the relationship between different categories or sets (R, M, and L).
- Statement I: All R are M. This means that the entire set of 'R' is contained within the set of 'M'. Every single element that is an R is also an M.
- Statement II: All L are M. This means that the entire set of 'L' is contained within the set of 'M'. Every single element that is an L is also an M.
From these statements, we know that both R and L are subsets of M. However, we don't have any direct information about the relationship between R and L. R and L could overlap, be separate, or one could be inside the other, as long as both are inside M.
Evaluating the Conclusions
Let's examine each conclusion to see if it must be true based on the given statements.
Conclusion I: Some M are not R.
This conclusion suggests that there are some elements in the set M that are not in the set R. Based on the statements, this is possible. For example, M could be a much larger set than R, with parts of M containing L, and other parts containing neither R nor L, but all of R being inside M. However, consider a scenario where the set M is exactly the same as the set R (i.e., R = M). In this case, Statement I ("All R are M") is true. Statement II ("All L are M") would mean All L are R. But then, if R=M, the conclusion "Some M are not R" becomes "Some R are not R", which is false. Since we are looking for conclusions that *logically follow* in all possible scenarios where the statements are true, and we found a scenario where this conclusion is false (R=M), Conclusion I does not logically follow.
Conclusion II: Some L are R.
This conclusion suggests that there is an overlap between the set L and the set R. Based on the statements, this overlap might exist, but it is not guaranteed. Statement I puts R inside M. Statement II puts L inside M. It is possible for L and R to be completely separate sets within M. For example, M could be "animals", R could be "dogs", and L could be "cats". All dogs are animals, and all cats are animals, but no cats are dogs. Since we can construct a scenario where the statements are true but Conclusion II is false (L and R are disjoint within M), Conclusion II does not logically follow.
Conclusion III: Some M are L.
This conclusion suggests that there are some elements in the set M that are also in the set L. Let's look at Statement II: "All L are M". This statement means that every single member of the set L is also a member of the set M. If this is true, then the set L is a subset of the set M. If L is a non-empty set (standard assumption in syllogisms unless explicitly stated otherwise) and it is contained within M, then the elements that make up L are themselves elements of M. Therefore, there must be 'some' elements in M that are L (specifically, all the elements of L). This conclusion logically follows directly from Statement II.
Summary of Conclusions
- Conclusion I: Some M are not R - Does not logically follow.
- Conclusion II: Some L are R - Does not logically follow.
- Conclusion III: Some M are L - Logically follows.
Based on our analysis, only Conclusion III necessarily follows from the given statements.
Revision Table: Key Syllogism Concepts
| Statement Type | Description | Example |
|---|---|---|
| Universal Affirmative (A) | All S are P | All dogs are mammals. |
| Universal Negative (E) | No S are P | No fish are birds. |
| Particular Affirmative (I) | Some S are P | Some students are athletes. |
| Particular Negative (O) | Some S are not P | Some food is not healthy. |
Additional Information: Drawing Inferences
In logical reasoning, especially with syllogisms, understanding how to draw valid inferences is key. A conclusion logically follows if it must be true whenever the statements are true. We don't rely on common sense or outside knowledge, only the relationships given in the statements.
- Conversion: Some statements can be converted. For example, "All L are M" can be converted to "Some M are L". This conversion is always valid. "All S are P" validly converts to "Some P are S". "No S are P" validly converts to "No P are S". "Some S are P" validly converts to "Some P are S". "Some S are not P" cannot be validly converted in this simple way.
- Venn Diagrams: Visualizing the statements using Venn diagrams can be very helpful. Draw circles representing the sets. "All R are M" means the R circle is inside the M circle. "All L are M" means the L circle is inside the M circle. Then, test each conclusion by seeing if it *must* be true in all possible valid diagrams of the statements.
Applying these principles, we confirm that "All L are M" implies "Some M are L".
Similar Questions
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
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 plates are bowls.
No bowl is a glass.
Some glasses are vessels.
Conclusions:
I. Some glasses are plates.
II. Some bowls are plates.
III. Some vessels are bowls.
Three statements are given followed by three conclusions numbered I, Il and lII. 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 fruits are wines.
Some wines are juices.
All juices are alcohols.
Conclusions:
I. Some alcohols are juices.
Il. Some wines are alcohols.
III. Some wines are fruits.
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:
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.
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 T are G.
II. All G are H.
Conclusions:
I. All T are H.
II. No G is T.
III. All H are G.
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 reds are pinks.
All whites are pinks.
All pinks are oranges.
Conclusions:
I. All whites are oranges.
II. All oranges are reds.
III. Some oranges are whites.
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 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.
II. No p is t.
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 Tim are Jack.
No Jack is Ben.
Some Ben is Ruth.
Conclusions:
I. No Ruth is Jack.
II. Some Ben are Jack.
III. No Ben is Tim.
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.
