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 know facts, decide which of the conclusions logically follow(s) from the statements. Statements: All wires are motors. All motors are pumps. Conclusions: i. All wires are pumps. ii. Some motors are wires.
Both the conclusions i and ii follow
Logical Reasoning: Analyzing Statements and Conclusions
This question falls under the category of Logical Reasoning, specifically dealing with Syllogisms or Deductive Reasoning. We are given two statements and two conclusions, and we need to determine which conclusion(s) logically follow from the given statements, assuming the statements are true.
Given Statements
- All wires are motors.
- All motors are pumps.
Given Conclusions
- All wires are pumps.
- Some motors are wires.
Analyzing the Statements Using Logical Deduction
Let's represent the categories:
- W = Wires
- M = Motors
- P = Pumps
Statement 1 says "All W are M". This is a universal affirmative proposition. It means that the set of Wires is entirely contained within the set of Motors.
Statement 2 says "All M are P". This is also a universal affirmative proposition. It means that the set of Motors is entirely contained within the set of Pumps.
We can visualize this relationship like nested sets or use syllogistic rules:
Statement 1: All W $\rightarrow$ M
Statement 2: All M $\rightarrow$ P
When we combine these two statements, we can see a direct link between W and P through M. If every Wire is a Motor, and every Motor is a Pump, then it logically follows that every Wire must also be a Pump.
Consider the conversion of universal affirmative propositions. The conversion of "All A are B" is "Some B are A". This conversion is valid for "All A are B". So, from Statement 1, "All wires are motors", we can logically infer "Some motors are wires".
Evaluating the Conclusions
Conclusion i: All wires are pumps.
Based on our analysis:
- All W are M (Statement 1)
- All M are P (Statement 2)
Combining these, we get: All W are P.
This conclusion perfectly matches our deduction from the combined statements. Therefore, Conclusion i logically follows from the given statements.
Conclusion ii: Some motors are wires.
Based on our analysis of Statement 1:
- Statement 1 is "All wires are motors".
The valid conversion of "All wires are motors" is "Some motors are wires". If every single wire is a motor, then within the group of motors, there must be at least some motors that are wires (specifically, all the wires). Therefore, Conclusion ii logically follows from Statement 1.
Conclusion Summary
Both Conclusion i (All wires are pumps) and Conclusion ii (Some motors are wires) logically follow from the given statements.
Revision Table: Logical Reasoning Concepts
| Term | Description | Example (from question) |
|---|---|---|
| Statement | A proposition assumed to be true for the purpose of deduction. | All wires are motors. |
| Conclusion | A proposition derived logically from the statements. | All wires are pumps. |
| Syllogism | A form of deductive reasoning where two statements (premises) lead to a conclusion. | Statement 1 + Statement 2 $\rightarrow$ Conclusion i |
| Universal Affirmative (A type) | A statement of the form "All A are B". | All wires are motors. |
| Particular Affirmative (I type) | A statement of the form "Some A are B". | Some motors are wires. |
| Conversion | Interchanging the subject and predicate of a proposition. Valid for A type (All A are B becomes Some B are A). | "All wires are motors" converts to "Some motors are wires". |
Additional Information: Understanding Syllogisms
Syllogisms are a fundamental part of deductive reasoning. They consist of two premises (statements) and a conclusion. The validity of a syllogism depends on its structure, not the truthfulness of the premises in the real world. In these types of logical reasoning problems, we strictly follow the given statements.
The structure encountered here is a classic valid syllogism form:
- All A are B.
- All B are C.
- Therefore, All A are C.
In our question:
- A = Wires
- B = Motors
- C = Pumps
So, "All wires are motors" and "All motors are pumps" logically leads to "All wires are pumps". This confirms Conclusion i.
Conclusion ii relies on the immediate inference through conversion. From "All A are B", we can always infer "Some B are A". This is because if the entire set A is inside the set B, then the elements of A are a part of B, and those elements are also A. Hence, there are some elements in B that are A.
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.
