Consider the following statements to be true even if they seem to be at variance from commonly known facts and decide which of the conclusions logically follows from the statements. Statements: All loves are hate. All hates are emotion. Conclusions: I: Some emotions are hates.
Both conclusions follow
Understanding Logic Syllogism: Statements and Conclusions
This question asks us to analyze given statements and determine which of the provided conclusions logically follow. We must assume the statements are true, even if they contradict common knowledge. This is a classic logic problem involving syllogisms.
Analysis of Statements
We are given two statements:
- Statement 1: All loves are hate.
- Statement 2: All hates are emotion.
Analysis of Conclusions
We need to evaluate if the following conclusions are valid based on the statements:
- Conclusion I: Some emotions are hates.
- Conclusion II: Some emotions are love.
Step-by-Step Logical Deduction
Let's represent the categories involved:
- L: Loves
- H: Hates
- E: Emotions
The statements can be represented as follows:
- Statement 1: All L are H ($\forall x (L(x) \rightarrow H(x))$)
- Statement 2: All H are E ($\forall x (H(x) \rightarrow E(x))$)
Evaluating Conclusion I: Some emotions are hates.
Conclusion I states: Some E are H ($\exists x (E(x) \land H(x))$). Statement 2 is "All hates are emotion" ($\forall x (H(x) \rightarrow E(x))$). In set theory terms, this means the set of Hates is a subset of the set of Emotions ($H \subseteq E$). If all members of set H are in set E, it logically follows that some members of set E must be members of set H (provided there is at least one hate). This is a valid conversion of an 'A' type proposition ("All H are E") to an 'I' type proposition ("Some E are H"). Therefore, Conclusion I logically follows from Statement 2.
Evaluating Conclusion II: Some emotions are love.
Conclusion II states: Some E are L ($\exists x (E(x) \land L(x))$). Let's combine the two statements:
- Statement 1: All L are H
- Statement 2: All H are E
If all L are H, and all H are E, then it logically follows that all L are E. This is transitive property of inclusion or subset relationship ($L \subseteq H$ and $H \subseteq E$ implies $L \subseteq E$). So, "All loves are emotion" ($\forall x (L(x) \rightarrow E(x))$). From "All loves are emotion", we can convert this 'A' type proposition ("All L are E") to an 'I' type proposition ("Some E are L"), provided there is at least one love. This means Some Emotions are Love. Therefore, Conclusion II logically follows from the combined statements.
Summary of Conclusions
Based on our analysis:
- Conclusion I ("Some emotions are hates") follows from Statement 2.
- Conclusion II ("Some emotions are love") follows from the combination of Statement 1 and Statement 2.
Since both conclusions logically follow from the given statements, the correct option is that both conclusions follow.
| Statement/Conclusion | Representation | Follows From | Reasoning |
|---|---|---|---|
| Statement 1 | All Loves are Hate | Given | Premise |
| Statement 2 | All Hates are Emotion | Given | Premise |
| Conclusion I | Some Emotions are Hates | Statement 2 | Conversion of "All H are E" to "Some E are H" |
| Conclusion II | Some Emotions are Love | Statement 1 & 2 | From "All L are H" and "All H are E", infer "All L are E"; then convert to "Some E are L" |
Revision Table: Key Concepts in Logic
| Concept | Description |
|---|---|
| Syllogism | A type of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. |
| Categorical Proposition | A statement that relates two categories or classes. Types include All S are P (A), No S are P (E), Some S are P (I), Some S are not P (O). |
| Conversion | An immediate inference formed by interchanging the subject and predicate terms of a proposition. Valid for A (to I), E (to E), and I (to I) propositions. "All S are P" converts to "Some P are S" (by limitation, assuming S exists). "All H are E" converts to "Some E are H". |
| Transitivity | If A is related to B, and B is related to C, then A is related to C. In syllogisms: If All A are B, and All B are C, then All A are C. |
Additional Information: Solving Syllogisms with Venn Diagrams
Venn diagrams can be a helpful tool to visualize syllogisms.
For this problem:
- Draw three overlapping circles representing Loves (L), Hates (H), and Emotions (E).
- Statement 1: "All loves are hate". Shade the part of the Love circle that is outside the Hate circle. This indicates there are no loves that are not hates.
- Statement 2: "All hates are emotion". Shade the part of the Hate circle that is outside the Emotion circle. This indicates there are no hates that are not emotions.
After shading according to the statements:
- Look at the area where H and E overlap. The statement "All hates are emotion" means the entire H circle is inside the E circle. Therefore, the overlap region between H and E (which is the entire H circle) contains entities. This supports Conclusion I ("Some emotions are hates"), as the shaded diagram shows H is within E, implying some E must be H.
- Look at the area where L and E overlap. Since the entire L circle is inside the H circle (from Statement 1), and the entire H circle is inside the E circle (from Statement 2), the entire L circle must be inside the E circle. This means all Loves are Emotions. If there are any Loves, then the overlap region between L and E (which is the entire L circle) contains entities. This supports Conclusion II ("Some emotions are love"), as the diagram shows L is within E, implying some E must be L (assuming Loves exist).
Venn diagrams visually confirm that both conclusions are supported by the statements.
Important Questions from 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:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
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 directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
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 follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
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 cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
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 employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
