Three conclusions after the three statements below ।, ।। and ।।। have been given. Assuming the given statements to be true even if they appear to be different from the known facts, determine which conclusion follows the statements rationally.
Statement:
All apples are bananas.
All grapes are bananas.
All bananas are plums.
The conclusion:
|. All apples are plums.
||. All grapes are plums.
III. Some grapes are bananas.
Syllogism Logic: Analyzing Statements and Conclusions
This problem requires us to analyze a set of logical statements and determine which of the given conclusions can be rationally derived from them. We must assume the statements are true, even if they seem unusual.
Analyzing the Statements
Let's break down the given statements and represent them using logical notation:
- Statement I: All apples are bananas. (Let A = Apples, B = Bananas)
Notation:A \rightarrow B - Statement II: All grapes are bananas. (Let G = Grapes)
Notation:G \rightarrow B - Statement III: All bananas are plums. (Let P = Plums)
Notation:B \rightarrow P
Evaluating the Conclusions
Now, let's examine each conclusion based on the statements:
Conclusion I: All apples are plums.
We have:
A \rightarrow B(from Statement I)B \rightarrow P(from Statement III)
Using the transitive property of logic (if A implies B, and B implies P, then A implies P), we can conclude that all apples are indeed plums.
Result: Conclusion I follows.
Conclusion II: All grapes are plums.
We have:
G \rightarrow B(from Statement II)B \rightarrow P(from Statement III)
Again, using the transitive property, if all grapes are bananas (G \rightarrow B), and all bananas are plums (B \rightarrow P), then it logically follows that all grapes are plums.
Result: Conclusion II follows.
Conclusion III: Some grapes are bananas.
Statement II states: All grapes are bananas (G \rightarrow B).
In logic, if a statement applies to *all* members of a category (All G are B), it inherently implies that the statement also applies to *some* members of that category (Some G are B), provided the category is not empty. Assuming grapes exist, this conclusion is valid.
Result: Conclusion III follows.
Final Determination
Based on the logical analysis:
- Conclusion I (All apples are plums) follows.
- Conclusion II (All grapes are plums) follows.
- Conclusion III (Some grapes are bananas) follows.
Therefore, all three conclusions rationally follow from the given 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.
