The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements. Statements: 1.) All pens are frogs. 2.) Some crows are frogs. Conclusions: I. No pen is a crow. II. Some pens are crows.
Either conclusion I or conclusion II follows.
Solving Syllogism Questions: Statements and Conclusions
This question asks us to analyze two statements and determine which of the given conclusions logically follow from them. The statements are considered true, even if they contradict known facts. This type of problem is common in logic reasoning tests and is often solved using Venn diagrams or analytical methods.
Analyzing the Statements
We have two statements:
- Statement 1: All pens are frogs.
- Statement 2: Some crows are frogs.
Let's represent these statements using circles (like in Venn diagrams) or by understanding the relationships they establish.
- Statement 1 tells us that the entire set of 'pens' is contained within the set of 'frogs'. If something is a pen, it must also be a frog.
- Statement 2 tells us that there is an overlap between the set of 'crows' and the set of 'frogs'. At least some crows are frogs.
Examining the Relationship Between Pens and Crows
The statements provide a direct relationship between Pens and Frogs, and between Crows and Frogs. However, they do not directly state any relationship between Pens and Crows. Let's consider the possibilities based on how the sets could overlap:
- We know all Pens are inside the Frogs circle.
- We know some Crows are in the Frogs circle.
The part of the Frogs circle where the Crows overlap could be:
- Entirely within the part of the Frogs circle that is outside the Pens circle. In this case, no Pen would be a Crow.
- Partially overlapping with the part of the Frogs circle that is inside the Pens circle (since Pens are inside Frogs). In this case, some Pens would be Crows.
Since both of these scenarios are possible based on the given statements, the relationship between 'Pens' and 'Crows' is not definite. There might be no pens that are crows, or there might be some pens that are crows. We cannot be sure either way.
Analyzing the Conclusions
Now let's look at the conclusions:
- Conclusion I: No pen is a crow.
- Conclusion II: Some pens are crows.
Let's check if each conclusion logically and definitely follows from the statements.
- Conclusion I: No pen is a crow.
This conclusion states there is absolutely no overlap between the set of Pens and the set of Crows. As we saw in possibility 1 above, this scenario is possible if the 'Some crows are frogs' overlap happens only in the part of the Frogs circle where there are no Pens.
However, is it definitely true? No. Possibility 2 shows a scenario where some pens *are* crows. So, Conclusion I does not *definitely* follow.
- Conclusion II: Some pens are crows.
This conclusion states there is at least some overlap between the set of Pens and the set of Crows. As we saw in possibility 2 above, this scenario is possible if the 'Some crows are frogs' overlap includes some part of the Pens circle (which is inside the Frogs circle).
However, is it definitely true? No. Possibility 1 shows a scenario where no pen is a crow. So, Conclusion II does not *definitely* follow.
Determining the Logical Outcome
Neither conclusion I nor conclusion II definitely follows on its own. However, notice that conclusion I ("No pen is a crow") and conclusion II ("Some pens are crows") are contradictory pairs (specifically, they form an 'Either/Or' case or complementary pair in some contexts of syllogism where the terms are the same). If one is true, the other must be false, and vice versa. Since we established that both scenarios (no overlap between pens and crows, or some overlap between pens and crows) are possible based on the statements, we cannot definitively say which one is true.
In such a case, where neither conclusion definitely follows, but one of the two contradictory conclusions must be true (either there's no overlap or there's some overlap), the answer is "Either conclusion I or conclusion II follows". The statements do not provide enough information to rule out one of the possibilities, but they establish a condition where one of these mutually exclusive possibilities must exist.
| Statements | Analysis |
|---|---|
| 1.) All pens are frogs. | Set of Pens <subseteq> Set of Frogs |
| 2.) Some crows are frogs. | Set of Crows ∩ Set of Frogs ≠ ∅ |
| Conclusion | Analysis | Definitely Follows? |
|---|---|---|
| I. No pen is a crow. | Set of Pens ∩ Set of Crows = ∅ | No (Possible, but not definite) |
| II. Some pens are crows. | Set of Pens ∩ Set of Crows ≠ ∅ | No (Possible, but not definite) |
Conclusion
Based on the analysis, neither conclusion I nor conclusion II is definitely true. However, since they represent the only two possible outcomes regarding the relationship between Pens and Crows (either there is some overlap or there is no overlap), one of them must be true. Therefore, either conclusion I or conclusion II follows.
Revision Table: Key Logic Reasoning Concepts
| Concept | Description |
|---|---|
| Syllogism | A form of logical reasoning where a conclusion is drawn from two given or assumed propositions (premises). |
| Statements (Premises) | The given propositions assumed to be true. |
| Conclusions | Propositions that logically follow from the statements. |
| "Definitely Follows" | The conclusion must be true in all possible scenarios based on the statements. |
| "Either/Or" Case | Occurs when two conclusions are contradictory, and while neither individually is definite, one of them must be true. |
Additional Information on Syllogism Types
Syllogisms involve different types of statements, which can be helpful in understanding the relationships:
- Universal Affirmative (All A are B): The entire set A is included in set B.
- Universal Negative (No A are B): There is no overlap between set A and set B.
- Particular Affirmative (Some A are B): There is at least one common element between set A and set B (overlap exists).
- Particular Negative (Some A are not B): There is at least one element in set A that is not in set B.
In this problem, we had a Universal Affirmative ("All pens are frogs") and a Particular Affirmative ("Some crows are frogs"). The relationship between the subjects of these two statements (Pens and Crows) is what we needed to determine. When the middle term (Frogs) is distributed in the Universal statement but not necessarily in the Particular statement's subject, and the conclusion involves the subjects of both statements, an 'either/or' scenario can arise if no definite link is established.
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.
