Statements:

Only doors are mats

Some doors are knob

Only a few knob is metal

No metal is aluminium

Conclusions:

I. No mats is aluminium

II. Some knob is not metal

III. Some mats being knob is a possibility

In each question below, some statements are given followed by some conclusions. You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows.

To determine which conclusions follow logically from the given statements, let us analyze the logical meaning of each statement step-by-step:

1. Analysis of Statements:
- "Only doors are mats": In syllogisms, "Only A are B" translates to "All B are A" with an exclusivity constraint. This means all mats are doors, and mats cannot belong to or intersect with any other category besides doors. Thus, mats are exclusive to doors.
- "Some doors are knob": This means there is a common intersection between the set of doors and the set of knobs.
- "Only a few knob is metal": The phrase "Only a few A is B" contains two pieces of information: "Some A are B" and "Some A are not B". Therefore, some knobs are metal, and some knobs are definitely not metal.
- "No metal is aluminium": This means that the set of metal and the set of aluminium have no intersection (they are completely disjoint).

2. Evaluation of the Conclusions:
- Conclusion I: "No mats is aluminium"
Since mats are exclusive to doors (due to "Only doors are mats"), mats cannot have any intersection with aluminium. Thus, it is definitely true that no mats are aluminium. This conclusion follows.
- Conclusion II: "Some knob is not metal"
From the statement "Only a few knob is metal", it is a direct rule that some knobs are not metal. Thus, this conclusion is definitely true and follows.
- Conclusion III: "Some mats being knob is a possibility"
As established, mats are exclusive to doors and cannot associate with any other category. Therefore, mats can never be knobs, which makes any possibility of mats being knob false. This conclusion does not follow.

Consequently, only conclusions I and II follow logically from the statements.

The correct option is Both I and II follow.