Statements: Some hills are rivers. Some rivers are deserts. All deserts are roads.

Conclusions:

1. Some roads are rivers.
2. Some roads are hills.
3. Some deserts are hills.

III is the converse of the second premise and so it holds.

Some pictures are frames. Some frames are idols.

Since both the premises are particular, no definite conclusion follows.

Some frames are idols. All idols are curtains.

Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some frames are curtains'. III is the converse of this conclusion and so it holds.

Some pictures are frames. Some frames are curtains.

Since both the premises are particular, no definite conclusion can be drawn.

Some towels are brushes. No brush is soap.

Since one premise is particular and the other negative, the conclusion must be particular negative (O-type) and should not contain the middle term. So, it follows that 'Some towels are not soaps'. No brush is soap. All soaps are rats.

Since the middle term is distributed twice, the conclusion must be particular. Since one premise is negative, the conclusion must be negative. So, it follows that 'Some brushes are not rats'. Since I and II involve the same terms and form a complementary pair, so either I or II follows.

III is the converse of the third premise and so it holds.

All fruits are vegetables. All vegetables are rains.

The conclusion must be universal affirmative and should not contain the middle term.

So, it follows that 'All fruits are rains'. Thus, I follows.

All pens are vegetables. All vegetables are rains.

Clearly, it follows that 'All pens are rains'. Thus, II follows.

Since both the premises are particular, no definite conclusion follows. However, III and IV form a complementary pair. Thus, either III or IV follows.

Since both the premises are particular, no definite conclusion follows. However, I and III involve only the extreme terms and form a complementary pair. So, either I or III follows. II is the converse of the first premise while IV is the converse of the second premise. Thus, both of them hold.

Some saints are balls. All balls are bats.

Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some saints are bats'. Thus, II follows. Some tigers are balls. All balls are bats.

Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some tigers are bats'. I is the converse of this conclusion and so it holds.

Some pens are books. All schools are books.

Since the middle term 'books' is not distributed even once in the premises, so no definite conclusion follows.

Some colleges are schools. All schools are books.

Since one premise is particular, the conclusion must be particular and should not contain the middle term.

So, it follows that 'Some colleges are books'. Thus, III follows.

Some pens are books. Some colleges are books.

Since both the premises are particular, no definite conclusion can be drawn.

Hence, only III follows.

All trains are buses. No room is bus.

Since both the premises are universal and one premise is negative, the conclusion must be universal negative (E-type) and should not contain the middle term. So, it follows that 'No train is room'. Thus, III follows.

All boats are rooms. No room is bus.

As discussed above, it follows that 'No boat is bus'.

II is the converse of this conclusion and so it holds. All trains are buses. No boat is bus.

Again, it follows that 'No train is boat'. I is the converse of this conclusion and so it holds.

Since each combination of premises shall contain two particular premises, no definite conclusion can be drawn. However, II and III are statements involving the extreme terms of the last two premises and form a complementary pair. Thus, either II or III follows.

Since each combination of premises has two particular premises, so no definite conclusion follows.