Statements: Some taxis have horns. Some taxis have lights.

Conclusions:

1. Every taxi has either horn or light.
2. Some taxis have neither light nor horn.
3. Some taxis have horns as well as lights.
4. No taxi has horn as well as light.

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.

Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some bottles are cups'. Thus, I follows. II is the converse of the second premise and so it also holds.

Since one premise is particular and the other negative, the conclusion must be particular negative and should not contain the middle term. Thus, III follows. I is the converse of the first premise and so it also holds.

Since the middle term 'terrorists' is distributed twice in the premises, the conclusion cannot be universal. So, it follows that 'Some guilty persons are criminals'. Thus, II holds.

Since both the premises are particular, no definite conclusion 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.

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 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 hills are rivers. Some rivers are deserts.

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

Some rivers are deserts. All deserts are roads.

Since one premise is particular, the conclusion must be particular and shouldn't contain the middle term. So, it follows that 'Some rivers are roads'. I is the converse of this conclusion and so it holds.

Some hills are rivers. Some rivers are roads.

Again, since both the premises are particular, no definite conclusion follows.

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.