Let us look at the following example:

(1) Water boils at 100 Celsius degrees.
(2) Water boils.

(2) is logically implied by (1). Suppose we would “make this explicit” by saying that if (1), then (2) or that (1) only if (2). The second would say,  in other words, that (2) is a necessary condition for (1). If water did not boil, it couldn’t boil at 100 Celsius degrees.

Now let us take another example:

(3) Earh is the third planet from the Sun and Earth has an atmosphere.
(4) The third planet from the Sun has an atmosphere.

It seems quite awkward to say that (4) is a necessary condition for (3), since we do not usually think of necessary conditions for a complex of independent facts.

I think we would rather say that given the fact that Earth is the third planet from the Sun, (4) is a necessary condition for the Earth having an atmosphere. However, we wouldn’t say that given the fact that Earth has an atmosphere, (4) is a necessary condition for the Earth being the third planet from the Sun.

The last sentence says that the identity of two objects depends on one of them having a certain property,  on the assumption that the other one has it. Although this is, logically speaking, correct,  it seems weird to assume that an object has a certain property and then think of what would be a necessary condition for another object being identical with it. That two objects share their properties is a necessary condition for their identity, but apart from that (and perhaps the existence of the two) we do not look for necessary conditions for identities under factual assumptions.

Perhaps things will get clearer if we look at a third example:

(5) It is raining and it rains only if there are clouds in the sky.
(6) There are clouds in the sky.

Again, it seems strange to say that (6) is a necessary condition for (5). By contrast, it seems natural to say that the presence of clouds in the sky is a necessary condition for the rain, given the fact that it rains only if there are clouds in the sky (one might wonder why this does not seem tautological, but this is unrelated).

The other version – the presence of the clouds in the sky is a necessary condition for the presence of the clouds being a necessary condition for rain, given the fact that it rains – seems completely twisted.

It is, indeed, necessary that a necessary conditioning relation should not be falsified by facts, in order for it to hold, but we do not look for further specific necessary conditions under particular scenarios in such cases.

This asymmetry seems to suggest that the conditional (and the identity statement) is not a premise in our reasoning, but an assumption under which we reason.

The suggestion, however, seems hardly acceptable, since it puts the previous example in the same class with:

(7) John has quit smoking.
(8) Somebody has quit smoking (under the assumption that John did smoke in the past).

To this one could reply by distinguishing presuppositions (necessary conditions for what was said being meaningful) from assumptions (which are not necessary for something being meaningful, but for what seems to follow from what was said being logically implied by what was said). But presuppositions seem to satisfy the second requirement for assumptions, and the first requirement is negative. It would appear, thus, that assumptions are just a special kind of presuppositions.

We should also note that some cases can put even this negative criterion for distinguishing between presuppositions and assumptions in discussion:

(9) John is a man.
(10) John is a man (under the assumption that John is identical with himself).

It can be argued that the identity assumption is necessary for (10) being a necessary condition for (9), but it could also be said that if John was not identical to himself, (9) would not be meaningful.

To conclude, viewing some premise of an argument as an actual assumption under which we reason might be unavoidable (this is also related to my remarks on our difficulty to distinguish enthymemes from complete arguments), but it also leads to further difficulties – as is the task of preventing such assumptions from being engulfed into the class of presuppositions.