I criticised Hume for being vague about just how contiguous C and E would need to be for us to believe that they have a causal relation, and I suggested that he needs to qualify 2 as follows:
2'. C and E are sufficiently spatiotemporally contiguous for us to believe that they have a causal relation (providing we also observe 1 and 3)
You're adding a term that is unnecessary. Any time we talk about cause and effect, we are, in essence, talking about one's belief
in a causal connection. There is, therefore, no need to mention the belief in a causal connection when discussing cause and effect: it's a given. Hume undoubtedly understood this, which is why he doesn't mention it.
It's not an unecessary term, and Hume does mention it. He uses the term 'idea' rather than 'belief' - perhaps I should have done the same. But Hume's argument in book 1, part, 3, section 14 of A Treatise of Human Nature
is about why we believe
that there is a necessary connection between certain pairs of events or objects, which we call causes and effects.
He could have talked about whether there actually is such a thing as necessary connection, or causality - whether we believe in them is not necessary relevant to that question. If I removed the reference to belief from what you've quoted, it would seem that I would be talking about something completely different - namely, whether causal relations exist
, rather than why we believe them to exist