What are natural ways to express 'contra-positively' in writing?

I often see 'conversely' being used when the meaning is to express the 'contra-positive'.

I know that the contra-positive of a statement is logically equivalent to the statement, but they're still different formulations in natural language.

Question(s): In general, is there a more natural sounding word to use instead of 'contra-positively' in (informal but possibly academic) written English? If not technically, then what could be a good alternative?

(Just as a made up example: If I am standing up, then I am awake. Contra-positively, If I am not awake, then I am not standing up.

'Conversely' would sound natural but feels wrong since it has such an otherwise specific meaning (in logic). Perhaps it is used more loosely in natural English, and I am being overly cautious.

None of the often proposed synonyms of different thesaurus searches feels natural to put instead, e.g. 'contrary' or 'oppositely'.)

Edits: The comments about my initial example being bad are all fair, which is why I (after several edits) instead chose to reformulate the question completely. (It was really more what got me thinking than a good example anyway.) I hope it is a bit more clear now. Also, I think the answers already provided are sufficient.

The phrase on the other hand is used to introduce a statement differing with one just made in some unspecified way:

on the other hand

in a way that is different from the first thing you mentioned:

• My husband likes classical music – I, on the other hand, like all kinds.
• [My husband likes classical music – I, on the other hand, don't.]
• [My husband likes classical music – I, on the other hand, play in an orchestra.]
• [My husband likes classical music – I, on the other hand, like cricket. So we toss up to see who gets to watch the television when there's a conflict.]

[Cambridge Academic Content Dictionary] [other examples added]

Conversely would be perfectly acceptable here, as the word in common parlance simply suggests some type of reversal or contrast, rather than the very specific meaning it takes in the domain of mathematical logic. There doesn't even need to be any kind of if-then statement to use the term:

Online sales went up last quarter, while in-store sales, conversely, went down.

As long as you have two statements that stand in stark contrast or represent some kind of reversal, the use of conversely is generally appropriate, even if it is not a true example of a logical converse. That said, if you are writing a thesis that deals extensively with logical converses, it may be better to use a different term.

If one is writing for an audience that knows the meaning of contrapositive(ly) it is, of course, best to use that term; no other term can convey quite the same idea. If one is writing for an audience that is unfamiliar with the term, one simply has to sacrifice some of the meaning that would be conveyed by it. Now, if two propositions are the contrapositives of each other, then they (1) look very different from each other, but are nevertheless (2) logically equivalent. The solutions proposed in the other answers emphasise (1), and sacrifice (2). If something needs to be sacrificed, it would be, however, better to sacrifice (1), which is obvious anyway, and choose an expression that, at least partially, conveys (2). Some of the expressions that may accomplish that are therefore and in other words.

One option is on the contrary, an idiomatic phrase that can be used to introduce the opposite idea (Merriam-Webster, under contrary):

As is often the case, one can get so involved in complex formalism, that one forgets the bigger picture. On the contrary, being too informal leads to interpretation and ambiguity.

Here is an example I found in a quick search on JSTOR, from Moquet et al., 2021:

The host range of specialist species that only share a few host species with B. dorsalis did not change significantly. On the contrary, we observed a significant shift in diversity or proportion of host range and climatic niches for the generalist species, such as Bactrocera zonata, Ceratitis Quilicii, and Ceratitis capitata.

The single-word version is contrariwise, which sounds more old-fashioned but literally means "on the contrary" (Merriam-Webster).

If the logical fact that two statements are contrapositives is important to emphasize, then I would call more attention to it. It is likely to be somewhat stilted, but perhaps that is worth it for the sake of making your argument logic explicit. It also makes an assumption that your audience understands the distinction and will benefit from having it pointed out. Here is an example:

All real men love watching curling. Note the contrapositive: if someone doesn't watch curling, it can be concluded that they are not real men.

If you can sacrifice some precision but want to preserve the sense that the second statement is just as true for the purpose of argument, then a more natural but more vague wording might be okay. Particularly if your reader doesn't remember the definition of contrapositive.

All ducks swim, so it follows that animals that don't swim are not ducks.