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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.

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    My choice would be Yet, being too informal leads to ... Your second thought is not contrary, and oppositely is not a word I know. If you make a big deal out of your contrast by saying "By contrast," there's a big chance your readers will think you are talking down to them. Feb 18 at 14:33
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    It seems to me the "contra-positive" in your example is something like If you keep the bigger picture in mind, you cannot get involved in complex formalism. Which doesn't look like a particularly useful thing to say. Feb 18 at 14:52
  • @FumbleFingers : Yes, hence the comment directly after: 'This is not perhaps...'. What I want to convey is that it is often the case that one of the activities gets the upper hand (in maths anyway), and balance is necessary. However, I am not aiming to get this across in one sentence, i.e. this is taken out of context. Hopefully, in context it will make sense. Feb 18 at 16:12
  • I'm not sure that being too informal is a useful way of referring to the "opposite" of [excessive] complex formalism. But contrariwise might be a useful word, if you don't mind at least some of your readers thinking that's a facetious nod in the direction of Tweedledee and Tweedledum in Alice in Wonderland. Feb 18 at 17:23
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    It looks to me like your proposed contrapositive statement is closer to the inverse, mathematically: the inverse of "if P then Q" being "if not P then not Q". (That being said, I don't think using "inversely" would be idiomatic.)
    – Carmeister
    Feb 19 at 0:02
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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]

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    I can't believe that phrase didn't pop into my head; I guess I was too involved in complex formalism ;). I would say it fits well enough, and at least I would prefer it to 'conversely', since this has such a specific meaning in formal mathematics; even though it may work as well. Feb 18 at 16:19
  • On the other hand implies an opposition between the two statements, as can be seen in the quoted examples. A contrapositive of something is, however, not in any kind of opposition to it; the two are logically equivalent.
    – jsw29
    Feb 18 at 17:04
  • @jsw2 It fits OP's example (I considered the previous analysis off-topic on ELU, dealing with formal logic). OP suggests 'what could best be described as the contra-positive' rather than supplying a really cast-iron example of a contrapositive. And obviously is looking for an expression that '[is] used a bit more loosely in everyday English, and would be fine here'. I'd say true contrapositives rarely occur in everyday English. OP goes from over-formality not to neutrality but to under-formality. Feb 18 at 19:14
  • @EdwinAshworth, fair enough, your answer responds to the example, and not to what is formulated, in general terms, as the question, while mine does the opposite. It is probably impossible to do both at the same time, because the example is not really an example of a contraposition (the OP's saying that it is 'not perhaps technically an example of a strict contra-positive' is an understatement). I do, however, think it is within the scope of this site to ask 'Is there an everyday term that is close in meaning to what the logicians call contrapositon?", which is what I took the question to be.
    – jsw29
    Feb 18 at 21:25
  • @jsw29 and Edwin: You are both right, but this answer just fitted best. This answer with jsw29's as a discussion at the end would have been the best possible fit imo. So, I accepted Edwins and upvoted both. I also edited my question somewhat. (Thank you both sincerely for taking my self-nitpicking so seriously; I value it greatly.) Feb 19 at 10:21
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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.

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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.

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    I think that the example sentence and the stated definition of contrapositive in the question are at odds. I'm unclear what the author is actually looking for. Either they're trying to say something like, A quadrilateral is a rectangle, therefore if the shape doesn't have two pairs of parallel sides, it's not a rectangle. or something like Rectangular rooms are boring. On the other hand, it's hard to place furniture in a triangular room.
    – ColleenV
    Feb 18 at 19:09
  • @ColleenV, granted, the OP's example does not really fit the question. This answer responds to what the OP has explicitly articulated as the question, without attempting to deal with the example.
    – jsw29
    Feb 18 at 21:13
  • I wasn't criticizing - I'm just pointing out that two very different answers can both be good answers based on the way that the question is phrased.
    – ColleenV
    Feb 18 at 21:19
  • I think this is a very good answer. The accepted (and perhaps simplest) answer just happened to fit best in context. But, you are both ( @ColleenV ) right, in that my ex. doesn't perfectly fit the question (which I also wrote). I guess it's implicitly a two-part question: (a) What would be appropriate in general, if you want to emphasize the contra-positive nature of the statement, but in a natural way? (b) What would best fit my example? As someone dealing with mathematics, I actually like this answer, but went with what would maximize being correct, avoiding confusion and sounding natural. Feb 19 at 9:42
  • Also, since English is not my first language, and it's not always that good, for all I knew, there could've been a perfectly natural sounding word equivalent in meaning to contra-positively, that I just didn't know or was able to find (especially since such a common expression as 'on the other hand' seems to have slipped my mind). (And yes, run-on sentences is a common problem for me; I just can't be bothered to fix them in comments ;). ) Feb 19 at 9:50
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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).

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  • I don't like "on the contrary" for what OP is trying to convey. The example from Moquet et al. strikes me as bad writing. I would have expected "on the contrary" to be followed by "the host range was nearly identical in all cases" or some other words emphasizing the fact that the host range in the first sentence did not change significantly. lexico.com/en/definition/on_the_contrary
    – David K
    Feb 19 at 19:07
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    This answer is wrong: "on the contrary" is always used to emphasize the original (negative) statement, not to introduce a change of subject. Dictionary examples: "There was no malice in her. On the contrary, she was very kind." / "It is not an idea around which the community can unite. On the contrary, I see it as one that will divide us." (Myself, I would use "Quite the contrary;" in place of "On the contrary," but anyway neither of these is the phrase for which the OP is looking.) As a Lewis Carroll fan, though, I do like contrariwise. Feb 20 at 15:43
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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.

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Similarly

From Wikipedia:

If a statement is true, then its contrapositive is true (and vice versa). If a statement is false, then its contrapositive is false (and vice versa).

So what you want is to express a positive correlation: If this is a square, then it has 4 sides. Similarly, if it doesn't have 4 sides, then it's certainly not a square.

Other options

  • In the same vein
  • It follows...
  • Therefore...
  • (We see/One sees/Clearly) then...

And so forth.

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Just prefix the contrapositive phrase with "In other words, ". Avoid the word "contrapositive" and its cognates unless you're a logician.

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"On the other hand" is a perfectly good solution, but "conversely" is fine as well. It is simply not true that "conversely" in English means that what follows is the logical converse of what precedes. The noun "converse" is an uncommon term that is generally used in its technical sense. The adverb "conversely" is quite common and means, to quote Merriam-Webster, "in a contrasting or opposite way." It is best not to think of it as the adverb form of the noun "converse" - that will lead you to think people are misusing it when they are not.

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