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In the following paragraph, is it appropriate to use the term "transversely" to describe something that has the opposite effect?

Tests have shown that the lower the range, the more likely that a submatrix will be singular. Transversely, the larger the range, the least likely that a submatrix will be singular.

If not, is there another word that I can use instead? I know that, in this instance, I cannot use the word "conversely" as this means something totally different.

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I'm not sure... why doesn't "conversely" work here? –  simchona Aug 13 '11 at 2:37
    
Hi @simchona, (I believe) the reason why "conversely" won't work here is because in mathematical speak, the converse of a statement A implies B, is B implies A. In the example that I have provided above, I have more or less negated the statement. I could, however, be wrong. –  Bill Aug 13 '11 at 2:47
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"Conversely" is, indeed, better. The converse "B implies A" may equivalently be given as "not A implies not B". (On another note, "least" should be "less".) –  GEdgar Aug 13 '11 at 3:05
    
Thanks @GEdgar, I wasn't certain whether "conversely" was appropriate in this case. You more or less confirmed it in my mind. –  Bill Aug 13 '11 at 3:14
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Correction: @GEdgar - no disrespect but, the explanation you give as to the meaning of "converse" is actually the "contrapositive". I only just realised that when I read your reply again. I think I may stick to "In contrast..." or "On the other hand..." as suggested below. –  Bill Aug 13 '11 at 3:25
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7 Answers 7

No, transversely is incorrect. You might consider "In contrast ..." or "On the other hand...".

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Thanks for that @Fraser Orr. I think I will stick to "In contrast..." just to be on the safe side. –  Bill Aug 13 '11 at 3:52
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I would use "inversely", as in "inversely proportional".

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Thanks for that @Hack Saw. Although I do believe your suggestion of using the term "inversely" more accurately reflects what I want to describe, I think I will stick to "In contrast..." just to be on the safe side. –  Bill Aug 13 '11 at 3:53
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If we have a statement:

The lower the range, the more likely that a submatrix will be singular.

the logical converse of the statement is:

The more likely that a submatrix will be singular, the lower the range.

and the logical inverse of the statement is:

The larger the range, the less likely that a submatrix will be similar.

So I would say that, if your aim is logical correctness, you should use the adverb inversely. This usage is also compatible with the common English meaning of the word inversely. The adverb transversely is just wrong here. From Merriam-Webster online dictionary:

Transversely: in a line or direction running from corner to corner.

But never use the adverb contrapositively. It doesn't exist.

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Thanks for that @Peter Shor. Like I noted in my reply to Hack Saw, I believe "inversely" is the more correct term to use but, just to be on the safe side, I have resorted to using "In Contrast..." as noted by Fraser. I definitely WON'T be using the term "Contrapositively..." LOL. –  Bill Aug 13 '11 at 3:59
    
I'm sure by strict logic the word is inversely, but in practice people (mathematicians and logicians aside) simply don't use this word when juxtaposing interrelated statements in this way. –  FumbleFingers Aug 13 '11 at 14:35
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Conversely is correct. In this context, "conversely" does not have the same meaning it has in logic. It most strongly means a claim that reverses one or more aspects of a previous claim. It has a weaker meaning akin to "on the other hand".

My dictionary has this example, "We provide tech-support 24-hours a day, seven days a week; conversely, our competitors only provide tech-support 8-hours a day, five days a week." I personally wouldn't use "conversely" that way and reserve it for cases where the order or direction of something in a causal chain is reversed. But it is not incorrect.

There is nothing wrong with something like this: "Study more and you will get higher grades; conversely, if you study less your grades will surely suffer." The direction of the amount of studying and the movement of the grades are reversed. So "conversely" works perfectly. So long as there is some sense of opposition, it flows quite naturally.

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I don't particularly object to conversely, though I find it odd that I'm okay with correspondingly as well, and superficially these words seem to be opposites. But I'm not keen on the "on the other hand" part, which implies that the second "rule" somehow contradicts the first. In fact, I feel it "almost" follows as a logical consequence. –  FumbleFingers Aug 13 '11 at 13:11
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I would simply use and - the problem with using words like transversely or inversely is that they may have very specific meanings in the topic of matrix maths which could cause confusion.

Even if they don't people might think they have a technical meaning and be trying to interpret it.

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I think the most appropriate word is correspondingly.

Substituting X for range is lower, and Y for submatrix is likely to be singular, we have

If X is true then Y is true. Correspondingly, if X is not true then Y is not true.

Although the second statement doesn't logically have to follow from the first, this will often be the case for many non-trivial propositions of the form "If X then Y". A shorter way of asserting both is "If and only if X, then Y".

Although it seems odd, I also endorse @David Schwartz's conversely, despite the fact that superficially this word is the direct opposite of correspondingly.

If we can use an expression rather than a single word, I'd say by the same token, which I think implies a relationship between the two statements without getting bogged down in whether it's inverse, converse, or corresponding.

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You don't want "transversely" or "conversely" or anything else. Those two sentences mean exactly the same thing! If you really feel the need to repeat yourself, I suppose you could write "To put it another way..."

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