I think you’re letting the notation confuse you. It would imply that, if A,B,C,D where e.g. real numbers, and that is the context the “<”-sign is mostly used in. But Orders can exist on sets other then sets of numbers. You can for example sort (order) the telephone book alphabetically, so that Cooper < Smith and still there is no k so that k*Cooper>Smith.
This is fairly confusing...in the telephone book example, you haven’t defined * as an operator. I frankly have no idea what you would mean by it. Using the notation kA > D implies a defined multiplication operation, which any reader should naturally understand as the one we all use everyday (and hence, we must assume that the set contains the sort of objects to which our everyday understanding of multiplication normally applies).
Now, this doesn’t mean that you are wrong to say that, on all such sets, kA > D does not follow necessarily from A < B < C < D for some k. It does not obtain, for instance, when A = 0. It also wouldn’t obtain on the natural numbers modulo 8, for A=2 and D=5 (just to take one example—it should be easy to create others). But neither of these have any relation to the context in which you made your claim.
So, the question is, can you find a plausible set of definitions for your set that makes your claim relevant to this conversation?
I think you’re letting the notation confuse you. It would imply that, if A,B,C,D where e.g. real numbers, and that is the context the “<”-sign is mostly used in. But Orders can exist on sets other then sets of numbers. You can for example sort (order) the telephone book alphabetically, so that Cooper < Smith and still there is no k so that k*Cooper>Smith.
This is fairly confusing...in the telephone book example, you haven’t defined * as an operator. I frankly have no idea what you would mean by it. Using the notation kA > D implies a defined multiplication operation, which any reader should naturally understand as the one we all use everyday (and hence, we must assume that the set contains the sort of objects to which our everyday understanding of multiplication normally applies).
Now, this doesn’t mean that you are wrong to say that, on all such sets, kA > D does not follow necessarily from A < B < C < D for some k. It does not obtain, for instance, when A = 0. It also wouldn’t obtain on the natural numbers modulo 8, for A=2 and D=5 (just to take one example—it should be easy to create others). But neither of these have any relation to the context in which you made your claim.
So, the question is, can you find a plausible set of definitions for your set that makes your claim relevant to this conversation?