Does "on" has the meaning of "by using" ?
1- I cant diagnose mental problems just by using this test. (or just by)
2- I cant diagnose mental problems just on this test.
Can I use the second one instead of the first one?
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Yes, "I can't diagnose mental problems just on this test" is perfectly good.
NB: comments suggest this might not be standard usage in medical circles. I suspect regional variations too, even for non-medical usage: "I can't decide just on this test."