What is a false negative?no_redirect=1

I think it is a matter of convention. I would not say that it is "wrong" to say $0$ is positive, as long as you properly define the meaning of "positive" from the onset.

Part of the reason I do not believe that "positive" has a universally accepted definition of meaning "strictly larger than $0$", is because of the (common) usage of the phrase "strictly positive number". Perhaps it is very redundant, or it illustrates the fact that "positive" does not have a universally accepted meaning.

Furthermore, there are examples throughout mathematics where the use of "positive" does not have a strict meaning. For example, a real-valued function $f:X\rightarrow \mathbb{R}$ is typically called a "positive function" if $f(X)\subseteq[0,\infty)$. In measure theory, one typically calls a "measure" $\mu: \mathcal{M} \rightarrow [0,\infty]$ on a measurable space $(X, \mathcal{M})$ a "positive measure" (to distinguish from so-called "complex measures" $\mu:\mathcal{M}\rightarrow\mathbb{C}$). In the first case, we still call $f$ positive even if there are $x\in X$ such that $f(x) =0$. In the second case, we still call $\mu$ positive even if there are $E\in\mathcal{M}$ such that $\mu(E)=0$.

There are many other examples in mathematics where "positive" is used to include $0$. So for these reasons, either the language is not universally accepted or is misleading. (I.e., misleading in the sense that does the word "positive" in the strict sense only apply to taking about the number $0$ itself, and not functions, measures, etc.?) I also admit that using language like "non-negative function" or "non-negative measure" is a bit more awkward than saying "positive function" or "positive measure". Nevertheless, I think it is too quick to jump to the conclusion that it is incorrect to view "positive" and "non-negative" as synonyms. I believe it is context dependent.

As a final remark, I do think using the word "positive" in the strict sense is more common. But, nevertheless, it does not seem incorrect to use "positive" in the non-strict sense as well and (dare I say) perhaps has merit in some contexts.

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answered Mar 25 '19 at 16:48

SatanaSatana

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