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Several times I have seen this advice (or words to this effect): "Never use \\ to break lines outside special contexts such as tabular." (for example, Pass a 's' (star) parameter to a macro). I cannot find a source for this advice.

Indeed, "The LaTeX Companion" (page 104) has this:

"Other ways of typesetting paragraphs are flush right and centered, with the flushright and center environments, respectively. In these cases the line breaks are usually indicated with the \\ command, whereas for ragged-right text (the flushleft environment discussed above) you can let LaTeX do the line breaking itself (if you are happy with the resulting quality)."

Moreover, instances of the use of \\ to format paragraphs are legion. Why is this use of \\ ill-advised?

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    But ` \\ ` doesn't create a new paragraph, it creates a newline. – Bernard Aug 2 '17 at 12:44
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    Related: When to use \par and when \\, or blank lines. – Alan Munn Aug 2 '17 at 12:50
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    The question asks about using it to format paragraphs, but in the comment above you say you know it doesn't create a new paragraph. If the question is about paragraphs, it is probably a duplicate of one of the questions linked above. If it is not about paragraphs, it needs to be clarified to say what it is about. – cfr Aug 2 '17 at 13:05
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    One of the interactions I didn't understand as a newbie was how the last fontsize active at the end of a paragraph defined the parameters for the whole paragraph, in spite of any \\ that would otherwise occur. It was very non-intuitive. \documentclass{article} \begin{document} \parbox{3in}{\normalsize This\\ \Huge is a test\\ \normalsize of the emergency\\ \tiny broadcast system}\qquad \parbox{3in}{\normalsize This\\ \Huge is a test\\ \normalsize of the emergency\\ \tiny broadcast system\normalsize} \end{document} – Steven B. Segletes Aug 2 '17 at 13:09
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    it isn't that using \\ to end a paragraph is ill-advised, it is just that it does not end a paragraph, so if you want to end a paragraph you need to do something else (leave a blank line normally) – David Carlisle Aug 2 '17 at 13:19