Hard SF is a term coined in 1957 by P. Schuyler Miller to distinguish between the science fiction promoted by John W. Campbell, editor of Astounding, and the SF stories being published in Galaxy and other magazines that were social satire or science fantasy.
Some users later took the term to mean SF based on the "hard" sciences (physics, chemistry, etc.) rather than the "soft" sciences (psychology, sociology, etc.). Others attributed "hardness" to any story that championed science.
There are several problems with these early definitions, of course. One is that some sciences (biology, psychology, etc.) have both "hard" and "soft" dimensions.
In 1992, Allen Steele wrote: "Hard sf is the form of imaginative literature that uses either established or carefully extrapolated science as its backbone." This seems to be the meaning held by most SF readers/fans today.
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