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At some point, I gained the notion that inflections of a word didn't constitute different words, but rather different forms of the same word. This Wikipedia article however, says the process of inflection is a part of word formation, which is the creation of new words. The linked-to Wikipedia article on word formation however, doesn't list inflection as a type.

So, are different inflections of a word different words, or different forms of the same word? Or are there different opinions here?

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The term "word" is a bit fuzzy here. When you want to be more exact, you may want to speak of word forms corresponding to the individual inflected forms and of lemmas or lexemes for equivalence classes of word forms. In corpus linguistics word is often used synonymously with word form (e.g., in corpus query languages), but I won't use word as a term at all when the distinction really matters.

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This is a classic "it depends on how you define it" problem. If you ask a large random sample of native English speakers across the globe, I expect that they would say that "child" and "children" are not different words, likewise "bring" and "brought", but "atom" and "atomic" are different words. I don't know of any large-scale surveys like that. If you ask linguists the same question, you will mostly get ideological responses rather than naive intuitions. We will typically defer a direct answer but probably blame the problem on "word", whereas I think the problem has to do with "same". Even if a person utters "child" twice, the result is not "the same", it is "the same in specific respects" even though the tokens are different in come respects. Given the question, about "inflection", there is a very limited possibility of divergence from absolute sameness.

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    I’d say the problem is not only ‘same’ and ‘word’, but also ‘be’ (or ‘are’ – case in point!). Jun 25 at 22:32
  • The problem would then still lie in the definition of "word", as "same" is being used with the assumption that a concrete line between "sufficient sameness" and "insufficient sameness" has been drawn to define the border between "the same word" and "a different word". The word "same" does not specify that in it of itself, but in the hypothetical scenario where "word" is defined with these borders, then the context would make "same" mean a concrete thing, instead of an ambiguous amount of similarity.
    – A. Kvåle
    Jun 27 at 19:10

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