TITLE: What are examples of theorems which were once "valid", then became "invalid" as standard definitions shifted?
QUESTION [25 upvotes]: That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in later times, be interpreted as constituting a false claim, due to changing fashions as to how to standardly formalize some of the relevant concepts.
I imagine this sort of thing has happened often (e.g., with shifting accounts of "polyhedra" a la Lakatos' "Proofs and Refutations", or a motley of different definitions of "continuity" before standardization on the one we use now), but I do not have enough awareness of history to be able to provide solid examples (e.g., it seems plausible to me that Darboux may have considered himself to have proven that every derivative is continuous, taking the intermediate value property to be defining for continuity, but I do not know if this is an accurate account of what he claimed).

REPLY [8 votes]: I am still not 100% sure I understand what the question is asking for, but it occurred to me today that one example might be the following statement:

$(*)$ If every S is P then some S is P.

Today, we would say that $(*)$ is not valid, because if S is vacuous then "every S is P" is true but "some S is P" is false.  However, for most of the history of Western civilization, $(*)$ was considered valid.  This is usually explained by saying that the "classical" statement that "every S is P" really means, in modern language, "there exists some S and every S is P."
This point is discussed in detail in the article on The Traditional Square of Opposition in the Stanford Encyclopedia of Philosophy, where it is also pointed out that if we additionally translate "some S is not P" into modern language as "if there exists some S then some S is not P" then we can recover the entire traditional square of opposition.
This seems to meet Sridhar Ramesh's request for an example not of changing standards of rigor but of "shifting standard formalizations of preformal concepts" (in this case, the concepts of "every" and "some").