TITLE: Why is $\alpha \rightarrow \forall x(\alpha)$ not generally correct in first-order logic?
QUESTION [5 upvotes]: Why is $\alpha \rightarrow \forall x(\alpha)$ not generally correct in first-order logic?
i.e., when there are free occurrences of $x$ in $\alpha$, and, on the same point, why is the formula scheme correct when there are no free occurrences of $x$ in $\alpha$?

REPLY [5 votes]: A good question. The difference between implication and rule of inference is clear in formal logic. However confusions in everyday reasoning are not rare because both constructions are usually pronounced as "if... then". That is why it is not easy to explain to pupils the difference between them but a few example may be useful. When we prove a theorem of geometry, it is a common manner to "take" a concrete triangle, say ABC. After investigating some its properties we formulate the theorem: "For all triangles.." However it would not be a correct analogy to "take" a concrete man, say Mack the Knife, and on the base of his life to conclude that "All men are killers". I think that such notices help understanding the logical principles and their teaching.