The distinction language/metalanguage is often used to explain the difference between logic and metalogic. My question is to know whether this explanation is sufficient. More precisely: : is it true that being formulated in metalanguage is a sufficient condition for a statement to belong to metalogic? ( Assuming the statement is correct).
The reason that leads me to doubt can be expressed by the following reasoning (which surely goes wrong somewhere) :
(1) Expressions as " being a valid formula", " being a logical consequence of," being derivable from" belong to metalanguage . ( There is no symbol in the object-language to express such properties or relations).
(2) Logicians constantly use these expressions, even at the elementary level. They don't seem interested in using their formal language/ formal system, but in saying things ABOUT them. In textbooks, exercices ask me to show that a given formula is a tautology, or that a given reasoning is valid, etc.
(3) So , if "belonging to metalanguage" is a sufficient condition to " belonging to metalogic", the discipline called " logic" is in fact " metalogic".
(4) But, if (3) is right, what would a "logic" book look like, in case there were absolutely no metalogic in it?
Origin of the question. Following the advice of a renowned logician in his online guide entitled Teach Yourself Logic, I once bought Jeffrey Hunter's Metalogic. The book crossed the ocean, arrived in France, and I began to study it. I have not been disappointed and found Hunter's book illuminating. However, after some reflection ( some will say : "... not deep enough reflection") I said to myself: " but if this is META-logic, could there be any room left for logic?"