Is the distinction between language and metalanguage strictly parallel the distinction between logic and metalogic?

Question

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?"

Answer 1

is it true that being formulated in metalanguage is a sufficient condition for a statement to belong to metalogic?

Not exactly. Consider the following example : an English book about Greek grammar is about a language : Greek, and it is written in English, the metalanguage. But it is not logic nor about logic.

But, yes,

the distinction between language and metalanguage strictly parallel the distinction between logic and metalogic.

See S.C.Kleene, Introduction to metamathematics (1952), page 62 :

To Hilbert is due the emphasis that strict formalization of a theory involves the total abstraction from the meaning, the result being called a formal system (or sometimes a formal theory); and second, his method of making the formal system as a whole the object of a mathematical study called metamathematics or proof theory.

In dealing with a particular formal system, we may call the system the object theory, and the metamathematics relating to it its metatheory.

We have that metamathematics is the mathematical discipline that studies the properties of those specific mathematical objects that are the formalized theories.

When the formalized theory is a logical calculus, like e.g. propositional calculus or predicate calculus, the metamathematical discipline studying them can be quite obviously called : metalogic.

This is consistent with what G.Hunter asserts in the Preface (p.xi) :

The main contents are : Proof of the consistency, completeness and decidability of a formal system of standard truth-functional propositional logic. The same for first-order monadic predicate logic.

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Considering specific examples, we have that :

(1) $p \to q$ is a formula of propositional calculus, i.e. an expression of the object language, written using the symbols of the language, like $\to$.

$\vDash (p \to p)$ is a statement in the metalanguage, asserting that formula $(p \to p)$ is valid.

The symbol $\vDash$ is not part of the calculus, i.e. is not part of the object language. It is a metalinguistic expression.

Regarding (2): exactly; a mathematical logic textbook is "made of" very few parts written in the object language : basically, only the initial examples of formulas and derivations, aimed at showing "how the calculus works".

The main part of the book is "metalogic" : it is metalogical the description of the syntax of the formal system. It is metalogical the proof of theorems like the Deduction Theorem and the Soundness and Completeness Theorems, i.e. the mathematical theorems expressing the properties of the formal system.

In conclusion, regarding (3): yes, a mathematical logic book is a metalogic book.

Compare with a textbook named "Physics": it is a book stating the physical theory, i.e. the theory regarding the facts of the physical realm.