# The language of Logical Theory as Metalanguage

I am reading the Handbook of Satisfiability, and in Chapter 1 (History of Satisfiability), I found the following statement:

Moreover, necessary and possible are predicates of the metalanguage (the language of logical theory) because they are used to describe sentences in the “object” language (the language that refers to entities in the world that is the object of investigation in logical theory).

I have two questions:

1. Could you help me please, to know what is "The language of logical theory" I have writen "'Language of Logical Theory'" in google and only found this in the same book.

2. Can I use necessary and possible as predicates? Could you give some example, please?

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The concept of metalanguage is central to logic. Here is an example. The following statement is in the language of propositional logic:

$$a \lor b$$

The following is in the metalanguage:

The formula $a \lor b$ is satisfiable.

In formal logic, we have some formalism for logical statements. This formalism specifies the syntax and semantics of logical statements. For example, in propositional logic we specify a set of logical connectives and how they are used to form logical statements (the syntax), and how a truth assignment to the variables yields a truth assignment of statements (the semantics). This constitutes the "object language". The metalanguage is just the usual English prose that we use, which might refer to the object language, as above.

In simple situations, we distinguish between the two by using logical symbolism for the object language, and natural language (like English) for the metalanguage. However, there are exceptions, the most interesting of which is the use of the metalanguage as a "shortcut" for a statement in the object language. For example, consider the following statement in the language of rings:

$\exists b . ab=1$

This states that $a$ is invertible. We could also write it in this way:

"$a$ is invertible"

What this means is that "$a$ is invertible" stands for some statement in the object language which states that $a$ is invertible. How is this different from stating in prose that $a$ is invertible? Well, suppose that we have some encoding function $\varphi$ which encodes formulas to integers. Then it is meaningful to consider $\varphi(\exists b.ab=1)$ or $\varphi(\text{"$a$is invertible"})$ (the two might even be equal if you choose $\exists b.ab=1$ as the denotation of "$a$ is invertible"), but not $\varphi(\text{$a$is invertible})$, the latter being as meaningless as $\varphi(\text{blue})$.

For further illustration of this difference, I recommend reading on Gödel's second incompleteness theorem.

Necessary and possible are logical concepts (concepts of the metalanguage) which can be thought of as properties of logical statements: a logical statement is necessary if it is always true (i.e., if it is a tautology), and it is possible if it could be true (i.e., if it is satisfiable). As such, you can think of them as predicates on the domain of logical statements. In the philosophical discipline of logic they have greater applicability, beyond propositional and predicate logic.

As an example, $a \lor \lnot a$ is possible and necessary, $a \lor b$ is possible but not necessary, and $a \land \lnot a$ is neither possible nor necessary. Every necessary statement is a fortiori possible.