eolas/zk/Object_language_and_meta_language.md

---
tags: [propositional-logic]
---

# Object and metalanguages

When we talk about a language we call that language the **object language**. A
**metalanguage** is a language used to describe some object language.

When we are developing a formal logical language (which we may call SL or PL for
'sentential' and 'propositional' logic respectively), the formal language is the
object language and natural language (e.g. English) is the metalanguage.

**Demonstration**

If we talk about German in English, German in the object language and English is
the metalanguage.

## Use and mention

There is an associated distinction: that of use and mention.

When we create an expression in a language we are said to _use_ that language.
When we remark upon said expression we are said to be _mentioning_ the language.
This distinction may correspond to the object and metalanguage difference above
but doesn't have to; use and mention can happen in the same language. For
example:

```
'London' is the word that denotes the capital of the UK.
```

## Metavariables

A metalinguistic variable (metavariable for short) is an expression in the
metalanguage that is used to talk generally about expressions of the object
language. The convention in these notes will be to embolden single letters when
these letters are used as metavariables.

For example, instead of saying _'P & Q' is an expression comprising two atomic
sentences and a conjunction_ we might say **\*P** is an expression comprising
two atomic sentences and a conjunction. In this instance **P** is a metavariable
in the metalanguage mentioning the expression P & Q in the object language