Editing Metalogic (section)

== Important distinctions ==

=== Metalanguage–object language ===

{{Main|Metalanguage}}

In metalogic, formal languages are sometimes called ''object languages''. The language used to make statements about an object language is called a ''metalanguage''. This distinction is a key difference between logic and metalogic. While logic deals with ''proofs in a formal system'', expressed in some formal language, metalogic deals with ''proofs about a formal system'' which are expressed in a metalanguage about some object language.

=== Syntax–semantics ===
{{Main|Syntax (logic)|Formal semantics (logic)}}

In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages. The term 'syntactic' has a slightly wider scope than 'proof-theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems. 'Semantic' is synonymous with 'model-theoretic'.

=== Use–mention ===
{{Main|Use–mention distinction}}

In metalogic, the words ''use'' and ''mention'', in both their noun and verb forms, take on a technical sense in order to identify an important distinction.<ref name="metalogic"/> The ''use–mention distinction'' (sometimes referred to as the ''words-as-words distinction'') is the distinction between ''using'' a word (or phrase) and ''mentioning'' it. Usually it is indicated that an expression is being mentioned rather than used by enclosing it in quotation marks, printing it in italics, or setting the expression by itself on a line. The enclosing in quotes of an expression gives us the [[name]] of an expression, for example:

{{poemquote|"Metalogic" is the name of this article.
This article is about metalogic.}}

=== Type–token ===
{{Main|Type–token distinction}}

The ''type-token distinction'' is a distinction in metalogic, that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the [[Type–token distinction|type]] of thing known as "The bicycle." Whereas, the bicycle in your garage is in a particular place at a particular time, that is not true of "the bicycle" as used in the sentence: "''The bicycle'' has become more popular recently." This distinction is used to clarify the meaning of [[symbol (formal)|symbols]] of [[formal language]]s.