> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/terms/syntax.md).

# Syntax

<https://en.wikipedia.org/wiki/Syntax_(logic>)

Syntax is concerned with the rules (grammar) governing the manipulation of expressions that constitute the *well-formed formulas* (wff) of a formal system and its formal language.

A well-formed formula (wff) is a syntactically valid string in a formal language, as opposed to the myriad of other, nonsense, strings.

Syntactic entities of formal languages:

* symbol
* formula
* system
* theorem
* proof
* interpretation

Symbols and strings may be divided into nonsense and well-formed formulas. *A formal language is identical to the set of its well-formed formulas*. The wff set may be divided into *theorems* and *non-theorems*.

> theorems ⊂ non-theorems ⊂ wff ⊂ all lang strings

symbols -> strings ↓ ↙ formulas --> atoms (atomic formulas) --> complex formulas ├─ nonsense formulas └─ well-formed formulas ├─ non-theorems └─ theorems