##  [Well-Formed Formula](/well-formed-formula-0) 

 Definition

A syntactically valid expression of a formal language constructed from atomic formulas and logical connectives according to the formation rules of that language; commonly abbreviated WFF.

 

 

 

 

 

 





## Principle

Principle

Defines the legal strings of the language by recursive formation rules: atomic formulas are WFFs, and if A and B are WFFs then certain combinations (e.g. ¬A, (A ∧ B), (A → B)) are WFFs; parentheses and operator arities are respected to avoid ambiguity.

 

 

 

 

 





## Demonstration

Demonstration

Examples of WFFs in propositional logic: p, ¬q, (p ∧ ¬q), ((p → q) ∨ r). A string like '∧ p q' or 'p q ∧' is not a WFF in usual infix notation without additional formation conventions.

 

 

 

 

## Misapplication

Misapplication

Assuming any intuitively meaningful string is a formula (for example omitting parentheses or ignoring operator arities) or attempting to read semantic content into a malformed string; using informal natural-language conjunctions without mapping them to formal connectives.

 

 

 

 

 





## Consequence

Consequence

Well-formed formulas guarantee that syntactic operations (proof rules, substitutions) and semantic evaluations (truth tables, valuations) are well-defined; only WFFs are inputs to proof systems and semantic evaluation functions.

 

 

 

 

## Reversal

Reversal

The inversion is an ill-formed string: sequence of symbols that fails the formation rules. Treating ill-formed strings as if they were formulas collapses syntax into semantically uninterpretable noise.

 

 

 

 

 





## Boundary

Boundary

Applies to the object language of a formal system; excludes meta-language comments, proof notations that are outside the grammar, and languages with different formation rules (e.g., some programming languages or type-theoretic syntaxes).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between readability/conciseness and strict formality: some notations compress parentheses or change associativity conventions for human convenience, which must be reconciled with the strict notion of well-formedness.

 

 

 

 

 





## Synthesis

Synthesis

A well-formed formula is any expression that meets the recursive syntactic formation rules of a formal language, ensuring it is a valid object for deduction and semantic evaluation within that system.