##  [Sort](/sort-0) 

 Definition

A syntactic label or category in many-sorted logics that partitions the domain into named subsets and restricts which terms, function symbols, and predicate symbols apply to which domain elements.

 

 

 

 

 

 





## Principle

Principle

Partition the universe of discourse into disjoint or overlapping named categories so that signatures and formulas are typed and ill-formed cross-category expressions are excluded.

 

 

 

 

 





## Demonstration

Demonstration

In a language with two sorts, Person and Number, a function symbol age: Person → Number can only be applied to terms of sort Person, preventing expressions like age(3) if 3 is of sort Number.

 

 

 

 

## Misapplication

Misapplication

Treating a sort as mere documentation and allowing arbitrary cross-sort application—e.g., applying a Person-only predicate to a Number term—undermines typing and can produce formulas that the intended models do not interpret.

 

 

 

 

 





## Consequence

Consequence

When used correctly, sorts enforce well-formedness, simplify model construction by separating domains, and can reduce the search space in automated reasoning by excluding ill-typed terms.

 

 

 

 

## Reversal

Reversal

The inverse concept is an untyped or single-sorted logic where there is a single undifferentiated domain and all symbols range over the same set of elements.

 

 

 

 

 





## Boundary

Boundary

Applies to languages and models explicitly defined with multiple sorts; it does not refer to ad hoc natural-language classes, and it excludes purely syntactic namespaces that do not constrain semantic domains.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between sorts as rigid semantic partitions (which restrict interpretation) and sorts used only as convenient syntactic namespaces (which do not affect satisfaction conditions).

 

 

 

 

 





## Synthesis

Synthesis

A sort is a named type in the signature that constrains which symbols and terms belong to which subdomain of the model, ensuring typed well-formedness and guiding interpretation.