Referent Variant Functions and Relations

1. Core Terminology

• Referent:
  
  The actual entity or object in the world that a linguistic or symbolic expression refers to.

• Reference:
  
  The act of pointing to or denoting a referent from within a system (e.g., language, logic, model).

• Variant:
  
  A change in form, role, or relation of the referent depending on context, function, or observer.

2. Function Types

A. Constant Reference Functions

• Definition: A function where the output (referent) does not change for a given input across contexts.
• Example:
  π() → 3.14159
  The referent of π is constant.

B. Contextual (Deictic) Reference Functions

• Definition: Functions whose referents vary based on the context of utterance (speaker, time, place).

• Example:
  
  I(talker) → Current speaker
  
  now() → Current time
  
  here() → Current location

• Formal Model:
  
  ref(deictic_term, context) → entity

C. Intensional Functions

• Definition: Maps a term to a function from possible worlds to referents.
• Example:
  "the president" may refer to different people in different possible worlds or times.
• Type:
  intension : Term → (World → Referent)

3. Relation Types

A. Referential Relation

• Binary relation: refers_to(expression, entity)
  Maps symbol to real-world entity.

B. Identity Relation

• Equivalence: a ≡ b means both symbols refer to the same referent.

C. Alias Relation (Synonymy)

• Multiple names or expressions point to the same referent.
• ref("H2O") = ref("Water")

D. Observer-Dependent Referent Function

• referent(observer, entity_label)
  → Changes referent depending on the agent's knowledge or belief.

4. Cognitive and Semantic Mapping

5. Examples of Referent Variant Relations

Summary Schema

Reference: Symbol → Referent
           (via function or relation)

Ref Function: f(context) → referent
Intensional Function: f(world) → referent
Observer-Variant: f(observer) → referent
Deictic: f(speaker, time, place) → referent