Type theory¶
The family of formal systems that classify expressions by types and govern how typed terms may be formed, transformed and interpreted.
Core Idea¶
Simple, dependent, polymorphic, intensional and homotopy type theories differ, type theory can be a logic, programming discipline or foundation and type checking need not imply semantic correctness. Judgments assign terms to types under contexts, formation introduction elimination and computation rules generate valid expressions and propositions-as-types can interpret proofs as terms. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Type theory belongs to logic and computation and is useful where the analyst can specify the typed logic and computation carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the syntax of terms types and contexts, judgment forms, formation introduction elimination and computation rules, substitution and definitional equality, typing derivations and metatheorems, propositions-as-types interpretation, semantic model and specific type-theory variant and foundational or programming use are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the syntax of terms types and contexts, judgment forms, formation introduction elimination and computation rules, substitution and definitional equality, typing derivations and metatheorems, propositions-as-types interpretation, semantic model and specific type-theory variant and foundational or programming use are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Type theory. Type theory compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed logic and computation carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
- Lock the constitutive rule. Express the syntax of terms types and contexts, judgment forms, formation introduction elimination and computation rules, substitution and definitional equality, typing derivations and metatheorems, propositions-as-types interpretation, semantic model and specific type-theory variant and foundational or programming use are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of logic and computation because they reuse the typed logic and computation carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Judgments assign terms to types under contexts, formation introduction elimination and computation rules generate valid expressions and propositions-as-types can interpret proofs as terms., and type the carrier, state every parameter and convention in the definition, test that the syntax of terms types and contexts, judgment forms, formation introduction elimination and computation rules, substitution and definitional equality, typing derivations and metatheorems, propositions-as-types interpretation, semantic model and specific type-theory variant and foundational or programming use are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Type theory Domain-specific
Parents (1) — outgoing typed relations
-
Type theory is a kind of Classification Prime
It remains its own entry because its identity is fixed by the syntax of terms types and contexts, judgment forms, formation introduction elimination and computation rules, substitution and definitional equality, typing derivations and metatheorems, propositions-as-types interpretation, semantic model and specific type-theory variant and foundational or programming use.
Children (1) — abstractions linked here by typed relations
-
Intuitionistic Type Theory Domain-specific is a kind of Type theory
Intuitionistic type theory is a constructive dependent specialization of type theory.
Hierarchy path (1) — routes to 1 parentless root
- Type theory → is a kind of Classification
Neighborhood in Abstraction Space¶
Type theory sits in a crowded region of the domain-specific corpus (2nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Formal Logic & Computation Systems (24 abstractions)
Nearest neighbors
- Typed lambda calculus — 0.94
- Identity type — 0.94
- Proof-theoretic semantics — 0.94
- Type signature — 0.94
- Independence of premise — 0.93
Computed from structural-signature embeddings · 2026-10-25