Math 2.0

Mathematical logic and foundations Definition

Higher-order logic

Builds on Predicate logicStructurePower setCartesian product

Definition

The relational types are defined inductively.

  1. ι\iota is a type, the type of individuals.
  2. If τ1,…,τn\tau_1, \dots, \tau_n are types with n≥1n \ge 1, then (τ1,…,τn)(\tau_1, \dots, \tau_n) is a type, the type of nn-ary relations between objects of types τ1,…,τn\tau_1, \dots, \tau_n.

Higher-order logic over a signature LL has, for each type τ\tau, variables of type τ\tau and quantifiers ∀\forall and ∃\exists over them. Its formulas are built as in first-order logic, with the atomic formula X(t1,…,tn)X(t_1, \dots, t_n) whenever XX has type (τ1,…,τn)(\tau_1, \dots, \tau_n) and each tit_i has type τi\tau_i. In the standard semantics over an LL-structure with domain MM, the domains DτD_\tau are given by Dι=MD_\iota = M and D(τ1,…,τn)=𝒫(Dτ1×⋯×Dτn)D_{(\tau_1, \dots, \tau_n)} = \mathcal{P}(D_{\tau_1} \times \dots \times D_{\tau_n}). Here 𝒫\mathcal{P} is the power set and ×\times the Cartesian product.

Remarks

The order of a type is 11 for ι\iota and one more than the largest order of τ1,…,τn\tau_1, \dots, \tau_n for (τ1,…,τn)(\tau_1, \dots, \tau_n). Logic of order kk allows quantifiers only over variables whose type has order at most kk. First-order logic is the case k=1k = 1, and second-order logic is the case k=2k = 2.

Example: a variable 𝒳\mathcal{X} of type ((ι))((\iota)) ranges over sets of subsets of MM, and ∀𝒳 ∃Y ∀x (Y(x)↔∃Z (𝒳(Z)∧Z(x)))\forall \mathcal{X} \, \exists Y \, \forall x \, (Y(x) \leftrightarrow \exists Z \, (\mathcal{X}(Z) \land Z(x))) is a third-order sentence. It says that each family of subsets has a union, and it is true in every structure.

Non-example: the expression X(X)X(X) is not a formula, because no type τ\tau equals a type (τ)(\tau) of relations on objects of type τ\tau.

Many presentations instead use the simply typed lambda calculus, with function types and a type of truth values; this is Church's simple theory of types.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Cartesian product, concept, no evidence levelCartesian productconceptPower set, concept, no evidence levelPower setconceptProposition, concept, no evidence levelPropositionconceptStructure, concept, no evidence levelStructureconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptHigher-order logic, definition, no evidence levelHigher-order logicdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cartesian product concept
  2. Power set concept
  3. Proposition concept
  4. Structure concept
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Negation definition
  9. Predicate definition
  10. Biconditional definition
  11. Propositional logic concept
  12. Predicate logic concept
  13. Higher-order logic definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Higher-order_logic
About this page
TagK40YJX8K1G
URIm2://K40YJX8K1G
Kinddefinition
WritingW0 stub
MSC03B16
DefinesHigher-order logic
UsesPredicate logic, Structure, Power set, Cartesian product
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.