Math 2.0

Mathematical logic and foundations Definition

Second-order logic

Builds on Predicate logicStructurePower set

Definition

Let LL be a signature. The formulas of second-order logic over LL are built like the formulas of first-order logic over LL, with these additions.

  1. For each n≥1n \ge 1 there are relation variables X,Y,…X, Y, \dots of arity nn, and X(t1,…,tn)X(t_1, \dots, t_n) is an atomic formula for terms t1,…,tnt_1, \dots, t_n.
  2. If φ\varphi is a formula and XX is a relation variable, then ∀X φ\forall X \, \varphi and ∃X φ\exists X \, \varphi are formulas.

In the standard semantics, a formula is evaluated in an LL-structure ℳ\mathcal{M} with domain MM as in first-order logic. An nn-ary relation variable ranges over the whole power set of MnM^n.

Remarks

Function variables may be added in the same way; they can be expressed by relation variables. Monadic second-order logic allows only relation variables of arity 11, that is, variables for subsets of MM.

Example: the sentence ∀X ((X(0)∧∀x (X(x)→X(S(x))))→∀x X(x))\forall X \, \big( (X(0) \land \forall x \, (X(x) \to X(S(x)))) \to \forall x \, X(x) \big) is a second-order sentence over the signature {0,S}\set{0, S}. It says that every subset of the domain that contains 00 and is closed under SS is the whole domain.

Non-example: a formula with a quantifier over sets of subsets of MM is not second-order, because no clause provides such variables; it belongs to third-order logic.

With the standard semantics, second-order logic has neither a complete proof system nor the compactness property. With Henkin semantics, in which the relation variables range over a chosen family of relations, both properties return.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Power 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 logicconceptSecond-order logic, definition, no evidence levelSecond-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. Power set concept
  2. Proposition concept
  3. Structure concept
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Negation definition
  8. Predicate definition
  9. Biconditional definition
  10. Propositional logic concept
  11. Predicate logic concept
  12. Second-order logic definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Second-order_logic
About this page
Tag47XHPD8XE2
URIm2://47XHPD8XE2
Kinddefinition
WritingW0 stub
MSC03B16, 03C85
DefinesSecond-order logic
UsesPredicate logic, Structure, Power set
Changed2 November 2026

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