Math 2.0

Mathematical logic and foundations Definition

Signature

Builds on Predicate logicFunction

Definition

A signature σ\sigma consists of:

  1. a set FF of function symbols;
  2. a set RR of relation symbols, disjoint from FF;
  3. an arity function ar⁡ ⁣:F∪R→ℕ\operatorname{ar} \colon F \cup R \to \N that gives the number of arguments of each symbol.

A function symbol of arity 00 is a constant symbol. The symbols of σ\sigma are the non-logical symbols of a first-order language; the variables, connectives, quantifiers and equality are not part of σ\sigma.

Remarks

The signature of rings is {+,⋅,−,0,1}\set{+, \cdot, -, 0, 1} with arities 2,2,1,0,02, 2, 1, 0, 0; it has no relation symbols. The signature of ordered sets has one relation symbol << of arity 22 and no function symbols. The list "a binary operation ⋅\cdot that is associative" is not a signature: associativity is an axiom, and a signature contains only symbols and their arities. Some books list the constant symbols as a third set instead of treating them as functions of arity 00. In Mathlib a signature is FirstOrder.Language.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Function, concept, no evidence levelFunctionconceptProposition, concept, no evidence levelPropositionconceptConjunction, 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 logicconceptSignature, definition, no evidence levelSignaturedefinition
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. Function concept
  2. Proposition concept
  3. Conjunction definition
  4. Disjunction definition
  5. Implication definition
  6. Negation definition
  7. Predicate definition
  8. Biconditional definition
  9. Propositional logic concept
  10. Predicate logic concept
  11. Signature definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Signature_(logic)
About this page
TagPWHJZYBNEY
URIm2://PWHJZYBNEY
Kinddefinition
WritingW0 stub
MSC03C07, 03B10
DefinesSignature
UsesPredicate logic, Function
Changed2 November 2026

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