TeXipedia

models

Indicates a semantic modeling or satisfaction relationship, commonly used in mathematical logic and set theory.

Overview

Essential in formal logic and mathematical proofs to express that one mathematical structure satisfies or models another structure or set of conditions.

  • Frequently used in model theory to show that a structure satisfies a set of axioms or formulas
  • Common in set theory and abstract algebra for describing relationships between mathematical objects
  • Appears regularly in computer science, particularly in formal verification and theoretical computer science
  • Often paired with other logical symbols in mathematical statements and proofs

Examples

Expressing that a set of axioms logically implies a theorem in mathematical logic.

Γ⊨ϕ\Gamma \models \phi 
\Gamma \models \phi

Showing that a mathematical structure satisfies a particular property.

M⊨TM \models T 
M \models T

Demonstrating model-theoretic satisfaction in formal semantics.

M,g⊨∀xP(x)\mathcal{M}, g \models \forall x P(x) 
\mathcal{M}, g \models \forall x P(x)