 ##  [Tarski-Vaught Test](/tarski-vaught-test) 

  ##  [Tarski-Vaught Test](https://mathlogic.quantumdictionary.io/tarski-vaught-test-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A syntactic criterion to determine when a substructure A of a structure M (in a given first-order language) is an elementary substructure: A is elementary in M iff for every formula φ(x,y) and every tuple a from A, if M satisfies ∃x φ(x,a) then there exists b in A such that M satisfies φ(b,a). Equivalently, A is closed under existential witness-finding in M for formulas with parameters from A.