A universal formula (of the form ∀x φ where φ is quantifier-free) is preserved under substructures: if a universal formula holds in a substructure, it holds in the parent. This preservation property characterizes which sentences describe properties that propagate upward through the substructure ordering and is fundamental to understanding model-completeness.
From first-order logic syntax, you know that formulas are built from atomic predicates using logical connectives (¬, ∧, ∨, →) and quantifiers (∀, ∃). A quantifier-free formula uses only connectives — no quantifiers at all. A universal formula (also written Π₁) prefixes a block of universal quantifiers in front of a quantifier-free matrix: ∀x₁ ∀x₂ … ∀xₙ φ(x₁,…,xₙ). The name comes from the fact that all variables are universally bounded. Universal formulas contrast with existential formulas (Σ₁), which prefix a block of existential quantifiers: ∃x₁…∃xₙ φ.
The core preservation fact is straightforward to prove once you think about it carefully. Suppose A is a substructure of B — meaning A is a structure with the same signature, A's domain is a subset of B's domain, and all relations and functions on A are restrictions of those on B. Now suppose the universal sentence ∀x φ(x) holds in A. Does it hold in B? Not necessarily: B might contain elements not in A that violate φ. But the reverse direction is what's true: if ∀x φ(x) holds in B, it holds in A. Because A ⊆ B, every element of A is also an element of B, and the quantifier-free formula φ is checked on those specific elements — and quantifier-free formulas are preserved under substructures since they only involve checking relations and functions, which A inherits from B. So ∀x φ holds in B ⇒ it holds in A. In other words, universal sentences are preserved going to substructures (from B down to A), while existential sentences are preserved going to superstructures (from A up to B).
A concrete example: the group axiom ∀x ∀y ∀z (x·(y·z) = (x·y)·z) is universal. If a set B with an operation is an associative structure, then any substructure A inherits associativity — you are just restricting which elements you check, and they still satisfy the universal claim. By contrast, the axiom ∃e ∀x (e·x = x) — the existence of an identity — is existential. A subgroup inherits the identity from the parent group, but not because universal-formulas force it; rather, this is an additional property that happens to be preserved for groups because the identity of the parent group lies in every subgroup (by definition). If you dropped that definition requirement, you could have a sub-semigroup without an identity.
The Łoś–Tarski theorem (preservation theorem) states the converse: a sentence is preserved under substructures if and only if it is logically equivalent to a universal sentence. This gives a semantic characterization of the universal fragment — preservation is not just a proof trick but the defining property. The connection to model-completeness is direct: a theory T is model-complete if and only if every formula is T-equivalent to a universal formula. Preservation under substructures thus becomes the bridge between the syntactic definition of universal formulas and the semantic notion of model-completeness, linking formula structure to structural behavior in models.