The formal relation |= (or ⊧) between a structure (or model), a variable assignment (where applicable), and a formula that holds exactly when the formula is true in that structure under that assignment according to the language's semantic rules.

Principle

Satisfaction is defined inductively on formula structure: atomic formulas are evaluated by applying the structure's interpretation to terms and relations; Boolean connectives combine truth values in the usual way; quantifiers quantify over domain elements via assignment updates; modal and other operators use the relevant semantic primitives (e.g., accessibility for Kripke semantics).

Demonstration

In first‑order logic, for structure M and assignment g, M,g |= ∃x P(x) iff there exists an element a in the domain of M such that M,g[x↦a] |= P(x). In propositional Kripke semantics, M,w |= □φ holds iff for all v with wRv we have M,v |= φ.

Misapplication

Treating the satisfaction relation as synonymous with syntactic provability (confusing |= with ⊢), ignoring the role of assignments for free variables, or applying satisfaction clauses from one logic to another without adjusting for different semantic primitives.

Consequence

The satisfaction relation underlies model-theoretic notions such as logical consequence, entailment, validity, elementary equivalence and the transfer of properties between structures; it provides the formal bridge between syntax and semantic truth.

Reversal

Provability (⊢) is the reversal: it is a syntactic relation between formulas (and possibly premises) and conclusions within a deductive system, not an evaluation in models and assignments; completeness theorems relate the two but they remain distinct concepts.

Boundary

Applies only where a precise semantic interpretation is defined (structures, valuations, domains); it does not capture informal notions of truth, pragmatic interpretations, or proof‑search processes unless explicitly formalized as structures and relations.

Semantic Tension

Tension exists between model‑theoretic satisfaction and proof‑theoretic entailment: satisfaction is semantic and reference‑preserving, while provability is syntactic and rule‑based; practical confusion arises when tools conflate checking truth in a model with proving a theorem.

Synthesis

The satisfaction relation is the inductively defined semantic predicate that specifies exactly when a formula holds in a structure under an assignment, serving as the foundation for model theory and for precise statements about truth, consequence and equivalence.