statementsI

StatementsI is a notational convention used in logic and philosophy to denote the collection of statements that are indexed by a set I. Formally, it refers to an indexed family {s_i} of statements, where each s_i belongs to a fixed language and i ranges over the index set I. The term emphasizes that the truth of each member can depend on the index i, and that discussions often concern properties that hold uniformly across all indices or for particular subfamilies.

The notation is a shorthand in formal discussions; it does not introduce a new kind of primitive object beyond the individual statements, but it clarifies when a discussion concerns an entire family rather than a single proposition. In practice, statementsI may be used to express universal or existential quantifications over indices, to formulate parametric proofs, or to study how logical properties vary with the index parameter.

In model theory and related fields, indexed families of formulas are central to arguments about compactness, completeness, and transfer principles. The framework for statementsI supports reasoning about families, counterexamples, and limit processes by treating the index set as a parameter space.

See also: indexed family, index set, first-order logic, model theory, parametric proofs, families of formulas.