independence

The term "independence" in logic refers to a property of axioms within a formal system. It means that a particular axiom cannot be proven from the other axioms in the system. This concept is crucial for understanding the structure and completeness of logical calculi.1

In a related but distinct context, "linear independence" applies to sets of mathematical objects like vectors or matrices. A set is linearly independent if no element can be expressed as a combination of the others, unless all coefficients are zero. This ensures that each element contributes uniquely to the set.2

Modern logic, particularly since 1900, has increasingly used symbolic techniques, leading to a deeper understanding of logic itself. Developments in proof theory and model theory have explored how proofs can be seen as attempts to construct counter-models, and how quantifiers like "every" and "some" function. This has even led to richer logical systems like "independence-friendly" logic, which can express concepts not possible in traditional logic.3