Underivable

The term 'underivable' describes something that cannot be logically or deductively obtained from a set of premises or initial conditions. It signifies the absence of a demonstrable path or sequence of reasoning that would lead to the conclusion or assertion in question. Something that is underivable is fundamentally independent of the starting points and therefore exists without being a direct consequence of them. This concept is particularly relevant in fields like mathematics, logic, philosophy, and computer science, where formal proofs and derivations are central to establishing truth and validity.

Underivable meaning with examples

• In Gödel's incompleteness theorems, certain mathematical statements within a formal system are proven to be underivable. This means there's no logical path within the system to prove them true or false, regardless of how complex the system becomes. They exist beyond the formal limits.
• Many of the core axioms in Euclidean geometry, such as the parallel postulate, are underivable from other axioms in the system. Attempts have been made to prove the parallel postulate from others, and those failed attempts led to non-Euclidean geometries. This independence defines the shape of geometry.
• The feeling of subjective experience, or qualia, is often considered underivable from purely physical explanations of the brain. Though neuroscience explains the physical components, there is no easy way to relate them to consciousness in a logical way.
• A particular piece of art's aesthetic qualities are underivable from knowing its materials and artist's biography. The beauty derived from an art piece often surpasses any logical description of its creation or context, making the beauty of it underivable.