Where does derivation collide?

In logic and proof theory, a derivation is a finite sequence or tree of formulas in which each step follows from earlier ones by the system's inference rules, ending in the conclusion it proves. Gentzen's 1935 investigations into logical deduction introduced natural deduction and the sequent calculus, giving derivations their modern tree form. The referent is a syntactic object certifying that a conclusion follows from premises within a formal system, unrelated to coining a new word or an algebraic operator despite the shared name.[1]

Unit

A finite rule-governed sequence or tree of formulas ending in a theorem

Recovery means

Not applicable

Measured by

Checked by verifying each step against the inference rules; analysed by proof length, structure, and normalisation

In abstract algebra, a derivation is a linear map D on an algebra that satisfies the Leibniz product rule, D(xy) = D(x)y + xD(y), abstracting the key property of differentiation to arbitrary algebras and central to the study of Lie algebras. Jacobson's Lie algebras develops derivations and their role in structure theory. The referent is an operator defined by an algebraic identity, not a logical proof or the formation of a new word, sharing with them only the inherited term.[2]

Unit

A linear operator on an algebra satisfying the Leibniz product rule

Recovery means

Not applicable

Measured by

Defined and verified by the Leibniz identity; studied via the derivation algebra and inner versus outer derivations

In morphology, derivation is the formation of new lexemes from existing ones, typically by affixation that may change meaning or word class (happy to happiness, nation to national), as distinct from inflection, which varies a single word's grammatical form. Bauer's English word-formation is a standard treatment of derivational processes. The referent is a word-building process in the lexicon, measured by productivity and the patterns of affixation, unrelated to a logical proof or an algebraic operator.[3]

Unit

The formation of a new lexeme from a base, usually via an affix

Recovery means

Not applicable

Measured by

Studied through affix productivity, output class and meaning shifts, and dictionary and corpus evidence of new words