The Principle of Definition-Dependent Provability: Provability as a Function of Definition

This paper introduces and defends the Principle of Definition-Dependent Provability (DDP), a metatheoretical principle asserting that provability is a function of definition. The principle provides a diagnostic tool for distinguishing two phenomena routinely conflated in philosophical discourse: genuine logical undecidability (where precisely defined sentences are neither provable nor refutable within a specified formal system) and definitional indeterminacy (where statements appear undecidable only because key terms lack sufficient formal specification). The paper develops a formal framework based on conservative definitional extensions, introducing a provability function f(d) that maps definitions to provability outcomes. Criteria for definitional admissibility (inferential role preservation, extensional adequacy, and theoretical fruitfulness) are proposed and given semi-formal characterization through the notion of conceptual profiles. The principle is demonstrated through detailed case studies spanning mathematics (randomness formalizations in algorithmic information theory, cryptography, and statistical testing), metaethics (competing definitions of objective moral facts), and metaphysics (free will, consciousness, mind-brain reduction). The continuum hypothesis is analyzed to clarify why genuine set-theoretic independence is not a counterexample to DDP. The paper situates DDP relative to logical pluralism, proof-theoretic semantics, meta-ontological deflationism, and conceptual engineering, clarifying its complementary role. Extensions to non-conservative cases involving theoretical embedding and framework change are discussed. A diagnostic checklist operationalizes the principle for evaluating claims of "in principle unprovability." The central thesis is methodological: there are no undecidable statements prior to definition—only statements that have not yet entered the formal space where proof is possible. DDP disciplines philosophical discourse by requiring that provability claims specify both the formal system and the precise formulation of the statement at issue. Keywords provability; definition; formal systems; undecidability; Gödel; metamathematics; conceptual engineering; philosophy of logic; definitional indeterminacy; conservative extensions Subject Areas Philosophy of Logic Metamathematics Philosophy of Language Formal Methods Conceptual Engineering Notes This work synthesizes established metatheoretical principles from Tarski, Suppes, Carnap, and contemporary conceptual engineering into a unified methodological framework. The principle is offered as a diagnostic tool for philosophical discourse rather than a completeness result or challenge to Gödel's incompleteness theorems.