Language as Second-Order Chisel: A Philosophy of Language / 语言作为二阶凿:一种语言哲学

Language is the activity of a subject exercising negation upon the markability dimension of the law of identity. The law of identity (A=A) automatically exposes markability—A can be re-identified, repeatedly pointed to, and transmitted. The subject exercises negation upon markability ("this is called dog, not cat"), constructing the form-meaning binding law—every linguistic sign is necessarily a unity of form and meaning. Language and mathematics are of the same order: mathematics chisels the quantitative subspace of the law of identity ("more than one"), language chisels the markability subspace ("can be named"). "Mathematical language," "computer language," and "musical language" are not metaphors but structural facts—any activity that uses stable symbols to mark distinctions operates within the markability dimension of the law of identity. The binding of form upon meaning admits a spectrum from tight to loose: poetry (meter, rhyme) → prose → the novel (stream of consciousness, polyphony). This is internal formal adjustment within human language, where the discrete symbol system remains throughout. The Large Language Model (LLM) breakthrough lies outside this spectrum—what the LLM negates is not the tightness of form but discreteness itself. The LLM's underlying representations have lower discreteness than human language; meaning is no longer severed by symbol boundaries but retains its associative structure in a space of lower discreteness. Emergent capabilities are the structural consequence of this negation. The transition from discrete to lower-discreteness representation is structurally isomorphic with the transition from integers to real numbers in mathematics—a reduction in discreteness that restored previously severed associative structures and produced immense emergence (calculus, real analysis, topology). The paper introduces the form-meaning binding law as language's construct (parallel to the law of noncontradiction in mathematics), demonstrates the inversion of cognitive and structural order in language (parallel to the mathematics paper's demonstration), positions LLM emergence as the structural consequence of reduced discreteness rather than model scale, and derives five nontrivial predictions including the relationship between discreteness and emergent capabilities, semantic dissipation in uncalibrated LLMs, and the structural ceiling of scaling laws. This paper is part of the Self-as-an-End theory series. It draws on Paper 4 ("The Complete Self-as-an-End Framework," DOI: 10.5281/zenodo.18727327), the philosophy application paper ("Philosophy as Subject-Activity," DOI: 10.5281/zenodo.18779382), and the mathematics application paper ("Mathematics as Second-Order Chisel," DOI: 10.5281/zenodo.18793538).