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L = Totality Of Language Games
Gᵢ ∈ L = A Specific Language Game
R(Gᵢ) = Rule Set Governing Gᵢ
(Gᵢ) = Allowable Actions In Gᵢ
S(Gᵢ) = Speech Acts Within Gᵢ
U(w, Gᵢ) = Use Function Assigning Meaning To Word w Under Game Gᵢ
Each Language Game, Denoted Gᵢ, Functions As A Discrete Rule-Governed Practice, Characterized Formally By A Rule Set R(Gᵢ), Allowable Actions A(Gᵢ), And Speech Acts S(Gᵢ). The Use Function U(w, Gᵢ) Assigns The Meaning Of A Word w Relative To Its Role Within Gᵢ, Thus Meaning(w | Gᵢ) = U(w, Gᵢ) ∈ A(Gᵢ). There Is No Universal Essence To Language Games; Instead, They Exhibit Overlapping Similarities Through Family Resemblances, Formally Described As L = ⋃ Gᵢ Where ∀ Gᵢ, Gⱼ: ∃ Shared Features F(Gᵢ, Gⱼ). Hence, ∄ E: ∀ Gᵢ (E ∈ Gᵢ); Instead, The Similarity(Gᵢ, Gⱼ) ≈ Σ δ(Fₖ), Yielding A Non-Essentialist, Cluster-Based Taxonomy Of Language Uses.
The Normativity Of Language Games Derives From Rule-Following; For Any Gᵢ, ∀ s ∈ S(Gᵢ), s Is Correct If And Only If s ∈ R(Gᵢ). To Master A Language Is To Internalize These Rules Via Participation In Its Form Of Life. Crucially, The Private Language Argument Demonstrates That A Language Whose Rules Rely Solely On Private, Inner Criteria Cannot Exist, Since ∃ R(G_PL): ∀ s, Correct(s) ⇔ s Appeals To Internal State Alone Leads To An Infinite Regress And Loss Of Rule-Governed Normativity. Thus, All Meaning Requires Public, Interpersonal Anchoring.
The Core Formula; Meaning(w) = Use(w | Gᵢ, R(Gᵢ), A(Gᵢ), Lebensform)
