Axiomatic extensions of n-valued Goguen propositional logic system denoted as ∏_~,Δ is first studied in this paper. Using induced function, the definition of Γ-k truth degree of formula relative to local finite theory Γ under the k conjunction is given. The MP rule, HS rule, and some correlation properties are also discussed. Finally, the definition of Γ-k similarity degree and Γ-k pseudo-metric in ∏_~,Δ between two formulas is presented, and some good properties about Γ-k similarity degree and Γ-k pseudo-metric relative to local finite theory Γ under the k conjunction are simultaneously obtained.