Rational Pavelka Logic

Origin. Pavelka (1979), Hájek (1998). Graded provability. Truth constants for rationals. Complete degree calculus. Foundation for graded deduction.

Models. Łukasiewicz logic with rational constants. r̄ denotes truth degree r ∈ [0,1] ∩ ℚ. Graded theorems. Fuzzy deduction theorem.

Formalism.

Language: Standard connectives plus constants. r̄ for each r ∈ [0,1] ∩ ℚ. Interpretation: v(r̄) = r.

Graded consequence: Γ ⊨_r φ: degree of φ at least r given Γ. |φ|_Γ = inf{v(φ) : v ⊨ Γ} (if Γ consistent)

Graded provability: ||φ||_Γ = sup{r : Γ ⊢ r̄ → φ} Syntactic degree.

Pavelka completeness: |φ|_Γ = ||φ||_Γ Semantic = syntactic degree.

Axioms (over Ł): r̄ ↔ s̄ if r = s (r̄ → s̄) ↔ min(1, 1-r+s) (r̄ & s̄) ↔ max(0, r+s-1)

Deduction theorem: Γ ⊢ r̄ → φ iff Γ, r̄ ⊢ φ Exact correspondence with constants.

Bookkeeping axioms: Axioms compute truth degrees. Infinitely many axioms.

Modus ponens: From φ and φ → ψ, infer ψ. Degree calculus: from r and r → s, get s.

Symbols.

Symbol	Unicode	Name	Meaning
r̄	—	Constant	Rational truth value
⊢_r	—	Graded	Provability degree
⊨_r	—	Graded	Semantic degree
\|\|·\|\|	—	Provability	Syntactic degree

Metatheory. Pavelka-style completeness. Graded soundness. Extends to other fuzzy logics. Infinitary axioms.

Applies to. Approximate reasoning. Graded beliefs. Expert systems. Fuzzy databases.

Limitations. Infinite axiom schemes. Computational complexity. Rational restriction. Philosophical status of grades.