Provability Logics for Relative Interpretability

In this paper the system IL for relative interpretability described in Visser (1988) is studied.1 In IL formulae A|> B (read: A interprets B) are added to the provability logic L. The intended interpretation of a formula A|> B in an (arithmetical) theory T is: T + B is relatively interpretable in T + A. The system has been shown to be sound with respect to such arithmetical interpretations (˘Svejdar 1983, Montagna 1984, Visser 1986, 1988P). As axioms for IL we take the usual axioms A→ A and ( A→A)→ A (Löb's Axiom) for the provability logic L and its rules, modus ponens and necessitation, plus the axioms: (1) (A→B)→(A|>B ) (2) (A|> B) ∧ (B|> C) → (A|> C) (3) (A|> C) ∧ (B|> C)→(A∨B|> C) (4) (A|> B)→( A→ B) (5) A|>A With respect to priority of parentheses |> is treated as →.
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