Provability Logics for Relative Interpretability

Abstract

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 →. Furthermore, we will consider the following extensions of IL: ILM = IL + M, where M is the axiom (A| > B)→(A∧ C| > B∧ C) ILP = IL + P, where P is the axiom (A| >B)→ (A| >B) 2 We will write | _ IL for derivability in IL, similarly for the other systems, but sometimes we may leave the subscript off. 1 We want to thank Albert Visser who inspired these investigations by asking us to try and find a useful semantics for the system ILM. We also thank Rineke Verbrugge for a number of corrections. 2 The scheme M is named after Franco Montagna who showed its soundness with respect to PA, even in the more general case when C is replaced by a Σ-formula. The background of the names of the schemes P and W is semantic and will be explained in the next section.

Unary interpretability logic.

“…By (2) and 72 it follows that T\-l(Ev <>E), which, by 73, implies Γ h IE and IE G Γ-but this contradicts the fact that IE £ Γ by the existence of an ^-critical successor of Γ.…”

“…Our modal completeness proofs use infinite maximal consistent sets instead of the finite ones used, for example, to prove L or IL complete (in Smoryήski [8] and de Jongh and Veltman [2], respectively). Our approach has the advantage that it can do without the large adequate sets employed there.…”

“…Proof: By [2] we have for all ,4 E £ (D, >), IL h A iff for all finite /L-models cM, cM (=^4. From this and 2.9 the proposition follows.…”

“…Recall that it employs infinite maximal consistent sets and a 'small' adequate set instead of finite maximal consistent sets that are contained in a 'large' adequate set (as used, for example, in [8] and [2]). Our method has also been used to prove the modal completeness of several of the binary interpretability logics mentioned in this paper (cf.…”

Problems in the Logic of Provability

“…Visser proposed a system ILM as a candidate for the interpretability logic of Peano Arithmetic and a system ILP as a candidate for the interpretability logic of Gödel-Bernays set theory. Veltman found a Kripke style semantics for these logics, which was studied in [28]. 26 Visser's first conjecture about arithmetical completeness was proved independently by Shavrukov [74] and by Berarducci [16].…”