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The logic of $$\Pi_ 1$$-conservativity. (English) Zbl 0713.03007
The modal propositional provability logic L (or G) has the following axioms: (L1) tautologies, (L2) $$\square (A\to B)\to (\square A\to \square B)$$, (L3) $$\square A\to \square \square A$$ and (L4) $$\square (\square A\to A)\to \square A$$. Its deduction rules are modus ponens and generalizations (if $$\vdash A$$, then $$\vdash \square A)$$. The interpretability logic IL extends the language with the binary modality $$\triangleright$$ and has the following extra axioms: (J1) $$\square (A\to B)\to (A\triangleright B)$$, (J2) $$(A\triangleright B \& B\triangleright C)\to (A\triangleright C)$$, (J3) $$(A\triangleright C \& B\triangleright C)\to (A\vee B\triangleright C)$$, (J4) $$A\triangleright B\to (\diamondsuit A\to \diamondsuit B)$$ and (J5) $$\diamondsuit A\triangleright A$$. ILM (Interpretability Logic with Montagna’s principle) results by adding the axiom $$(A\triangleright B)\to (A \& \square C)\triangleright (B \& \square C).$$
$$I\Sigma_ 1$$ is the fragment of Peano arithmetic with induction restricted to $$\Sigma_ 1$$-formulas. Let $$T\supseteq I\Sigma_ 1$$ be $$\Sigma_ 1$$-sound. An arithmetical p(artial) c(onservativity) interpretation of ILM in T is by definition a mapping * associating with each formula of ILM a sentence of T such that (1) * commutes with the connectives, (2) $$(\square A)^*:=\Pr_ T(A^*)$$, where $$\Pr_ T$$ is the provability predicate of T, and (3) $$(A\triangleright B)^*:=(\forall z\Pi_ 1$$-sentence) $$(\Pr_ T(B^*\to z)\to \Pr_ T(A^*\to z))$$, i.e., the $$\Pi_ 1$$-consequences of $$T+A^*$$ include the $$\Pi_ 1$$-consequences of $$T+B^*.$$
ILM is sound for arithmetical pc-interpretations, i.e., if ILM$$\vdash A$$, then $$T\vdash A^*$$ for each *. The authors prove in this paper the following arithmetical completeness theorem. If $$T\supseteq I\Sigma_ 1$$ is $$\Sigma_ 1$$-sound, then ILM is complete with respect to arithmetical pc-interpretations, i.e., if not ILM$$\vdash A$$, then there is a pc- interpretation * such that not $$T\vdash A^*$$.
Reviewer: H.C.M.de Swart

##### MSC:
 03B45 Modal logic (including the logic of norms)
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##### References:
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