### A cut-free sequent system for Grzegorczyk logic, with an application to the Gödel–McKinsey–Tarski embedding

Roy Dyckhoff, Sara Negri
2013 Journal of Logic and Computation
It is well-known that intuitionistic propositional logic Int may be faithfully embedded not just into the modal logic S4 but also into the provability logics GL and Grz of Gödel-Löb and Grzegorczyk, and also that there is a similar embedding of Grz into GL. Known proofs of these faithfulness results are short but model-theoretic and thus non-constructive. Here a labelled sequent system Grz for Grzegorczyk logic is presented and shown to be complete and therefore closed with respect to Cut. The
more » ... ompleteness proof, being constructive, yields a constructive decision procedure, i.e. both a proof procedure for derivable sequents and a countermodel construction for underivable sequents. As an application, a constructive proof of the faithfulness of the embedding of Int into Grz and hence a constructive decision procedure for Int are obtained. 1 Here in translation, including use of the name S4 in place of the name he used. 2 As a referee remarked, Gödel's translation, taken literally, is seen not to be faithful by considering the intuitionistically unprovable formula (P⊃P)⊃(P ∨¬P) which is translated to the S4 provable formula 2(P⊃P)⊃2(P ∨¬P); however one can assume that in his note the recursive part of the definition was left implicit. Downloaded from 2 Cut-free sequent system for Grzegorczyk logic of Int into what were later called the provability logics GL and Grz of Gödel-Löb and (resp.) of Grzegorczyk. Dummett and Lemmon proved that Int + A if and only if S4+ 2 A 2 . A similar result, with Grz in place of S4, was stated (without proof) in  . See also  for a survey. The provability logic GL extends the normal modal logic K by Löb's axiom schema 2(2A⊃A)⊃2A (from which the 4 schema 2A⊃22A follows, and the T schema 2A⊃A does not follow); its semantics is given by irreflexive transitive Noetherian frames. In 1976, Solovay  emphasized the importance of GL by presenting it as the logic that characterizes arithmetic provability, i.e. he showed that, for any modal formula A, GL A if and only if, for every realization r of the atoms in A as sentences of PA (Peano Arithmetic), one has PA P r (A), where, for formula X, the sentence P r (X) of PA is defined on atoms X as r(X), routinely on conjunctions, disjunctions, negations, implications and absurdity, but for modal formulae 2B one has P r (2B) ≡ Bew( P r (B) ), where · is a fixed Gödel numbering of sentences of PA as numerals and Bew (short for Beweisbar) is a defined unary predicate with Bew( · ) capturing arithmetic provability of sentences. The Löb axiom schema 2(2A⊃A)⊃2A is then interpreted as saying, for any sentence S of PA, that, if PA Bew( S )⊃S, then already PA S, i.e. Löb's Theorem. It had been observed already by Gödel in 1933 that a naive 'provability' interpretation of Int would clash with the alethic interpretation of necessity: S4 proves 2(2A⊃A), but the instance of this with ⊥ in place of A, translated by a provability interpretation into PA, expresses the provability of consistency, Bew( ¬Bew( ⊥ ) ), which by the second incompleteness theorem fails in any system containing arithmetic. Grzegorczyk defined in  , along the lines of the semantic topological method of McKinsey and Tarski, a special class of topological (point-free) spaces associated to finite reflexive transitive and antisymmetric frames (i.e. finite partial orders) and showed that Heyting algebras can be embedded in these frames. He also provided an axiomatization of the logic (now called Grz) characterized by such frames, as the extension of S4 by the axiom ((A =2B) =2B) ∧ ((¬A =2B) =2B) =2B), where C = D abbreviates 2(C ⊃D), and proved semantically that it is a proper extension of S4, not contained in S5 but in which Int is faithfully embedded by means of the translation · 2 . Segerberg later  gave a simpler axiomatization over S4 using the schema 2(G(A)⊃A)⊃A, where G(A) ≡ 2(A⊃2A). Several authors independently proposed [2-4, 6, 15, 18, 19] a modified interpretation (the provability-truth interpretation) of modality in terms of arithmetic provability; this uses a translation · + from Grz to GL, in which (2A) + is defined as A + ∧ 2(A + ); for its motivation see [20, 21] . Provability of A in Grz is then equivalent to provability in GL of its translation A + and therefore to provability in PA of every P r (A + ). That Int can be embedded into GL then follows from a modification · of the translation · 2 used for the embedding into S4 (and Grz); this modification · interprets atoms P as P ∧2P and implicational formulae A⊃B as (A ⊃B )∧2(A ⊃B ). The modal interpretation · 2 of Int, together with the translation · + of Grz into GL, thus gives the embedding · of Int into GL. However, unlike the proofs of soundness, the syntactic proofs of faithfulness of these embeddings are not entirely straightforward, as witnessed in section 9.2 of  for the relatively simple case of the embedding of Int into S4. Semantic arguments can be found in, e.g., [8, 15] . In the case of the syntactic proofs, careful invention of a strengthened induction hypothesis, stronger than that which has to be proved, is required. This is simplified in that simple case by the use of labelled systems  ; we show in Section 6 how to attack the problem for the faithfulness of the embedding · 2 of Int into Grz. A first step to establishing such a faithfulness result consists in the formulation of a cut-free sequent system for the logic (in this case, Grz) that is the target of the embedding. A (traditional) by guest on July 19, 2013 http://logcom.oxfordjournals.org/ Downloaded from 1. , are multisets of labelled formulas from Int, possibly with relational atoms also in ; 2. , are multisets of labelled atomic formulas; 3. is a multiset of labelled formulae of the form G(P) or G(A 2 ⊃B 2 ); and that G3Grz 2 , , ⇒ 2 , . by guest on