Grzegorczyk Logic Research Articles

Interconnection of the Lattices of Extensions of Four Logics

We show that the lattices of the normal extensions of four well-known logics—propositional intuitionistic logic \(\mathbf {Int}\), Grzegorczyk logic \(\mathbf {Grz}\), modalized Heyting calculus \(\mathbf {mHC}\) and \(\mathbf {K4.Grz}\)—can be joined in a commutative diagram. One connection of this diagram is an isomorphism between the lattices of the normal extensions of \(\mathbf {mHC}\) and \(\mathbf {K4.Grz}\); we show some preservation properties of this isomorphism. Two other connections are join semilattice epimorphims of the lattice of the normal extensions of \(\mathbf {mHC}\) onto that of \(\mathbf {Int}\) and of the lattice of the normal extensions of \(\mathbf {K4.Grz}\) onto that of \(\mathbf {Grz}\). The link between \(\mathbf {Int}\) and \(\mathbf {Grz}\) is a well-known isomorphism established by the Blok–Esakia theorem.

The Lyndon property and uniform interpolation over the Grzegorczyk logic

We consider versions of the interpolation property stronger than the Craig interpolation property and prove the Lyndon interpolation property for the Grzegorczyk logic and some of its extensions. We also establish the Lyndon interpolation property for most extensions of the intuitionistic logic with Craig interpolation property. For all modal logics over the Grzegorczyk logic as well as for all superintuitionistic logics, the uniform interpolation property is equivalent to Craig’s property.

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

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 Godel-Lob 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 completeness 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.

Cut-elimination for Weak Grzegorczyk Logic Go

We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Godel---Lob logic GL (provability logic) and Grzegorczyk's logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here.

FORMULAS IN MODAL LOGICS4

Here, we provide a detailed description of the mutual relation of formulas with finite propositional variablesp1, …,pmin modal logicS4. Our description contains more information onS4than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to find a normal form equivalent to a given formulaAby clarifying the behavior of connectives and giving a finite method to list all exact models.

Problem of restricted interpolation in superintuitionistic and some modal logics

A restricted interpolation property IPR is investigated in modal and superintuitionistic logics. The problem of description of logics with IPR over the intuitionistic logic Int and the modal Grzegorczyk logic Grz is solved. It is proved that in extensions of Int or Grz IPR is equivalent to the projective Beth property PB2. It follows that IPR is decidable over Int and strongly decidable over Grz.

Restricted interpolation property in superintuitionistic logics

The restricted interpolation property IPR in modal and superintuitionistic logics is investigated. It is proved that in superintuitionistic logics of finite slices and in finite-slice extensions of the Grzegorczyk logic, the property IPR is equivalent to the projective Beth property PB2.

Projective Beth Property in Extensions of Grzegorczyk Logic

All extensions of the modal Grzegorczyk logic Grz possessing projective Beth's property PB2 are described. It is proved that there are exactly 13 logics over Grz with PB2. All of them are finitely axiomatizable and have the finite model property. It is shown that PB2 is strongly decidable over Grz, i.e. there is an algorithm which, for any finite system Rul of additional axiom schemes and rules of inference, decides if the calculus Grz+Rul has the projective Beth property.

On the complexity of the disjunction property in intuitionistic and modal logics

In this article we study the complexity of disjunction property for intuitionistic logic, the modal logics S4 , S4.1 , Grzegorczyk logic, Gödel-Löb logic, and the intuitionistic counterpart of the modal logic K . For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs. The techniques we use do not require proving structural properties of the calculi in hand, such as the cut-elimination theorem or the normalization theorem. This is a key point of our approach, since it allows us to treat logics for which only Hilbert-style characterizations are known.

Definability in Normal Extensions of S4

A projective Beth property, PB2, in normal modal logics extending S4 is studied. A convenient criterion is furnished for PB2 to be valid in a larger family of extensions of K4. All locally tabular extensions of the Grzegorczyk logic with PB2 are described. Superintuitionistic logics with the projective Beth property that have no modal companions with this property are found.

Refutations, Proofs, and Models in the Modal Logic K4

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk's logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.

Least fixed points in grzegorczyk's logic and in the intuitionistic propositional logic

Least fixed points in grzegorczyk's logic and in the intuitionistic propositional logic