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### Description of modal logics inheriting admissible rules for S4

V. Rybakov
1999 Logic Journal of the IGPL
Using this condition we describe all tabular modal logics inheriting inference rules admissible for S4.  ...  We give a necessary and sufficient condition for any modal logic with fmp to inherit all inference rules admissible in S4.  ...  First we recall a simple general fact: Lemma 4.1 A tabular modal logic λ := λ(F 1 · · · F n ) extending S4, preserves all inference rules admissible for S4 iff each logic λ(F i ) does. Proof.  ...

### Uniform Interpolation and Propositional Quantifiers in Modal Logics

Marta Bílková
2007 Studia Logica: An International Journal for Symbolic Logic
The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.  ...  We shall present such a proof of the uniform interpolation theorem for normal modal logics K and T. It provides an explicit algorithm constructing the interpolants.  ...  Ghilardi and Zawadowski in  proved that uniform interpolation fails for modal logic S4. It follows immediately that it fails for modal logic K4 as well.  ...

### Page 2074 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews
We show that a modal logic 4 with FMP above S4 preserves all inference rules admissible for S4 iff 4 has the so-called co-cover property. It turns out that there are 2*° logics of this kind.  ...  (RS-KRAS; Krasnoyarsk) Modal logics preserving admissible for S4 inference rules. (English summary) Computer science logic (Kazimierz, 1994), 512-526, Lecture Notes in Comput.  ...

### Page 6407 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews
Summary: “The main result of this paper is a complete semantic description of all modal logics with the finite model property (fmp) that preserve all inference rules admissible for \$4: a modal logic A  ...  (RS-KRAS; Krasnoyarsk) Preserving of admissible inference rules in modal logic. (English summary) Logical foundations of computer science (St. Petersburg, 1994), 304-315, Lecture Notes in Comput.  ...

### Rules admissible in transitive temporal logic TS4, sufficient condition

2010 Theoretical Computer Science
The paper 1 develops a technique for computation inference rules admissible in temporal logic T S4 .  ...  The main result of the paper is a sufficient condition for admissibility inference rules in T S4 .  ...  So, T S4 , despite being defined by the same class of frames as the modal logic S4, does not always admit rules admissible in S4.  ...

### Logics with the universal modality and admissible consecutions

2007 Journal of Applied Non-Classical Logics
In this paper 1 we study admissible consecutions (inference rules) in multi-modal logics with the universal modality.  ...  We consider extensions of multi-modal logic S4 n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic (as a set of theorems) is closed.  ...  Acknowledgements I would like to thank Valentin Shehtman for his idea to extend our previous research (Golovanov et al., 2005) to arbitrary multi-modal logics with universal modality. References  ...

### A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4

Yosuke Fukuda, Akira Yoshimizu, Michael Wagner
2019 International Conference on Rewriting Techniques and Applications
We propose a modal linear logic to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it.  ...  While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality  ...  We follow the proof for propositional linear logic by Lincoln et al.  . To show the admissibility of Cut, we consider the admissibility of the following cut rules: Γ !A Γ , (!A) n B !  ...

### Page 2193 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews
We also prove that self-admissible rules are always admissible for what are, in a sense, canonical log- ics such as S4 or IPC regarding the type of algebra generating rules.” 2001d:03056 03B45 Sturm, Holger  ...  The main result of our paper is a complete description of all self-admissible quasi-characterizing inference rules.  ...

### Page 3851 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews
The authors investigate admissible rules of inference in proposi- tional logics, that is, those which are stable with respect to any substitution and preserve the set of theorems of logic. As A. I.  ...  It is known that three of the five pretabular extensions of the modal logic \$4 and one of the three such extensions of Int have finite bases for admissible rules, while the remaining pretabular logics  ...

### Page 802 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews
., Dordrecht, 1999; see MR 2002b:03001] for classical proposi- tional logic and labelled rules for S4 of M.  ...  Admissibility of logical inference rules, North-Holland, Amsterdam, 1997; MR 981:03035].  ...

### A Linear-logical Reconstruction of Intuitionistic Modal Logic S4 [article]

Yosuke Fukuda, Akira Yoshimizu
2019 arXiv   pre-print
We propose a "modal linear logic" to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it.  ...  While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality  ...  We follow the proof for propositional linear logic by Lincoln et al.  . To show the admissibility of Cut, we consider the admissibility of the following cut rules: Γ !A Γ , (!A) n B !  ...

### Hintikka multiplicities in matrix decision methods for some propositional modal logics [chapter]

Serenella Cerrito, Marta Cialdea Mayer
1997 Lecture Notes in Computer Science
This work is a study of the inter-translatability of two closely related proof methods, i.e. tableau (or sequent) and connection based, in the case of the propositional modal logics K, K4, T, S4, paying  ...  in order to reduce as much as possible the search space for proofs.  ...  The authors wish to thank the anonymous referees of previous works for their suggestion to work in this direction and Alain Heuerding for useful discussions on this topic.  ...

### Frege systems for extensible modal logics

Emil Jeřábek
2006 Annals of Pure and Applied Logic
We prove a similar result for an infinite family of normal modal logics, including K4, GL, S4, and S4Grz .  ...  Mints and Kojevnikov  have recently shown p-equivalence of Frege systems for the Intuitionistic Propositional Calculus (IPC ) in the standard language, building on a description of admissible rules  ...  In a similar spirit, bases of admissible rules for some modal logics were constructed by Jeřábek  .  ...

### Taming Displayed Tense Logics Using Nested Sequents with Deep Inference [chapter]

Rajeev Goré, Linda Postniece, Alwen Tiu
2009 Lecture Notes in Computer Science
We outline such a procedure for the deep-sequent system DKt and its S4 extension.  ...  Our first calculus SKt is a variant of Kashima's calculus for Kt, which can also be seen as a display calculus, and uses "shallow" inference whereby inference rules are only applied to the top-level nodes  ...  Acknowledgment: We thank the anonymous referees for their helpful comments on an earlier draft of the paper.  ...

### Does the deduction theorem fail for modal logic?

Raul Hakli, Sara Negri
2011 Synthese
Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems  ...  A contractionand cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of 2A from A.  ...  Acknowledgements We thank Giovanna Corsi, Per Lindström, Paolo Maffezioli, Per Martin-Löf, Grisha Mints, Peter Pagin, Stephen Read, Gabriel Sandu, Sergei Soloviev, and Göran Sundholm for discussions and  ...
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