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### Proof Theory and Algebra in Substructural Logics [chapter]

Kazushige Terui
2011 Lecture Notes in Computer Science
Proving cut elimination for such a generalized sequent calculus is closely related to proving that a variety is closed under the corresponding generalized completions.  ...  Finding a new variant of sequent calculus (such as hypersequent calculus) amounts to finding a new variant of MacNeille completions (generalizations of Dedekind's completion Q → R).  ...  Proving cut elimination for such a generalized sequent calculus is closely related to proving that a variety is closed under the corresponding generalized completions.  ...

### Equivalential Gentzen systems and the reduced matrices of the Gentzen systems associated with finitely valued logics

Àngel J. Gil, Jordi Rebagliato
2006 Reports on Mathematical Logic
Font for his valuable comments on a draft of the paper. Also, we would like to thank an anonymous referee whose comments were really helpful.  ...  exactly the consequence relation determined by the sequent calculus.  ...  Let L be a finite L-algebra. A VL-Gentzen system is a Gentzen system determined by a VL-sequent calculus.  ...

### The categorical equivalence between disjunctive sequent calculi and algebraic L-domains [article]

Longchun Wang, Qingguo Li
2020 arXiv   pre-print
This paper establishes a purely syntactic representation for the category of algebraic L-domains with Scott-continuous functions as morphisms.  ...  The central tool used here is the notion of logical states, which builds a bridge between disjunctive sequent calculi and algebraic L-domains.  ...  We only illustrate this for the rule (Lwk): Proposition 3 . 5 . 35 For the disjunctive sequent calculus (L(P D ), ⊢ D ) associated with a given algebraic L-domain (D, ≤), we have (1) a formula ϕ is a  ...

### Cut-free display calculi for relation algebras [chapter]

Rajeev Goré
1997 Lecture Notes in Computer Science
We extend Belnap's Display Logic to give a cut-free Gentzen-style calculus for relation algebras.  ...  It also appears to be the rst purely propositional Gentzen-style calculus for relation algebras.  ...  Acknowledgments: Thanks to Lev Gordeev, Wolfgang Heinle, Greg Restall and Timothy Surendonk for many helpful discussions.  ...

### Program Logics of Renominative Level with the Composition of Predicate Complement

Mykola S. Nikitchenko, Oksana Shkilniak, Stepan S. Shkilniak
2019 International Conference on Information and Communication Technologies in Education, Research, and Industrial Applications
We introduce a special consequence relation for this logic, construct a sequent calculus, and prove its soundness and completeness.  ...  One of the methods involves extension of program algebras with the composition of predicate complement. This permits to modify rules of the logic making them sound.  ...  We construct a sequent calculus C QCR for relation |= IR  . The main objects of this calculus are sequents. Here we consider only the case with finite sequents.  ...

### From Kripke Models to Algebraic Counter-Valuations [chapter]

Sara Negri, Jan von Plato
1998 Lecture Notes in Computer Science
Starting with a derivation in the refutation calculus CRIP of Pinto and Dyckhoff, we give a constructive algebraic method for determining the values of formulas of intuitionistic propositional logic in  ...  The values of compound formulas are computed pointwise from the values on atoms, in contrast to the non-local determination of forcing relations in a Kripke model based on classical reasoning.  ...  In Pinto and Dyckhoff (1995) , a related refutation calculus CRIP was given, for showing underivability of a sequent Γ ⇒ ∆.  ...

### Lambek-Grishin Calculus: Focusing, Display and Full Polarization [article]

Giuseppe Greco, Valentin D. Richard, Michael Moortgat, Apostolos Tzimoulis
2020 arXiv   pre-print
LG-algebras: this amounts to a semantic proof of the so-called completeness of focusing, given that the standard (display) sequent calculus for Lambek-Grishin logic is complete w.r.t.  ...  We introduce a novel focused display calculus fD.LG and a fully polarized algebraic semantics FP.LG for Lambek-Grishin logic by generalizing the theory of multi-type calculi and their algebraic semantics  ...  maps sending an element to its copy in the appropriate sub-algebra of A G , and finally, for each A, B in the appropriate sub-algebra of A G , for each weakening relation R in A G , and for each binary  ...

### Decision methods for linearly ordered Heyting algebras

Roy Dyckhoff, Sara Negri
2005 Archive for Mathematical Logic
The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented  ...  , inspired by related ideas in Gödel-Dummett logic.  ...  (a Fellowship for the first author).  ...

### Page 3271 of Mathematical Reviews Vol. , Issue 85h [page]

1985 Mathematical Reviews
See a paper by D. J. Foulis and C. H. Randall {ibid. 14 (1984), no. 1, 65-81] for an opposing view. Jeffrey Bub (London, Ont.) Maddux, Roger (1-IA) 85h:03067 A sequent calculus for relation algebras.  ...  A. Tarski provided a double foundation for the calculus of rela- tions. His semantic approach led to the universal algebraic concept of a relation algebra.  ...

### Page 7344 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews
It is then shown (Theorem 3.1) that consequence relations form a category MLS (for Multi-Lingual Sequents) whose objects (called coherent sequent calculi) are algebras L endowed with a consequence relation  ...  The strong proximity lattice associated to a coherent sequent calculus is ob- tained by factoring out logical equivalence and taking implication for the approximation relation.  ...

### Multi Lingual Sequent Calculus and Coherent Spaces

Achim Jung, Mathias Kegelmann, M. Andrew Moshier
1997 Electronical Notes in Theoretical Computer Science
We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system.  ...  We propose a di erent cut rule which addresses this problem. The new cut rule can be used as a basis for composition in a suitable category of logical systems.  ...  The previous three lemmas together entail the following result: Theorem 8.7 A binary relation on nite sets from a term algebra L over generators B is a coherent sequent calculus if and only if j B B has  ...

### Unified correspondence and proof theory for strict implication

Minghui Ma, Zhiguang Zhao
2016 Journal of Logic and Computation
Lambek calculus (BDFNL).  ...  Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.  ...  Alessandra Palmigiano (Delft University of Technology) for her comments on drafts of this paper, and the reviewers for their helpful comments on the submitted version.  ...

### Sequent Calculi for Semi-De Morgan and De Morgan Algebras [article]

Minghui Ma, Fei Liang
2016 arXiv   pre-print
A contraction-free and cut-free sequent calculus G3SDM for semi-De Morgan algebras, and a structural-rule-free and single-succedent sequent calculus G3DM for De Morgan algebras are developed.  ...  G3DM is embedded into a sequent calculus for classical propositional logic. G3SDM is embedded into the sequent calculus G3ip for intuitionistic propositional logic.  ...  A Sequent Calculus for De Morgan Algebras In this section, we shall present a single-conclusion sequent calculus G3DM for De Morgan algebras.  ...

### Page 3921 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews
To provide a fitting for the associativity of the concatenation 96¢:03107 in the sequent calculus the following should hold in channel the- ory: (Su)(s-4u and ux) <=> (3v)(s4x andt4v).  ...  The central part of the paper gives a sequent calculus which is sound and complete with respect to this algebra. Moreover, it has the cut property and is also decidable.  ...

### A Mechanised Proof System for Relation Algebra Using Display Logic [chapter]

Jeremy E. Dawson, Rajeev Goré
1998 Lecture Notes in Computer Science
Our implementation is the rst mechanisation of any display calculus, but also provides a useful interactive proof assistant for relation algebra.  ...  We describe an implementation of the Display Logic calculus for relation algebra as an Isabelle theory.  ...  The second author is supported by the Australian Research Council via a Queen Elizabeth II Fellowship.  ...
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