ANALYTIC CUT AND INTERPOLATION FOR BI-INTUITIONISTIC LOGIC

2016 ◽  
Vol 10 (2) ◽  
pp. 259-283 ◽  
Author(s):  
TOMASZ KOWALSKI ◽  
HIROAKIRA ONO
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Property ◽  
Separation Property ◽  
Variable Separation ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Sequent Systems ◽  
Subformula Property

AbstractWe prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Halldén completeness follows.

Representability and amalgamation for Heyting polyadic algebras

2011 ◽  
Vol 48 (4) ◽  
pp. 509-539 ◽  
Author(s):  
Tarek Ahmed
Keyword(s):  
Intuitionistic Logic ◽  
Heyting Algebra ◽  
Interpolation Property ◽  
Amalgamation Property ◽  
Infinite Dimension ◽  
Locally Finite ◽  
Craig Interpolation ◽  
Polyadic Algebras ◽  
Craig Interpolation Property

We prove that every (not necessarily locally finite) polyadic Heyting algebra of infinite dimension is representable in some concrete sense. We also show that this class has the super amalgamation property. As a byproduct we infer that a certain infinitary extension of predicate intuitionistic logic, or equivalently, the intuitionistic fragment of Keisler’s infinitary logics, is complete and enjoys the Craig interpolation property.

scholarly journals Von Wright’s truth-logic and around

2013 ◽  
Vol 19 ◽  
pp. 39-50
Author(s):  
А.С. Карпенко
Keyword(s):  
Modal Logic ◽  
Interpolation Property ◽  
Tense Logic ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Truth Logic

In this paper von Wright’s truth-logic T__ is considered. It seems that it is a De Morgan four-valued logic DM4 (or Belnap’s four-valued logic) with endomorphism e2. In connection with this many other issues are discussed: twin truth operators, a truth-logic with endomorphism g (or logic Tr), the lattice of extensions of DM4, modal logic V2, Craig interpolation property, von Wright–Segerberg’s tense logic W, and so on.

scholarly journals Interpolation in Normal Extensions of the Brouwer Logic

2016 ◽  
Vol 45 (3/4) ◽  
Author(s):  
Zofia Kostrzycka
Keyword(s):  
Special Kind ◽  
Interpolation Property ◽  
Craig Interpolation ◽  
Craig Interpolation Property

The Craig interpolation property and interpolation property for deducibility are considered for special kind of normal extensions of the Brouwer logic.

scholarly journals Interpolation and Amalgamation for Arrays with MaxDiff

2021 ◽  
pp. 268-288
Author(s):  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Deepak Kapur
Keyword(s):  
Semantic Analysis ◽  
Index Theory ◽  
Interpolation Property ◽  
Hierarchical Approach ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Interpolation Algorithms ◽  
Free Level

AbstractIn this paper, the theory of McCarthy’s extensional arrays enriched with a maxdiff operation (this operation returns the biggest index where two given arrays differ) is proposed. It is known from the literature that a diff operation is required for the theory of arrays in order to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom). Our maxdiff operation significantly increases the level of expressivity; however, obtaining interpolation results for the resulting theory becomes a surprisingly hard task. We obtain such results via a thorough semantic analysis of the models of the theory and of their amalgamation properties. The results are modular with respect to the index theory and it is shown how to convert them into concrete interpolation algorithms via a hierarchical approach.

scholarly journals A Remark on Maksimova's Variable Separation Property in Super-Bi-Intuitionistic Logics

2017 ◽  
Vol 14 (1) ◽  
Author(s):  
Guillermo Badia
Keyword(s):  
Intuitionistic Logic ◽  
Separation Property ◽  
Algebraic Characterization ◽  
Variable Separation ◽  
Intuitionistic Logics

We provide a sucient frame-theoretic condition for a super bi-intuitionistic logic to have Maksimova's variable separation property. We conclude that bi-intuitionistic logic enjoys the property. Furthermore, we offer an algebraic characterization of the super-bi-intuitionistic logics with Maksimova's property.

scholarly journals Algebraic Characterization of the Local Craig Interpolation Property

2018 ◽  
Vol 47 (1) ◽  
Author(s):  
Zalán Gyenis
Keyword(s):  
Algebraic Logic ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Algebraic Characterization ◽  
Local Version ◽  
Ongoing Research ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Superamalgamation Property

The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka– Németi and Sain. 

scholarly journals Failure of Interpolation in Constant Domain Intuitionistic Logic

2013 ◽  
Vol 78 (3) ◽  
pp. 937-950 ◽  
Author(s):  
Grigori Mints ◽  
Grigory Olkhovikov ◽  
Alasdair Urquhart
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Constant Domain ◽  
Constant Domains

AbstractThis paper shows that the interpolation theorem fails in the intuitionistic logic of constant domains. This result refutes two previously published claims that the interpolation property holds.

Intuitionistic logic freed of all metarules

2007 ◽  
Vol 72 (4) ◽  
pp. 1204-1218 ◽  
Author(s):  
Giovanna Corsi ◽  
Gabriele Tassi
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Proof Search ◽  
Subformula Property

AbstractIn this paper we present two calculi for intuitionistic logic. The first one. IG, is characterized by the fact that every proof-search terminates and termination is reached without jeopardizing the subformula property. As to the second one, SIC, proof-search terminates, the subformula property is preserved and moreover proof-search is performed without any recourse to metarules, in particular there is no need to back-track. As a consequence, proof-search in the calculus SIC is accomplished by a single tree as in classical logic.

scholarly journals Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

Synthese ◽  
2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
General Introduction ◽  
Main Method ◽  
Philosophical Importance ◽  
Subformula Property

AbstractThis paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformula property.

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