Craig's interpolation theorem for the intuitionistic logic and its extensions—A semantical approach

Studia Logica ◽  
1986 ◽  
Vol 45 (1) ◽  
pp. 19-33 ◽  
Author(s):  
Hiroakira Ono
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Theorem ◽  
Craig’S Interpolation Theorem

Interpolation in Conditional Equational Logic1

1991 ◽  
Vol 15 (1) ◽  
pp. 80-85
Author(s):  
P.H. Rodenburg
Keyword(s):  
Interpolation Theorem ◽  
Equational Logic ◽  
Craig’S Interpolation Theorem ◽  
Natural Formulation

In a natural formulation, Craig’s interpolation theorem is shown to hold for conditional equational logic.

Harmonious logic: Craig’s interpolation theorem and its descendants

Synthese ◽  
2008 ◽  
Vol 164 (3) ◽  
pp. 341-357 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Interpolation Theorem ◽  
Craig’S Interpolation Theorem

scholarly journals On weak and strong interpolation in algebraic logics

2006 ◽  
Vol 71 (1) ◽  
pp. 104-118 ◽  
Author(s):  
Gábor Sági ◽  
Saharon Shelah
Keyword(s):  
Algebraic Logic ◽  
Amalgamation Property ◽  
Weak Form ◽  
Interpolation Theorem ◽  
Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Finite Dimensional ◽  
Craig’S Interpolation Theorem ◽  
Superamalgamation Property

AbstractWe show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi [12].

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.

Interpolation theorems for Lk,k2+

1978 ◽  
Vol 43 (3) ◽  
pp. 535-549 ◽  
Author(s):  
Ruggero Ferro
Keyword(s):  
Natural Deduction ◽  
Interpolation Theorem ◽  
Positive Answer ◽  
Strong Limit ◽  
First Order ◽  
Limit Cardinal ◽  
Saturated Models ◽  
Order Language ◽  
Craig’S Interpolation Theorem

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].

scholarly journals Model Theory in a Paraconsistent Environment

2021 ◽  
Vol 27 (2) ◽  
pp. 216-216
Author(s):  
Bruno Costa Coscarelli
Keyword(s):  
Model Theory ◽  
Paraconsistent Logic ◽  
Interpolation Theorem ◽  
Point Of View ◽  
The Third ◽  
Reasoning System ◽  
Paraconsistent Reasoning ◽  
E Mail ◽  
Craig’S Interpolation Theorem ◽  
Completeness Theorems

AbstractThe purpose of this thesis is to develop a paraconsistent Model Theory. The basis for such a theory was launched by Walter Carnielli, Marcelo Esteban Coniglio, Rodrigo Podiack, and Tarcísio Rodrigues in the article ‘On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency’ [The Review of Symbolic Logic, vol. 7 (2014)].Naturally, a complete theory cannot be fully developed in a single work. Indeed, the goal of this work is to show that a paraconsistent Model Theory is a sound and worthy possibility. The pursuit of this goal is divided in three tasks: The first one is to give the theory a philosophical meaning. The second one is to transpose as many results from the classical theory to the new one as possible. The third one is to show an application of the theory to practical science.The response to the first task is a Paraconsistent Reasoning System. The start point is that paraconsistency is an epistemological concept. The pursuit of a deeper understanding of the phenomenon of paraconsistency from this point of view leads to a reasoning system based on the Logics of Formal Inconsistency. Models are regarded as states of knowledge and the concept of isomorphism is reformulated so as to give raise to a new concept that preserves a portion of the whole knowledge of each state. Based on this, a notion of refinement is created which may occur from inside or from outside the state.In order to respond to the second task, two important classical results, namely the Omitting Types Theorem and Craig’s Interpolation Theorem are shown to hold in the new system and it is also shown that, if classical results in general are to hold in a paraconsistent system, then such a system should be in essence how it was developed here.Finally, the response to the third task is a proposal of what a Paraconsistent Logic Programming may be. For that, the basis for a paraconsistent PROLOG is settled in the light of the ideas developed so far.Abstract prepared by Bruno Costa Coscarelli.E-mail: [email protected]: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

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