Interpolation in Extensions of First-Order Logic

We prove a generalization of Maehara\u2019s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig\u2019s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders

Interpolation in extensions of first-order logic

We prove a generalization of Maehara's lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig's interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.Comment: In this up-dated version of the paper a more general notion of singular geometric theory is provided allowing the extension of our interpolation results to further fundamental mathematical theorie

Towards Automated Reasoning in Herbrand Structures

Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally exhibit the induction scheme, thus providing a congenial, computationally-oriented framework for formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof system for them incomplete. Furthermore, the fact that they are not compact poses yet another prooftheoretic challenge. This paper offers several layers for coping with the inherent incompleteness and non-compactness of these logics. First, two types of infinitary proof system are introduced—one of infinite width and one of infinite height—which manipulate infinite sequents and are sound and complete for the intended semantics. The restriction of these systems to finite sequents induces a completeness result for finite entailments. Then, in search of effectiveness, two finite approximations of these systems are presented and explored. Interestingly, the approximation of the infinite-width system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the infinite-height system

Interpolation via translations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Godel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama-Ono style translation with features of a Kolmogorov-Gentzen-Godel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. (c) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim555515534Fundagdo para a Ciencia e a Tecnologia (FCT)EU FEDER via SQIG (Security and Quantum Information Group) at Instituto de TelecomunicacoesQuantLog [POCTI/MAT/55796/2004]QSec [PTDC/EIA/67661/2006]KLog [PTDC/MAT/68723/2006]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)QuantLog [POCTI/MAT/55796/2004]QSec [PTDC/EIA/67661/2006]KLog [PTDC/MAT/68723/2006]FAPESP [2004/14107-2