We study the topic of realization from classical justification logics in the context of the recently introduced Gödel justification logics. We show that the standard Gödel modal logics of Caicedo and Rodriguez are not realized by the Gödel justification logics and moreover, we study possible extensions of the Gödel justification logics, which are strong enough to realize the standard Gödel modal logics. On the other hand, we study the fragments of the standard Gödel modal logics, which are realized by the usual Gödel justification logics. We prove the corresponding realization theorem by using Fitting’s merging of realizations as well as appropriate hypersequent calculi on the modal side, adapting the work by Metcalfe and Olivetti. For these hypersequent calculi, we also show a cut-elimination theorem. We provide natural semantical characterizations for all of these newly introduced logics.

中文翻译：

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Gödel证明逻辑和实现

我们在最近介绍的Gödel证明逻辑的背景下研究经典证明逻辑中的实现主题。我们证明了Caicedo和Rodriguez的标准Gödel模态逻辑不是由Gödel证明逻辑实现的，此外，我们研究了Gödel证理逻辑的可能扩展，这些扩展足以实现标准Gödel模态逻辑。另一方面，我们研究标准Gödel模态逻辑的片段，这些片段是通过普通的Gödel证明逻辑来实现的。我们使用Fitting的实现合并来证明相应的实现定理以及模态方面的适当的继发性结石，改编了Metcalfe和Olivetti的工作。对于这些继发性结石，我们还显示了切消定理。我们为所有这些新引入的逻辑提供自然的语义特征。