论文标题
具有限制性实现的可兑现性和解释性逻辑
Provability and interpretability logics with restricted realizations
论文作者
论文摘要
理论t的可预订性逻辑是一组模态公式，在任何算术中实现下，t在t中都可以证明。我们通过要求算术实现来自指定的$γ$来稍微修改这个概念。我们对可解释性逻辑进行了类似的修改。这是2012年的一篇论文。我们首先研究了具有限制性实现的可宝律逻辑，并表明，对于理论T和限制的各种自然候选者，$γ$中的每个句子都在t中有很好的理解（meta） - t中的数学内容，结果是线性框架的逻辑。但是，对于理论原始递归算术（PRA），我们定义了一个片段，通过利用PRA和I $σ_1$之间的经过良好研究的关系，从而产生更有趣的可预订性逻辑。然后，我们研究可解释性逻辑，获得了IL（PRA）的一些上限，其表征仍然是可解释性逻辑的主要开放问题。同样，该上限与线性框架相对紧密。该技术还用于产生IL（PRA）$ \ subset $ ilm的非平凡结果。
The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $Γ$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $Γ$, where each sentence in $Γ$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and I$Σ_1$. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) $\subset$ ILM.
