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Provability Interpretation of Propositional and Modal Logics. (arXiv:1704.07677v1 [math.LO])
来源于:arXiv
In 1933, G\"{o}del introduced a provability interpretation of the
propositional intuitionistic logic to establish a formalization for the BHK
interpretation. He used the modal system, $\mathbf{S4}$, as a formalization of
the intuitive concept of provability and then translated $\mathbf{IPC}$ to
$\mathbf{S4}$. His work suggested the problem to find a concrete provability
interpretation of the modal logic $\mathbf{S4}$. In this paper, we will try to
answer this problem. In fact, we will generalize Solovay's provability
interpretation of the modal logic $\mathbf{GL}$ to capture other modal logics
such as $\mathbf{K4}$, $\mathbf{KD4}$ and $\mathbf{S4}$. Then we will use these
results to find a formalization for the BHK interpretation and we will show
that with different interpretations of the BHK interpretation, we can capture
some of the propositional logics such as Intuitionistic logic, minimal logic
and Visser-Ruitenburg's basic logic. Moreover, we will show that there is no
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