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A non-compact deduction rule for the logic of provability and its algebraic models | | Yoshihito Tanaka
; | | Date: |
12 Feb 2020 | | Abstract: | In this paper, we introduce a proof system with a non-compact deduction rule,
that is, a deduction rule with countably many premises, to axiomatize the logic
$mathbf{GL}$ of provability, and show its Kripke completeness in an algebraic
manner. As $mathbf{GL}$ is not canonical, a standard proof of Kripke
completeness for $mathbf{GL}$ is given by a Kripke model which is obtained by
changing the binary relation of the canonical model, while our proof is given
by a submodel of the canonical model of $mathbf{GL}$ which is obtained by
making use of an infinitary extension of the J’{o}nsson-Tarski representation.
We also show the three classes of algebras defined by $Box xleqBoxBox x$
and one of the following three conditions,
$igwedge_{ninomega}Diamond^{n}1=0$, the non-compact deduction rule and the
L"{o}b formula, are mutually different, while all of them define
$mathbf{GL}$. | | Source: | arXiv, 2002.4782 | | Services: | Forum | Review | PDF | Favorites | |
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