PP-2003-17: Goris, Evan (2003) Extending ILM with an operator for $\Sigma_1$-ness. [Report]
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Abstract
In this paper we formulate a logic $\Sigma$ILM.
This logic extends ILM and contains a new unary modal operator $\Sigma_1$.
The formulas of this logic can be evaluated on Veltman frames.
We show that $\Sigma$ILM is modally sound and complete with respect to a
certain class of Veltman frames.
An arithmetical interpretation of the modal formulas can be obtained by reading
the $\Sigma_1$ operator as formalized $\Sigma_1$-ness in PA
and |> as formalized $\Pi_1$-conservativity between finite extensions of PA.
We show that under this arithmetically interpretation $\Sigma$ILM is sound and
complete.
The main motivation for formulating $\Sigma$ILM at all
is that one counterexample for interpolation in ILM seems to emerge because of
the lack of ILM to express $\Sigma_1$-ness. We show that $\Sigma$ILM does not
have interpolation either.
Our counterexample seems to emerge because of the inability of $\Sigma$ILM to
express $\Sigma$-interpolation.
(A formula A -> B has a $\Sigma_1$-interpolant if there exist some
$\Sigma_1$ formula S such that PA |- A -> S and PA |- S -> B.)
Item Type: | Report |
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Report Nr: | PP-2003-17 |
Series Name: | Prepublication (PP) Series |
Year: | 2003 |
Uncontrolled Keywords: | Interpolation, Interpretability logic |
Subjects: | Logic |
Date Deposited: | 12 Oct 2016 14:36 |
Last Modified: | 12 Oct 2016 14:36 |
URI: | https://eprints.illc.uva.nl/id/eprint/101 |
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