PP200317: 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 

Report Nr:  PP200317 
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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