# PP-2003-17: Extending ILM with an operator for \$\Sigma_1\$-ness

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 |&gt; 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 -&gt; B has a \$\Sigma_1\$-interpolant if there exist some
\$\Sigma_1\$ formula S such that PA |- A -&gt; S and PA |- S -&gt; B.)

Item Type: Report PP-2003-17 Prepublication (PP) Series 2003 Interpolation, Interpretability logic Logic 12 Oct 2016 14:36 12 Oct 2016 14:36 https://eprints.illc.uva.nl/id/eprint/101

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