Extending ILM with an operator for $\Sigma_1$-ness
Evan Goris
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.)
Keywords: Interpolation, Interpretability logic