PP200317: Goris, Evan (2003) Extending ILM with an operator for $\Sigma_1$ness. [Report]

Text (Full Text (PDF))
PP200317.text.pdf Download (581kB)  Preview 

Text (Full Text (PS))
PP200317.text.ps.gz Download (419kB) 

Text (Abstract)
PP200317.abstract.txt Download (1kB) 
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 
Actions (login required)
View Item 