MoL-2008-12: Intuitionistic Subframe Formulas, NNIL-Formulas and n-universal Models

MoL-2008-12: Yang, Fan (2008) Intuitionistic Subframe Formulas, NNIL-Formulas and n-universal Models. [Report]

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In this thesis, we investigate intuitionistic subframe formulas and NNIL- formulas by using the technique of n-universal models. Intuitionistic sub- frame formulas axiomatize subframe logics which are intermediate logics characterized by a class of frames closed under subframes. Zakharyaschev in- troduced the subframe formulas by using [^,->]-formulas, which contain only ^ and -> as connectives. It then follows that subframe logics are axiomatized by [^,->]-formulas. NNIL-formulas are the formulas that have No Nesting Of Implications to the left. Visser, de Jongh, van Benthem and Renardel de Lavalette proved that NNIL-formulas are exactly the formulas preserved under taking submodels. The topic of this thesis was inspired by N. Bezhanishvili who used the insight that NNIL-formulas are then preserved under subframes as well to introduce subframe formulas in the NNIL-form. It was proved that NNIL-formulas are su±cient to axiomatize subframe logics. This thesis is set up in a way to be able to connect the results on subframe formulas defined by [^,->]-formulas and NNIL-formulas by using n-universal models as a uniform method. Our original intention to throw new light on subframe logics by the use of NNIL-formulas was barely realized, but we do provide new insights on the NNIL-formulas themselves. Chapter 2 gives a background on intuitionistic propositional logic and its Kripke, algebraic and topological semantics. In Chapter 3, we discuss n-universal models U(n) of IPC by giving proofs of known theorems in a uniform manner including a direct and very perspicuous proof of the fact that the n-universal model of IPC is isomorphic to the upper part of the n-Henkin model. This then also gives a method for a new proof (Theorem 3.4.9) of Jankov's theorem on KC. In Chapter 4, we summarize classic and recent results on subframe logics and subframe formulas. In Chapter 5, we investigate properties of the [^,->]-fragment of IPC consisting of [^,->]- formulas only. This chapter is based on the results in Diego, de Bruijn and Hendriks. We redefined the exact model defined by using the n-universal models of IPC and give a uniform treatment of known results. In Chapter 6, we give an algorithm to translate every NNIL-formula to a [^,->]-formula in such a way that they are equivalent on frames. We study subsimulations between models and construct representative models for equivalence classes of rooted generated submodels of U(n) induced by two-way subsimulations. We construct finite n-universal models U(n)^NNIL for NNIL-formulas with n variables by the representative models and prove the related properties. As a consequence, the theorem that formulas pre- served under subsimulations are equivalent to NNIL-formulas becomes a natural corollary of the properties of U(n)^NNIL. Finally, we obtain the subframe logics axiomatized by two-variable NNIL-formulas by observ- ing the structure of U(2)NNIL. Although it is not yet clear how to general- ize the result for the model U(2)NNIL and the subframe logics axiomatized by NNIL(p,q)-formulas to the models U(n)^NNIL for any n 2 ->, this result clearly suggests that the U(n)^NNIL models are a good tool for future work on subframe logics.

Item Type: Report
Report Nr: MoL-2008-12
Series Name: Master of Logic Thesis (MoL) Series
Year: 2008
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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