Partial Up and Down

This paper presents logics for reasoning about extension and reduction of partial information states. This enterprise amounts to non-persistent variations of certain constructive logics, in particular the so-called logic of constructible falsity of Nelson 1949]. We provide simple semantics, sequential calculi, completeness and decidability proofs. The most simple logical means for knowledge representation is the semantic concept of partial truth-assignment. Propositions with a deenite truth-value reeect the knowledge of a chosen agent. Propositions which are mapped to 1 are the things that the agents knows to be true, while propositions which have value 0 cover the information that the agents knows to be false. Propositions whose truth-values are left underspeciied denote the agent's ignorance. In this paper we develop dynamic extensions over these simple static representations, that is formalisms which provide logical means for reasoning about changing partial information states. We will follow Van Benthem and De Rijke's style of dynamic modal logic van Benthem 1991] de Rijke 1992], where such formalisms are deened on the basis of total information states. We will focus on two kinds of changes: enrichment and reduction. These kinds of manipulations of states can easily be deened using a structural extension order which evolves naturally from the deenition of partiality. Given the static meaning '] ] of a proposition ' , i.e. the partial states which support this proposition, the dynamic meaning '] ] dy is induced by the extension order: fhs; ti j s t & t 2 '] ]g. It represents a relational description of what happens to a state s when it is extended with the information '. In an analogous way we specify the negative dynamic meaning '] ] ? dy of ', that is, the ways a situation s can shrink when the information ' has been removed from it: fhs; ti j t s & t 6 2 '] ]g. These two dynamic denotations are the basic relations for dynamic modal reasoning over