范杰报告:Almost Necessary(“几乎必然”)

发布时间:2014-12-22 来源:本站原创 作者:本站编辑   浏览次数:

题目:Almost Necessary(“几乎必然”)

报告人:范杰(北京大学博士生)

时间:2014年12月24日(周三)晚上7:00

地点:逻辑与智能研究中心教室


摘要:A formula is contingent if it is possibly true and possibly false. A formula is non-contingent if it is not contingent, i.e., if it is necessarily true or necessarily false. In an epistemic setting, `a formula is contingent' means that you are ignorant about its value, whereas `a formula is non-contingent' means that you know whether it is true. Although non-contingency is definable in terms of necessity as above, necessity is not always definable in terms of non-contingency, as studied in the literature. We propose an `almost-definability' schema AD for non-contingency logic, the logic with the non-contingency operator as the only modality, making precise when necessity is definable with non-contingency. Based on AD we propose a notion of bisimulation for non-contingency logic, and characterize non-contingency logic as the (non-contingency) bisimulation invariant fragment of modal logic and of first-order logic. A known pain for non-contingency logic is the absence of axioms characterizing frame properties. This makes it harder to find axiomatizations of non-contingency logic over given frame classes. In particular, no axiomatization over symmetric frames is known, despite the rich results about non-contingency logic  obtained in the literature since the 1960s. We demonstrate that the `almost-definability' schema AD can guide our search for proper axioms for certain frame properties, and help us in defining the canonical models. Following this idea, as the main result, we give a complete  axiomatization of (uni-modal) non-contingency logic over symmetric frames.We will also extend the axiomatization result to the multi-modal case.