Frame semantics for negation on the Australian Plan accommodates many different negations, but it falls short on accommodating subminimal negation when the language contains conjunction and disjunction. In this paper, I will present a multi-relational frame semantics –multi-incompatibility frame semantics– that can accommodate subminimal negation. I will first argue that multi-incompatibility frames are in accordance with the philosophical motivations behind negation on the Australian Plan, namely its modal and exclusion-expressing nature. Then, I will prove the soundness and completeness results of a subminimal logic that consists of the multi-incompatibility semantics and a proof system with operational rules that characterize subminimal negation, conjunction and disjunction. Lastly, I will prove some key correspondence theorems that relate frame conditions to certain principles that are associated with stronger negations, which will give rise to a new kite of negations that includes subminimal negation.

澳大利亚计划中的否定框架语义可以容纳许多不同的否定，但是当语言包含连词和析取词时，它就不能容纳最小的否定。在本文中，我将提出一种多关系框架语义——多不兼容框架语义——它可以适应最小否定。我将首先论证多重不相容框架符合否定澳大利亚计划背后的哲学动机，即其模态和排斥表达的性质。然后，我将证明由多重不相容语义和具有操作规则的证明系统组成的次极小逻辑的健全性和完整性结果，该操作规则表征极小否定、合取和析取。最后，