Since the invention of Notation3 Logic, several years have passed in which the theory has been refined and applied in different reasoning engines like Cwm, EYE, and FuXi. But despite these developments, a clear formal definition of Notation3’s semantics is still missing. This does not only form an obstacle for the formal investigation of that logic and its relations to other formalisms, it has also practical consequences: in many cases the interpretations of the same formula differ between reasoning engines. In this paper we tackle one of the main sources of that problem, namely the uncertainty about implicit quantification. This refers to Notation3’s ability to use bound variables for which the universal or existential quantifiers are not explicitly stated, but implicitly assumed. We provide a tool for clarification through the definition of a core logic for Notation3 that only supports explicit quantification. We specify an attribute grammar which maps Notation3 formulas to that logic according to the different interpretations and thereby define the semantics of Notation3. This grammar is then implemented and used to test the impact of the differences between interpretations on practical cases. Our dataset includes Notation3 implementations from former research projects and test cases developed for the reasoner EYE. We find that 31% of these files are understood differently by different reasoners. We further analyse these cases and categorise them in different classes of which we consider one most harmful: if a file is manually written by a user and no specific built-in predicates are used (13% of our critical files), it is unlikely that this user is aware of possible differences. We therefore argue the need to come to an agreement on implicit quantification, and discuss the different possibilities.

自从Notation3 Logic发明以来，经过了数年的时间，对该理论进行了完善并应用于各种推理引擎，例如Cwm，EYE和FuXi。但是，尽管有了这些发展，Notation3语义的明确形式化定义仍然缺失。这不仅构成对该逻辑及其与其他形式主义的关系进行形式研究的障碍，而且还具有实际后果：在许多情况下，推理引擎之间同一公式的解释不同。在本文中，我们解决了该问题的主要来源之一，即隐式量化的不确定性。这是指Notation3使用绑定变量的能力，对于这些变量，没有明确说明通用或存在量词，而是隐式假定了这些变量。我们通过定义Notation3的核心逻辑（仅支持显式量化）来提供澄清工具。我们指定一个属性语法，该语法根据不同的解释将Notation3公式映射到该逻辑，从而定义Notation3的语义。然后实施此语法，并将其用于测试解释之间的差异对实际案例的影响。我们的数据集包括来自以前的研究项目的Notation3实现和为推理机EYE开发的测试用例。我们发现这些文件中有31％被不同的推理者以不同的方式理解。我们会进一步分析这些情况，并将其归类为最有害的类别：如果文件是由用户手动编写的，并且未使用特定的内置谓词（占关键文件的13％），该用户不太可能意识到可能存在的差异。因此，我们认为有必要就隐式量化达成协议，并讨论不同的可能性。