We explore a kind of first-order predicate logic with intended semantics in the reals. Compared to other approaches in the literature, we work predominantly in the multiplicative reals [0,\infty], showing they support three generations of connectives, that we call non-linear, linear additive, and linear multiplicative. Means and harmonic means emerge as natural candidates for bounded existential and universal quantifiers, and in fact we see they behave as expected in relation to the other logical connectives. We explain this fact through the well-known fact that min/max and arithmetic mean/harmonic mean sit at opposite ends of a spectrum, that of p-means. We give syntax and semantics for this quantitative predicate logic, and as example applications, we show how softmax is the quantitative semantics of argmax, and R\'enyi entropy/Hill numbers are additive/multiplicative semantics of the same formula. Indeed, the additive reals also fit into the story by exploiting the Napierian duality -log \dashv 1/exp, which highlights a formal distinction between 'additive' and 'multiplicative' quantities. Finally, we describe two attempts at a categorical semantics via enriched hyperdoctrines. We discuss why hyperdoctrines are in fact probably inadequate for this kind of logic.

翻译：我们探索一种具有实数预期语义的一阶谓词逻辑。相较于文献中的其他方法，我们主要在乘法实数[0,∞]中展开研究，证明其支持三代连接词，分别称为非线性、线性加性与线性乘性连接词。均值与调和均值作为有界存在量词与全称量词的自然候选者出现，事实上我们观察到它们与其他逻辑连接词的交互行为符合预期。我们通过一个众所周知的事实解释这一现象：最小/最大值与算术平均/调和平均分别位于p-均值谱系的两端。我们为此定量谓词逻辑提供了语法与语义规范，并通过示例应用展示了softmax如何作为argmax的定量语义，以及Rényi熵/希尔数如何成为同一公式的加性/乘性语义。实际上，加法实数亦可通过利用奈皮尔对偶性-log ⊣ 1/exp融入该理论框架，这凸显了“加性”与“乘性”量之间的形式区别。最后，我们通过富超基尝试构建两种范畴语义，并论述了为何超基理论可能不适用于此类逻辑。