  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.
