Decision-making about economy-environment systems is often characterized by deep uncertainties. We provide an axiomatic foundation of preferences over lotteries with known payoffs over known states of nature and unknown probabilities of these outcomes (“Knightian uncertainty"). We elaborate the fundamental idea that preferences over Knightian lotteries can be represented by an entropy function (sensu Lieb and Yngvason 1999) of these lotteries. Based on nine axioms on the preference relation and three assumptions on the set of lotteries, we show that there uniquely (up to linear-affine transformations) exists an additive and extensive real-valued function (\entropy function") that represents uncertainty preferences. It represents non-satiation and (constant) uncertainty aversion. As a concrete functional form, we propose a one-parameter function based on Rényi's (1961) generalized entropy. We show that the parameter captures the degree of uncertainty aversion. We illustrate our preference function with a simple decision problem and relate it to other decision rules under Knightian uncertainty (maximin, maximax, Hurwicz, Laplacian expected utility, minimum regret).