As noted by Samuelson in his introduction of the Strong Independence axiom, essentially the same set of axioms rationalize an Expected Utility representation of preferences over lotteries with (i) a scalar payoff such as money and (ii) vector payoffs such as quantities of different commodities. Assume a two-good setting, where an individual's preferences satisfy the Strong Independence axiom for lotteries paying off quantities of each good separately. This paper identifies the incremental axioms required for the preference relation over lotteries paying off the vector of goods to also satisfy the Strong Independence axiom. The key element of this extension is a Coherence axiom which requires a particular "meshing together" of certainty preferences over commodity bundles and preferences over non-degenerate lotteries for individual goods. The Coherence axiom is shown to have interesting theoretical implications for Allais paradox-like behavior when confronting lotteries over multiple goods.