Homogeneity inferences arise whenever an assertion implies a universal positive, while its denial implies a universal negative. In the lexicon of natural language, there are no simple words that express a particular negative quantifier or connective, such as hypothetical *nall and *nand. I present an account of homogeneity inferences that is intended to also shed light on cross-linguistic lexicalization data. The account is based on two assumptions which together constrain the behavior of negation: rejection is weak, and vacuous configurations may be omitted. When both assumptions are enforced, we obtain a bilateral state-based logic in which all, some, and no may be expressed, but the negation of universal collapses on no. Similarly, under the same assumptions, for the connectives: and, or, and nor are definable but not the negation of conjunction. In the Logic of the Lexicon, under the stated assumptions, particular negatives disappear. This logic is the basis for an account of homogeneity inferences based on illocutionary force. In the conclusion, I sketch the consequences of adopting such a logic for a theory of cognitive simplicity that contributes to explaining lexicalization data.