Operator grammar is a mathematical theory of human language that explains how language carries information. This theory is the culmination of the life work of Zellig Harris, with major publications toward the end of the last century. Operator grammar proposes that each human language is a self-organizing system in which both the syntactic and semantic properties of a word are established purely in relation to other words. Thus, no external system (metalanguage) is required to define the rules of a language. Instead, these rules are learned through exposure to usage and through participation, as is the case with most social behavior. The theory is consistent with the idea that language evolved gradually, with each successive generation introducing new complexity and variation. Operator grammar posits three universal constraints: dependency (certain words depend on the presence of other words to form an utterance), likelihood (some combinations of words and their dependents are more likely than others) and reduction (words in high likelihood combinations can be reduced to shorter forms, and sometimes omitted completely). Together these provide a theory of language information: dependency builds a predicate–argument structure; likelihood creates distinct meanings; reduction allows compact forms for communication. Dependency The fundamental mechanism of operator grammar is the dependency constraint: certain words (operators) require that one or more words (arguments) be present in an utterance. In the sentence John wears boots, the operator wears requires the presence of two arguments, such as John and boots. (This definition of dependency differs from other dependency grammars in which the arguments are said to depend on the operators.) In each language the dependency relation among words gives rise to syntactic categories in which the allowable arguments of an operator are defined in terms of their dependency requirements. Class N contains words (e.g. John, boots) that do not require the presence of other words. Class ON contains the words (e.g. stumble) that require exactly one word of type N. Class ONN contains the words (e.g. wear) that require two words of type N. Class OOO contains the words (e.g. because) that require two words of type O, as in John stumbles because John wears boots. Other classes include OO (e.g. is possible), ONNN (e.g. put), OON (e.g. with), ONO (e.g. know), ONNO (e.g. ask) and ONOO (e.g. attribute). The categories in operator grammar are universal and are defined purely in terms of how words relate to other words, and