According to the conjunction rule, the probability of A and B cannot exceed the probability of either single event. This rule reads and in terms of the logical operator ∧, interpreting A and B as an intersection of two events. As linguists have long argued, in natural language "and" can convey a wide range of relationships between conjuncts such as temporal order ("I went to the store and bought some whisky"), causal relationships ("Smile and the world smiles with you"), and can indicate a collection of sets rather than their intersection (as in "He invited friends and colleagues to the party"). When "and" is used in word problems researching the conjunction fallacy, the conjunction rule, which assumes the logical operator ∧, therefore cannot be mechanically invoked as a norm. Across several studies, we used different methods of probing people's understanding of and-conjunctions, and found evidence that many of those respondents who violated the conjunction rule in their probability or frequency judgments inferred a meaning of and that differs from the logical operator ∧. We argue that these findings have implications for whether judgments involving ambiguous and-conjunctions that violate the conjunction rule should be considered manifestations of fallacious reasoning or of reasonable pragmatic and semantic inferences.