An aposiopesis is a rhetorical tool that suggestively leaves a line or sentence incomplete. It can be used to suggest fear of discussing something taboo or sadness while speaking of tragedies. Confusion, cowardice, disgust, and horror are other emotions that aposiopesises often imply. One's imagination is often left working to fill out the rest of the unspoken or unwritten phrase. In Greek, the language of its etymologic origin, it means "becoming silent".

Some examples:

Another example is a line from Shakespeare's King Lear. However, there is a bit of controversy surrounding it. The word anacoluthon denotes a very similar rhetorical usage as aposiopesis. In short, what the speaker/writer intended to say/write isn't obvious with an anacoluthon, while the meaning and omitted words are obvious with an aposiopesis. The line in question is

"I will have such revenges on you both That all the world shall--I will do such things,"
in Act 2.4, concerning Lear's vengeful attitude towards his traitorous daughters. Personally, I think it falls under the aposiopesis category. It seems to me Lear was going to end "That all the world shall" with "know". However, the vastly more qualified professors that wrote the footnotes in "The Oxford Companion to the English Language" disagree. They classify that line as an anacoluthon.

Thanks to: m_turner, Starke. Sources: Murfin, Ross. The Bedford Glossary of Critical and Literary Terms. Boston, MA: Beford, 1998.