  The paper proposes and studies temporal logics for attributed words, that is,
data words with a (finite) set of (attribute,value)-pairs at each position. It
considers a basic logic which is a semantical fragment of the logic
$LTL^\downarrow_1$ of Demri and Lazic with operators for navigation into the
future and the past. By reduction to the emptiness problem for data automata it
is shown that this basic logic is decidable. Whereas the basic logic only
allows navigation to positions where a fixed data value occurs, extensions are
studied that also allow navigation to positions with different data values.
Besides some undecidable results it is shown that the extension by a certain
UNTIL-operator with an inequality target condition remains decidable.
