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The idea of the centaur does exist in our imagination, and inside our heads, and the name of it in our mouths.
In the Centaur battle, having been crushed by rocks and trunks of trees, he was changed into a bird; or he disappeared into the depths of the earth unharmed.
No existential form suits a judgment such as " A centaur is a fiction," when we do not believe that there is a centaur, or that reality includes a centaur.
Sagittarius, figured later as a Centaur, stood for the Babylonian Mars.
In a correspondence with Mill, Brentano rejoined that the centaur exists in imagination; Bradley says, " inside our heads."