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SYNTAX: an A-position to which a theta-role is assigned. EXAMPLE: in John buys the apples both the arguments John and the apples are in a theta-position being theta-marked as Agent and Theme. But the verb seems in (i) does not assign a theta-role to its grammatical subject John. Therefore, the matrix subject position, although an A-position, is not a theta-position.

(i) John seems [ti to have left]
The positions in (i) containing John and its trace ti are both A-positions, but only the position of the trace is a theta-position. An A-position to which no theta-role is assigned (such as the one in (i) which is taken by John) is called a theta-bar position (theta'-position). Another name for theta-bar position is non-theta position.
LIT. Chomsky, N. (1986a)
Chomsky, N. (1981)