This movement won't cause Brian to get a theta-role from the verb want because 1) want only assigns a proposition role to its complement, and that role is already assigned to the clause Brain to leave. 2) Brian in such cases moved to the specifier position of AgrOP rather than directly into the complement position of want. Carnie (2007, p. 411) Carnie, Andrew (2007). Syntax: A Generative Introduction (2 ed.). Massachusetts, USA: Blackwell Publishing. ISBN978-1-4051-3384-5.
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Such selection (theta-role assignment) is strictly local and only occurs between sister nodes. A lexical head assigns a theta-role to its complement, and the subject of this XP can receive a theta-role from the X' including the head and its complement. The argument in subject position is called external argument, and the one in object position is called internal argument.(Williams 1981) Adjuncts don't receive theta-roles so are not restricted by theta-criterion. Williams, Edwin (1981). "Argument Structure and Morphology". The Linguistic Review. 1 (1): 84–114. doi:10.1515/tlir.1981.1.1.81. S2CID170491269.
Such selection (theta-role assignment) is strictly local and only occurs between sister nodes. A lexical head assigns a theta-role to its complement, and the subject of this XP can receive a theta-role from the X' including the head and its complement. The argument in subject position is called external argument, and the one in object position is called internal argument.(Williams 1981) Adjuncts don't receive theta-roles so are not restricted by theta-criterion. Williams, Edwin (1981). "Argument Structure and Morphology". The Linguistic Review. 1 (1): 84–114. doi:10.1515/tlir.1981.1.1.81. S2CID170491269.