Minimality Condition

The Minimality Condition ensures that there can be no ambiguity of government. The 'absolute' form of the Minimality Condition states that a projection of an intermediate head serves as a barrier for government by another proper governor; it is defined as follows in Chomsky (1986b):

(i) gamma is a barrier for beta if gamma is a projection or the immediate projection of delta, a zero-level category distinct from beta

The choice between 'a projection' and 'the immediate projection' depends on whether a specifier should be governed from outside or not. Thus, in (ii)

(ii)	   X'' /|      spec X'            | X

only X' is a barrier when 'the immediate projection' is chosen in (i), leaving the specifier position open to government from outside, while both X' and X'' are barriers for government when 'a projection' is chosen in (i). The Minimality Condition is intended to exclude ambiguity of government, meaning that barriers created by (i) are barriers for government only, not for movement. In other words, Minimality does not play a role in Bounding theory, while it plays a crucial role in determining whether or not the ECP is violated. Consider for instance (iii).

(iii) how did John announce [NP a plan [CP t2 to [ t1 fix the car t]]]

Here t2 cannot be antecedent-governed by how, resulting in an ECP violation, because the projection of the N0-head plan serves as a Minimality barrier. In Rizzi (1990) the concept of Minimality is relativized, so that government is only blocked by an intermediate governor of the same kind (rather than by any intermediate head, as in (i)). This means that government by a constituent in an A' specifier position will be blocked if there is an intermediate A' specifier, government by a constituent in an A specifier position will be blocked if there is an intermediate A specifier, while head-government will be blocked by an intervening head.

Link
Utrecht Lexicon of Linguistics