Abstract
The present paper examines the grammatical function of the English preposition under with specific focus on its cognitive foundations. Under has the grammatical function as a head of a protasis of the related main clause (e.g., Under the agreement, agricultural prices would be frozen.). In order to examine this grammatical function of under, the present paper discusses the asymmetrical semantic extensions between over and under, which constitutes an antonymic pair, arguing the following points: first, the grammatical function of under is derived from its spatial sense through the control sense; second, though both over and under have the control sense, they exhibit different types of control; third, the asymmetry along the vertical axis of the perceived world motivates not only asymmetrical semantic extensions but also establishes the grammatical function of under.
