This paper argues for Kayne's (1984) Connectedness Condition which is aimed at accounting for the island effect first discovered by Ross (1967). An island is a structure out of which an element cannot escape. A variable which is confined in an island is not related to the operator outside the island. The study of islands, or the boundary condition for structures that allow operator-variable relations, is important, because the displacement builds operator-variable relations and because displacement is a characteristic of human natural language. Displacement is a phenomenon in which the actual position of an element in a sentence is distinct from the position where it is interpreted.The island effect is a long-standing formidable problem that resists a simple and elegant explanation. Linguists have tried hard and proposed various conditions and principles to explain the island effect, such as the Subjacency Condition, the ECP (Empty Category Principle), the CED (Condition on Extraction Domain), barriers, Relativized Minimality, the MLC (Minimal Link Condition), the PIC (Phase Impenetrability Condition), etc. Various factors have been considered;(a) the nature of empty category e (trace t or original) that has been left by movement (of the copy),e. g. , whether e is a sister of a lexical category L, or whether e is properly governed , (b) the structure of islands, e. g. , whether islands contain adjunction structure, or whether the specifier position is occupied, (c) the position of islands, e. g. , whether the island is a sister of L, (d) the manner of movement, e. g. , whether the movement crosses more than two bounding nodes (barriers) at a time, whether each step in a movement is the shortest possible, whether the movement is internal Merge, whether the movement takes place cyclically, whether the movement takes place within the same structure-building space, or takes place between distinct spaces (sideward movement), (e) the timing of movement, e. g. , whether the movement takes place before Spell-Out, etc.Kayne's (1984) take on islands is new and insightful: the language system in the human brain is solving a legibility problem of classic topology, i. e. , is it possible from e to reach the antecedent by drawing with one stroke of the brush? A legibility problem is posed by external systems (the thought system and the perception-motor system) which are connected to the language system, which the language system must solve in order for it to be identified by external systems. An example of classic topological problems is Euler's path : is it possible to cross seven bridges by passing each bridge once?, a hard problem posed by a citizen of the then Konigsberg in1736 (now Kaliningrad in Russia), which is the origin of graph theory in mathematics.Euler's Path (Can one draw this with one stroke of the brush? If not, prove it. )Euler proved that it is not possible to draw it with one stroke, providing necessary and sufficient (iff) conditions for the path solution.Kayne's (1984) Connectedness Condition is a legibility problem that the language system solves in the optimal way. An acceptable sentence is the optimal solution to the Connectedness Condition, topological in nature.

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