What is the difference between "a desert island" and "a deserted island"? Are they synonyms?
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I think people are being misled by overly literal interpretations. Let's first note this definition from Oxford Dictionaries (it's also the one automatically shown by Googling define desert island)...
I would say that word "typically" is crucial, in that it implies a desert island might not in fact be uninhabited. Particularly since the usage example implies at least one inhabitant (the castaway lives there, even if he hopes his residence will be temporary).
More importantly, I can find any number of online definitions for desert island (that's 5 just there). To all intents and purposes, no dictionaries explicitly define deserted island - if you want to know what that means, you have to look up each word separately and work it out yourself.
Actually, that's not strictly true. If you search for define "deserted island" Google appears to show a definition for those exact words, but in fact it's just redirecting to the desert island definition. Wkipedia does something similar, in that the page http://en.wikipedia.org/wiki/Deserted_island goes to the same page as http://en.wikipedia.org/wiki/Desert_island.
I hope that's enough to show desert island is a well-known collocation with numerous specific connotations to most Anglophones (notably including Robinson Crusoe, Desert Island Discs, and humorous postcards). Whereas deserted island is just two words that might occur consecutively.
In short, whether there's any etymological justification for it or not, most Anglophones understand desert island as tropical, sandy-beached, [probably] without permanent inhabitants, etc., etc. They certainly wouldn't use that collocation in respect of uninhabited islands of Scotland, for example (which they might well be prepared to call deserted islands).
TL;DR: desert island evokes arid, sandy, desert-like, deserted evokes uninhabited, abandoned.
They are often used as if they were synonyms… But, "a desert island" is "an island which has never been inhabited, which is and has always been uninhabited", whereas "a deserted island" should mean "an island which once was inhabited but whose inhabitants left for one reason or another, whose inhabitants deserted it".
Although I am sure that this should, this normative attitude towards languages, is definitely not everybody's cup of tea! And tea-totallers are right in the end…
This reminds me of a game show I once saw on (French) TV: the people on set were given a phrase like 'not to beat about the bush' and given four different meanings the phrase might have.
The presenters had asked a hundred people in the street to choose between the different meanings of the phrase. To win the game, the people on set did not have to give the right answer, but the answer with the highest percentage.
At first this sounded preposterous to me… but then it dawned on me that this is exactly the way languages work in the long run: a mistake taken up by a majority of people ends up not being one!
An example: the sour cherry, in French, is l'agriotte from aigre, meaning 'sour' ('eager' belongs to the same word family as aigre). A lot of people mistook this for la griotte… which is the correct form today!
Similarly, when enough people have used nonnamous instead of anonymous, when Greek is Greek to enough of us, and when humourous has become plain, nonnamous might become correct!
I think it is one of those phrases which has gotten misused with time. Deserted island became desert island because people left off the "ed". There us no basis of comparison for an island with a desert because the word desert has an explicit meaning which is unrelated to an island unless the topography is like that of a desert and the fact that it is surrounded by water dismisses some if that slight similarity even.
Deserted implies uninhabited, not necessarily once, but no longer, inhabited. Desert, as in "desert island," is an otherwise obsolete form of deserted. Compare "For oh, for oh, the hobby horse is forgot," (instead of "forgotten.")
I live in the desert heat island of Phoenix, Arizona. It doesn't seem very tropical or sandy-beached, but as far as "without permanent inhabitants", we do get a lot of Snow Birds in the winter from Chicago and Canada.