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Quine, like many others before him, thought that the meaning of words depends on the context they are in.

But what compelled Quine to hold that in light of this there is an ambiguity as to what any given word references?

Couldn't someone admit that the same word can reference different things in different contexts while still holding that reference is a viable aspect of language?

What else logically motivated Quine to conclude that there is no fact of the matter about what any word references besides the fact that words' referents are context-dependent?

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Yes. He believed he saw indeterminacy of translation in actual languages. Of course his ideal case of "radical translation" involves languages with no common history at all--which never happens on this earth. English speakers can (and do, and must) draw on vast historical precedent in translating a closely related language such as French. But many language pairs are a lot less related than that. And all sciences build theories on non-ideal evidence.

Quine was in fact an avid language learner. He lectured in Portuguese in Brazil, and in WW II he worked in naval intelligence decoding German messages. I have heard that he also worked in Asian languages for intelligence but I cannot find confirmation of that now. Anyway,Tom Tillemans article "Count nouns, mass nouns, and translatability" (in Chakrabarti et al eds. Comparative Philosophy Without Borders) quotes Quine's argument for one kind of indeterminacy, namely that of his "gavagai" example, based on actual features of Asian languages. Tillemans is skeptical of Qune's analysis of the Asian languages but he shows how Quine intended to describe actual situations in translating between languages (naturally with more evident indeterminacy for languages that are less related to each other historically).

Apart from Tillemans' article, and my avowedly undocumented claim about working n Asian languages for intelligence, this can all be sourced from Wikipedia and Mactutor.

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    Here's a source regarding the Asian languages and Quine. Apparently an article Quine wrote in 1968 makes some claims about Asian languages. – virmaior Jun 22 '16 at 0:52
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Quine's thesis about the inscrutability of reference has nothing to do with context-dependence. As you noticed yourself, the context is just one more means to fix a reference. It is not a reason for skepticism about the nature of reference.

Rather, Quine's inscrutability of reference thesis comes from one aspect of the gap, in inferences from the particular to the universal. Faced with a particular object, how do I find the right general terms to describe it? Is this a "rabbit", or maybe "undetached rabbit parts", as in Quine's famous "gavagai" example?

An inference from the particular to the universal has been called induction. And indeed, Quine's inscrutability of reference thesis is akin to Nelson Goodman's New Riddle of Induction. Put briefly, the "old" riddle of induction was: faced with some particular green (e.g.) objects, can I justify a thesis about green objects in general? The "new" riddle of induction is: faced with some green objects, is it correct to describe them as 'green', or rather as (e.g.) 'grue', for purposes of generalization? When grue = green until date XYZ, blue hence. Quine argued that many possible conceptualizations are equally correct. That there was no fact of the matter which conceptualization was more correct. Goodman tried to explain, how we in fact choose one candidate conceptualization over another: the logic of induction.

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