Consider this [1] article on language, discussing the properties of language. It seems that what has been discovered fails this in at least two ways. We haven't demonstrated, or even claimed, any sort of grammar; and what's being communicated isn't symbolic, but directly representational.
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Languages are not just sets of symbols. They also contain a grammar, or system of rules, used to manipulate the symbols. While a set of symbols may be used for expression or communication, it is primitive and relatively unexpressive, because there are no clear or regular relationships between the symbols. Because a language also has a grammar, it can manipulate its symbols to express clear and regular relationships between them.
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Another important property of language is the arbitrariness of the symbols. Any symbol can be mapped onto any concept (or even onto one of the rules of the grammar). For instance, there is nothing about the Spanish word nada itself that forces Spanish speakers to use it to mean nothing. That is the meaning all Spanish speakers have memorized for that sound pattern. But for Croatian speakers nada means hope.
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[1] http://psychology.wikia.com/wiki/Language#Properties_of_lang...