Is there any case of a "convergent language family"?

Question

As far as I know, language families originate from a process of divergent, tree-like evolution. All the languages within a family or subfamily can be traced back to the same proto-language, which was spoken several centuries ago. At some point, the various regions in which that proto-language was spoken started developing their own particular innovations. This process went on and on, to the point that, after some time, they were not mutually intelligible anymore and could thus be considered as “different languages”.

But is there any case of the opposite trajectory, i.e., languages that started out very different from each other, then started to converge because of language contact and ended up very similar, much like actual language families?

Now, I am aware of the existence of language areas (Sprachbunds), in which languages share several features by virtue of the high level of contact among them. But even in Sprachbunds, languages maintain their familial “identity”, so to speak. In the Balkans, for example, we can talk about Slavic languages, Romance languages, etc. There is no such thing as a “Balkanic familiy”.

Answer 1

I think there would be an epistemic block (i.e. a part of our methodology which prevents us from discovering something, regardless of what the data may say) against ever discovering such a thing. Let's imagine two languages, X and Y, which do not have a common ancestor. If, by traditional criteria of assessing relatedness (apparent shared innovations), we determine that X and Y have a common ancestor, then we will say that X and Y are related, even though they are not related. However, we have no way of discovering our mistake, since so far there is no method for assessing relatedness which has overtaken the comparative method in reliability.

But since not all language families are equally well substantiated, and none is 100% certain to be a genetic unit, it is highly probable that at least one such case exists: a language family which is called a language family but is really a case of convergence by contact. "Khoisan" is such a case. Once proposed as a family by Greenberg and then accepted as one by scholars, it has subsequently been discredited (see work by Tom Gueldemann on this topic) and is no longer held by specialists to constitute a valid genetic grouping. Since non-specialists may not be uniformly aware of the non-validity of Khoisan as a genetic unit, there are those who will still refer to the Khoisan family, even though it is not a family, but a group of families that resemble each other partly due to contact. But it would be hard to call it a family of convergence, since you would have to simultaneously believe that it's a family and that it's not really a family.

Putting this issue aside, there are many cases of languages which are unrelated but are strikingly similar. Compare Cantonese and Vietnamese: they both have lexical tone, a division between checked and non-checked tones, mostly monosyllabic morphemes, verb serialization, roughly the same segmental phonotactics, similar word order patterns, a variety of sentence-final emphatic particles, share a large proportion of their vocabulary, etc. But they are known to be unrelated, and have a good enough written record to not have ever been confused by serious researchers.

Answer 2

From Wikipedia:

The evidence of linguistic relationship is observable shared characteristics that are not attributed to borrowing.

From this it follows that your two languages will never become part of the same language family, because all borrowed features should be ignored when determining membership of a linguistic family. If an Indo-European language has adopted so many features from a Semitic language that both its vocabulary and its grammatical constructions are 99 % borrowed from the Semitic language, leaving only a barely recognizable 1 % Indo-European features, it should by this definition still be considered an Indo-European language. Even if this becomes 100 %, it follows that the old language should in principle still be said to be Indo-European!

This the absurd but necessary consequence of the definition shows the limits of its usefulness. It is the only way to classify a language with absolute certainty.

So a language is treated here as something entirely separate from its speakers: the essential factor to consider is its past. This poses the following problem: how do we determine which language has borrowed from which? In the above case, it could be said that the speakers have simply gradually adopted the Semitic language and stopped speaking their previous, Indo-European language. It just depends on whether you treat the pair as one "original" Indo-European language influenced by a Semitic superstrate, or an "original" Semitic superstrate that has moved to a new community of speakers and made the old Indo-European languages extinct as to its having any living speakers. The choice seems rather arbitrary.

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Another possible definition could be this (my own version):

A language belongs to a language family if the features typical of this family comprise the greatest part of its own features.

So a language whose vocabulary and grammar is 40 % inherited or borrowed from Indo-European languages, 30 % Semitic, and 20 % from other languages would be said to be Indo-European. However, this results in serious difficulties. How does one measure these percentages? Count words in its dictionary? Count words actually used in recorded conversations? And what if it has more words from family A but more grammatical features from family B? Which is more important? What to do with pragmatic and narrative features? How much weight do we attach to all these features and categories of features? What if one group of its speakers use more features from A, but another group uses more from B?

As an alternative, we could simply say each language can belong to several families at once. In most languages, a main component can probably be reasonably identified, but some should perhaps be said to belong to several families to an equal or comparatively undeterminable degree. If you take this interpretation, your question more or less becomes moot: languages can belong to the same family to various degrees, according as they contain more or fewer features from the same family.