On finding the sources of shared items, OR: The irrelevance of anteriority

Similarities between different languages are data. It's easy to come up with any of several wildly different measures of such similarities, typically by applying edit distances to wordlists (as in the ASJP*) or texts, but the result should not be mistaken for an analysis - it's just a measurement, a compression of the data. It doesn't tell you anything about the causes of these similarities on its own. Historical linguistics is not the measurement of similarities, but the effort to find the hypothesis about past events that best explains them. Your H₀, of course, is always "coincidence". Once you've rejected that, you're left with the trickier task of disentangling contact from common ancestry - trickier because, quite often, they partially overlap.

To understand linguistic causation in the past, an essential starting point is to look at it in the present. Suppose that you are a native speaker of English:

1. If you say "football" or "garage" to your child while speaking English, it's because you grew up speaking English, and you know that this is what other English speakers say. The fact that French speakers happen to call it "football" too, if you're even aware of it, has nothing to do with your choice of words.
2. If you say "football" or "garage" to your child while speaking French, it's because you later studied French, and you know that this is what French speakers say. The fact that it's also what English speakers say no doubt made it easier to memorise, but if French speakers had named them something else, you would be doing the same.

We thus see that, for shared words, inheritance from either of two radically different languages can yield precisely the same outcome. The fact that English and French share these words in the first place is obviously due to contact (in each direction). The fact that your child is growing up with them, however, is because you're faithfully passing on the existing norms of one or the other language, not because you're combining them. In historical linguistic jargon, the use of the word "football" is at this point being inherited, not borrowed. Thus, if an English-monolingual Cajun says "stupid", it's not because he's managed to hold on to his ancestors' French word "stupide", it's because that happens to be the English word for it.

So, if we have a word in language A, and find the same word in two potential source languages B and C, we can't determine which it came from by looking at which language was spoken in the area earlier, or which was spoken by the speakers' ancestors. We can only determine which it came from by determining which language (if either) was transmitted as a whole, and the evidence for that can only come from forms that aren't shared between B and C. I leave the application of this to Levantine ʕāmmiyya as an exercise for the reader.

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* It's beating a dead horse at this point, but: this Automated Similarity Judgement Program? It, too, finds that Levantine is way closer to Standard Arabic than to Aramaic, just like any historical linguist could have told you from the start.