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It is a well-known and widely repeated fact that the linguistic reconstruction associated with the comparative method is no longer effective for large temporal depths (informally estimated to be between 6000 and 1000 years, depending on the number of languages, linguistic branches and ancient documentation available). The problems are of various types:

  1. There is a broad semantic shift, whereby the primary meaning of lexemes is changing until the original meaning is almost unrecognizable and cognates are difficult to identify.
  2. Lexical replacement whereby a lexeme is substituted by another etymologically uncorrelated lexeme to mean similar things.

However, I have not seen too many studies or opinions that dare to answer more concrete questions such as:

  1. at what percentage of cogands shared among several languages does it become statistically impossible to find regular correspondences between etymologically related lexemes with related meanings (cognates).
  2. What is the weight of each specific factor, and which of all the factors weighs more when identifying cognates between very remotely correlated languages.
  3. Does the existence of many branches influence the possibility of reconstructing a proto-language (in European linguistics, many roots only appear in some branches and most of the reconstructed terms do NOT appear in each and every linguistic branch of the Indo-European family).

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You might be interested in Beyond lumping and splitting: probabilistic issues in historical linguistics by Baxter and Ramer (1999). From their abstract:

In this paper, we argue that the temporal reach of historical linguistics can probably be extended significantly in some cases, but only if the probabilistic issues are faced squarely, and new techniques of handling them are developed and widely understood.

The gist of the paper is that the concrete questions you describe can be answered, but the current (as of 1999) tools and techniques of historical linguistics are generally not able to do so. They argue that more quantitative, statistical work is needed.

Unfortunately, I have not seen any papers from the 20-odd years since then that follow up on this and try to come up with the hard numbers you're looking for. For the most part, historical linguistics research has remained qualitative rather than quantitative, with the effectiveness of the comparative method evaluated more by how reasonable the conclusions seem to experts than by a statistical model.

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