Subject: n - ary vs . binary

lloyd anderson asks how n-ary ( for n greater than 2 ) comparison could ever be worse than binary . consider the problem of heads and tails when we toss coins . what are the chances of two coins ( standing for two languages obviously ) both coming up the same ( i . e . , both heads or both tails ) when tossed once ? since there are four possible outcomes of the binary toss , namely , hh , ht , tt , and th , and only two where both come up the same , the chances are 50 % . but now consider what happens when we toss three coins ( standing for three languages ) . since a coin only has two sides , in each possible outcome at least two of the coins come up the same . so the chances of 2 out of 3 coming up the same ( which would correspond to saying let two out of three languages agree in something and then they are related ) are 100 % ( which means this is not a valid test for relatedness ) . of course , if we had wanted all 3 out 3 to come up the same , then the situation would be drastically different , but in linguistics n-ary comparison never to my knowledge involves such a requirement . however , the only reason that n-ary comparison does so poorly here is that ( a ) there are only two possible outcomes per language , i . e . , languages come in only two varieties , and ( b ) the number of languages being compared is small ( only three ) . the ( a ) part is the one where real linguistic applications are drastically different from our little coin-tossing game ( since when you look for language relationships , you are looking at hundreds or thousands or maybe even more possibilities , not two , because you are looking at phonological shapes of morphemes mostly , and these allow lots of possibilities , at least thousands ) . so in the real situations that alone insures that n-ary comparison is better than binary . but it is also true that if you increase the number of coins ( languages ) in ( b ) , that also has the same effect . but you still have to be careful : the main concern is that given n languages being compared you must worry about how many out of the n are required to agree and about not making that number too small ( if you do , then again chance tends to take over ) . which raises a question : is there any published work on comparing languages which explicitly calculates these numbers ( and does it right ) ? alexis mr
