Words are cognate if they stem from a common root. How can we tell? If we have two sets of words S and T, then we need to postulate a third set R of 'proto-word

Words are cognate if they stem from a common root. How can we tell? If we have two sets of words S and T, then we need to postulate a third set R of 'proto-words' as well as two mappings R → S and R → T where the mappings can (within reasonable margins of error) predict the form of the words given the roots. For example, take German and Latin: I assert the words Vater and pater (both 'father') are cognate, but to do so, I need to examine a whole set of corresponding words in both languages, so I take the sets {Vater, Fisch, Fuß} and {pater, piscis, pes}. From these, I can see clear correspondences (e.g. a German f sound always corresponds to a Latin p sound[1], the Latin sc seems to correspond to the German sch) and some things that are unchanged. Next, I come up with a set of roots like {pater, piš, pes} and a set of rules to apply like {p → f in German, š → sc in Latin, š → sch in German}, and I can state with some confidence that these words descend from a common root. In practice, you'd do this with way* more than three words—preferably you'd use most of the language—and there is some wiggle room for related meanings or exceptional changes, e.g. the German word alt and English word old seem unrelated to the Latin word senere 'to be old', because they actually stem from words for 'to grow up', and are therefore cognate to the Latin altus 'high, tall'. Additionally, some rules are more plausible than others based on research and phonological considerations, so the change of s → h is well-attested in actual languages and is consistent with similarities between the two sounds, while the change of r → b is highly unlikely. And while there is some room for error, there is a limit e.g. one might argue that English and Japanese are cognate because in Japanese, 'to eat' is taberu and 'to see' is miru, and in English we eat off a table and see in a mirror. This doesn't get borne out with more thorough phonological rules and doesn't seem to reflect regular semantic processes, either. tl;dr: Two words are cognate if they come from a common root; without direct knowledge of the parent language (e.g. Latin for Italian and French), we postulate one and show that regular processes can derive the child languages' vocabulary. [1]: If you don't speak German, know that the German v is pronounced like an f.