Robert Lowth is a household name in English grammatical theory. How has his writings positively influenced today's grammatical discourses?. Were there any weaknesses in his postulations?
He has made no contribution whatsoever to modern grammatical theory, except as an example modern linguists hold up of how to entirely misunderstand dialectal variation and historical change in language. Lowth just made up "rules" of English grammar that never existed historically and are still not followed in informal speech today, for example that two negatives make a positive. To illustrate this, I confidently assert that there is no fluent speaker of English today (or at any time in the history of the language) who might have misunderstood the statement "I can't get no satisfaction" to indicate "I can get some satisfaction." Bishop Lowth has been singled out as a force in creating the ideology that "good grammar" might mean something other than "the kind of grammar I, as a man of authority, insist that everyone use", but modern grammatical theory has completely debunked all of his ideas on "correctness". That is, his ideas have no connection whatsoever to grammatical theory, but they are rather a means by which people in power can assert that those who speak differently are "wrong" and thereby deserve to be mocked and derided and treated as though the grammatical patterns of their dialect of English were a product of ignorance, itself a reflection of their generally low moral and inellectual capacity.
Here is a quote from History of the English Language, by Pyles & Algeo (1979), describing Bishop Lowth and several other "experts" like him:
Present-day conceptions of “correctness” are to a large extent based on the notion, prominent in the 18th century, that language is of divine origin and hence was perfect in its beginnings but is constantly in danger of corruption and decay unless it is diligently kept in line by wise men who are able to get themselves accepted as authorities, such as those who write dictionaries and grammars. Latin was regarded as having retained much of its original “perfection”. When English grammars came to be written, they were based on Latin grammar, even down to the terminology. The most important eighteenth century development in the English language was its conscious regulation by those who were not really qualified for the job, but who managed to acquire authority as linguistic gurus.
If you are interested in exploring how modern grammatical theory sees these issues, I strongly recommend reading a book like English with an Accent, byt Rosani Lippi-Green, which was just released in a second edition in 2012.