On should better

A correspondent writes to ask about should better, which he has encountered from time to time, and wonders whether it is idiomatic English. He gives two examples, both taken from published books: 'The means by which one can solve the definitional equations are some very simple properties, which one should better specify in advance, and these are the properties of and and of yields.''One should better pay attention to what Darwin and Wallace had to say about the same problem. When faced with the monumental task of classifying natural life, both biologists came to the conclusion that all divisions were arbitrary.' He asks: are they synonymous with had better do X or with specify/pay attention in a better way? I think the context suggests the latter, in both cases. The first quote is from an Italian logician, Giovanni Sambin, in his One Hundred Years of Intuitionism, p.305, and he makes it very clear in the surrounding paragraphs what the 'better specification' is. Better is an adverbial modifier here. The string means 'which it would be better to specify in advance'. The second quote is evern clearer, when we examine the context. It's from a book called A Scientific Model of History, by Juan J Gomez-Ibarra, p. 28, and in the previous paragraph we read: 'Should we reduce the figure of twenty-one civilizations down to twenty because of... Or, should we better rename the two societies as...' The inverted order suggests that better is modifying rename - 'we should rename in a better way'. In which case, the 'we should better' usage follows on naturally. He is using better to modify pay attention. He doesn't mean 'ought to'. So why was my correspondent uncertain? It's because there is interference from the had better ('ought to') construction, which has led to the use of a modal should better as a blend (of should X and had better X). I've heard this usage in several regional dialects, but it hasn't (yet) established itself as idiomatic standard English. I've also heard it quite a lot from learners of English as a foreign language. It's a usage which usually poses no problem of interpretation in speech, and is probably already a feature of English as a lingua franca. But, as we see from these examples, it is waiting in the wings to upset any adverbial use of better following should. For this reason, I'd avoid it myself, and go for an alternative syntactic solution, such as replacing better by rather or rephrasing (as above) with a more explicit adverbial phrase.