I am going through some of my old school notes about English idioms and the text describes idioms as having a non productive syntactic structure. I don't understand and couldn't find anything more about the 'non productive' part of the phrase.
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It isn’t true that idioms have a non-productive syntactic structure. An idiom is an expression that can be understood only as a whole, and not by interpreting each of its parts. He kicked the bucket means ‘He died’, but you wouldn’t know that if all you knew were the words he, kicked, the and bucket. There is, however, nothing non-productive about the grammar. It follows the normal English word order, and the pronoun, he, is in the form used when it is the subject of a sentence.
What is an idiom? As explained in the Glossary of linguistic terms by Eugene E. Loos,
Alternatives 2 & 3 above are grammatical and make sense (semantically correct), but do not make an idiom — because the idiom cannot have its words substituted or rearranged.
Today, however, we see this rule regularly violated. Every idiom has been turned on its head in various contexts for various effects.
Productivity is concerned with whether a grammatical process is still being used in the formation of new words- i.e., it is involved in the production of new words. A non-productive process is one which has previously been used, but is no longer being used such that we can see evidence of its use, but no new words or phrases have been produced for a long time using that process.
So your non-productive syntactic structure refers to a syntax structure that was used in the past, so that the language has words or phrases in use that exhibit that syntactic structure, but is no longer being used to form new words or phrases.