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We are able to extract full rates of convergence by extending the use of a logical metatheorem recently proved by Kohlenbach. In recent case studies we were able to find such explicit rates of convergence in two concrete cases. Our novel method now provides an explanation in logical terms for these findings. This amounts, loosely speaking, to general conditions under which we in this specific setting can transform a \u2200\u2203\u2200-sentence into a \u2200\u2203-sentence via an argument involving product spaces. This reduction in logical complexity allows us to use the existing machinery to extract quantitative bounds of the sort we need.<\/jats:p>","DOI":"10.2178\/jsl\/1254748697","type":"journal-article","created":{"date-parts":[[2009,10,5]],"date-time":"2009-10-05T17:05:40Z","timestamp":1254762340000},"page":"1401-1428","source":"Crossref","is-referenced-by-count":6,"title":["Logical aspects of rates of convergence in metric spaces"],"prefix":"10.1017","volume":"74","author":[{"given":"Eyvind 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