On 4/3/13 11:46 PM, Albert Y. C. Lai wrote:
On 13-04-03 07:39 PM, Alexander Solla wrote:
There's your problem. Mathematicians do this specifically because it is helpful. If anything, explicit quantifiers and their interpretations are more complicated. People seem to naturally get how scoping works in mathematics until they have to figure out free and bound variables.
Quantifiers are complicated, but I don't see how explicit is more so than implicit.
When the quantifiers are implicit, we can rely on the unique human ability to DWIM. This is a tremendous advantage when first teaching people about mathematical concerns from a logical perspective. However, once people move beyond the basics of quantification (i.e., schemata, rank-1 polymorphism, etc), things get really hairy and this has lead to no end of quibbles in philosophy and semantics, where people seem perversely attached to ill-specified and outdated logics. On the other hand, with explicit quantification you can't rely on DWIM and must teach people all the gritty details up front--- since the application of those details is being made explicit. This is an extremely difficult task for people who are new to symbolic reasoning/manipulation in the first place. In my experience, it's better to get people fluently comfortable with symbolic manipulations before even mentioning quantifiers. -- Live well, ~wren