The view of logic presented here is at odds with one conventional approach, which reduces the science to a kind of superficial manipulation of strings of symbols that have no necessary relationship to reality. Logic is said to deal, not with the content of these strings, not with how (if at all) they might relate to the real world, but only with their external form, and particularly with the identification of valid formal progressions from string to string. The latter approach grew out of nineteenth-century mathematics, a field which itself came to be seen by its practitioners as a similar kind of formal manipulation. The shortcomings in this notion of logic became apparent in the early twentieth century, when it was shown that such formalistic logic could not suffice even for many mathematical derivations, much less for grasping truths about the concrete world (
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