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Let us begin with the basic words of logic, the individual and the universal. For the individual, Schnfinkel used the letters A, B, and S. A list of such individual letters is defined by selecting them from a list, so that s[A,B,S] is the list A,B,S. The universal is expressed through a combination of the list operator SP with the individual, so s[SP[A,B],S] is the list A,B. A list of propositions is represented by the union of the list of individual elements and the union of the list of the universal elements, so the list s[A,B,SP[S,A]]. It was the universal S that had to have Schnfinkel bring in the more mathematical tools of the day.
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what of those who would deny that mathematics is based on computation? he answers that they are ignorant of the nature of mathematics and incapable of understanding the problem. even the recent faulty example of a proof by enumerative calculation of some theorem of number theory. mathematica v12 crack 2020! to be really convincing, schnfinkel has to introduce a three-tiered system of proof-generating, proof-checking and proof-evaluating. and he says, with apologies, that he is not going to do that. that would take too long to explain.
schnfinkel then returns to combinators. he says that they are more than simply a convenient device for making proofs. he says that they are a characteristic of the logical construction of the universe and of the formalization of the true process of mathematical thinking. indeed, the expression combinator should be the business of literary criticism.