By Dr. Hans Freudenthal

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1110~101+ and so on. 1 102. # a S b . t t . a = b . a,b + and so on. 1 1 1 I. + a = 1 1 I . + . a~11001: a 2 10. h . C G ~ ~ l l ' e - * c Z = l OV.. a = 1 1 . a=111~ and so on. 1 1 1 2. ~ a = l l l . t t . a ~ l l l . n . a ~ l l l : a = b . t t - a $ b . b_la and so on. + 1 1 1 3. # a I : l l l . / \ . b ~ l O . + . a + b . ~ l O O l : a > 110. A . b 2 10. -+ - a b > 1000 : + . a 5 b . b+d: a>b. b+d 9 1 1 1 4 . a=b' a= b A . c = d . +- c = d . A . a = b' . a = b . e=f : t t : a = b .

I think Carnap is right when he confines his analysis to an artificial language given in advance and precisely described. though I have the impression that his languages are just not rich enough to disclose essential problems. Formalist semantics is atomistic. It makes the assumption that some elementary expressions have a definite meaning, and that the meaning of composed expressions is derived by certain rules from that of the elementary expressions. If this could be realized, the meaning of any expression could be made independent of the context.

EInt. x = 0 + - . - - - -+. - - . . - . - - - - - A connective written A and meaning “for every” has arisen from an infinite conjunction. This seems to be the most natural way of introduction. Some phonetic resemblance between ‘ A ~and ‘A’ would be desirable. Texts such as # U = l . + . U1o> 1 0 ‘ A * C & = 11. r\-Etc: . A a - a ~ N u m+. d o > 0 + could hardly serve to introduce ‘A7. d o > 0 as a (true) proposition. This means that we have tacitly agreed to generalize over free variables. ) If we should start with the last program text, t,he receiver would not understand why we at once add the word written ‘A’, and he would be unable to guess its meaning.

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