The Decline of String-Manipulation Logic and Mathematics (optional material, p. 2)
A Paradox Arising from "String-Manipulation" Logic

  1. We assume here certain basic logical premises:
    1. ~(a~a)  —A proposition cannot be both true and false.
    2. a(aa)  —If a is true, then "a and a" is true.
    3. ((ab)(bc))(ac)  —If a implies b and b implies c, then a implies c.
    4. (ab)(~b~a)  —If a implies b, then not-b implies not-a.

  2. Define x as the string '~x'.

  3. Then x(xx)  —by (1b)
             (x~x)  —by substitution from (2)

  4. Hence x(x~x)  —by application of (1c) to (3)

  5. Consequently ~(x~x)~x  —by application of (1d) to (4)

  6. ~(x~x)  —by (1a)

  7. Therefore ~x  —from (5) and (6)

  8. Therefore x  —by substitution from (2)

  9. Hence x~x  —combining (7) and (8)

  10. But (9) is impossible!  —by (1a)
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