This log was inspired by "How to Read Wittgenstein" and "Ludwig Wittgenstein: the duty of genius" by Ray Monk. It is based on reading Tractatus Logico-Philosophicus by Ludwig Wittgenstein translated by D. F. Pears & B. F. McGuinness (Routledge and Kegan Paul:1963)

Thursday, April 3, 2008

Every truth-function an be obtained by successively negating elemental propositions.

Successive applications of the operator (-----T)(ξ,....) to elemental propositions can generate every truth-function. Wittgenstein calls it the negation of all the propositions in the right-hand pair of brackets.

Let the sign 'ξ' indicate a variable whose terms are propositions. Let ξ‾ indicate the list of all those values where the order of the terms is indifferent. The bar indicates that the variable represents of all its values. The declaration describes the propositions the variable represents. If ξ has the three values P, Q, R, then (ξ‾)=(P, Q, R).

What the values of the terms are are must be fixed, but how the each term of the bracketed expression are described is indifferent. There are three ways that it can be done:
  1. Direct enumeration, in which case we can simply substitute constant values for the variable.
  2. Stating a function fx whose values for all values of x are the propositions to be described.
  3. Stating a formal law that governs the construction of the propositions. In that case the bracketed expression has all the terms of a series of forms as its members.
So instead of '(-----T)(ξ,....) we can write 'N(ξ‾)' where N(ξ‾) is the negation of all the values of ξ. If ξ‾ has only one value, then N(ξ‾) = ~p (not p); if it has two values, then N(ξ‾) = ~p . ~q. (neither p nor g).

Tt is now clear that this operation may be used to construct propositions, but exactly how it can be done must be made clear as well.

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