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)

Saturday, April 5, 2008

The General Form of a Truth Function.

Wittgenstein did not fully explain his symbolism in the Tractatus. The following uses text from Russell's introduction.

The general form of a truth function is: [p-, ξ-, N(ξ-)] where
p- stands for all elemental propositions,
ξ- stands for any set of propositions, and
N(ξ-) stands for the negation of all members of ξ-.

This general form of a proposition simply says that every proposition generated by successive negations of elemental propositions. The whole symbol [p-, ξ-, N(ξ-)] is an algorithm:
• select a set of any of the atomic propositions,
• negate them all,
• then take any selection of the set of propositions now obtained,
• together with any of the originals
• -- and so on indefinitely.

This describes a procedure which, when performed on given elemental propositions, can generate all other propositions that are not elemental. The process depends upon:
(a) Sheffer's proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of ``not-p and not-q'';
(b) Wittgenstein's theory of the derivation of general propositions from conjunctions and disjunctions; and
(c) The assertion that a proposition can only occur in another proposition as argument to a truth-function.

If the general form of constructing propositions is given, then how one proposition can be generated from another is also given. Thus, the general form of an operation Ω'(η-) can be written as: [ξ-, N(ξ-)]'(η-) or as [η-,ξ-,N(ξ-)]. This is the most general form of transition from one proposition to another.

To apply this method to arrive at numbers one defines:
x=Ω^(0)'x Def.
and
Ω'Ω^(ν)'x=Ω^(ν+1)'x Def.

Now, according to the rules of our notation, one write the series
x, Ω'x, Ω'Ω'x, Ω'Ω'Ω'x,...,
so that
Ω^(0)'x, Ω^(0+1)'x, Ω^(0+1+1)'x, Ω^(0+1+1+1)'x, .....

Therfore, one can rewrite the general form [x, ξ, Ω'ξ]) as
[Ω0'x, Ων'x, Ων+1'x]
and define:
0+1=1 Def.,
0+1+1=2 Def.,
0+1+1+1=3 Def.,
etc.
One sees from this that a number is the exponent of an operation.

The concept number is just the general form of a number which is common to all numbers; it is the variable number. Also, the concept numerical equality is the general form of all particular cases of numerical equality.

So the general form of an integer is: [0, ξ, ξ+1].

The theory of classes is completely superfluous in mathematics due to the fact that the generality required in mathematics is not accidental generality.

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