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)

Wednesday, April 2, 2008

A proposition is a truth function of elemental propositions.

An elemental proposition is a truth function of itself. The truth value arguments of a proposition are elemental propositions.

The arguments of functions are readily confused with the indices of names because both arguments and indices enable me to recognize the meaning of the signs containing them. When Russell writes '+c', the 'c' is an index and the sign as a whole indicates addition of cardinal numbers. But this use is arbitrary and it would be quite possible to choose a simple sign instead of '+c'. In (~p) however, 'p' is not an index but an argument: the sense of (~p) cannot be understood unless the sense of 'p' has been understood already. (An index always describes the object to whose name we attach it. In the name Julius Caesar, 'Julius' is an index: e.g. the Caesar of the Julian gens.) According to Wittgenstein, Frege's theory about the meaning of propositions and functions is based on the confusion between an argument and an index. Frege regarded the propositions of logic as names, and their arguments as the indexes of those names.

Truth functions can be ordered as rows of a table. Probability theory is based on this.

The truth functions of any given number of elemental propositions can be written out in a table like this one for all binary combinations:

(TTTT)(p,q) (If p, then p, and if q, then q.) (p⊃p.q⊃q) Tautology
(FFFF)(p,q) (p and not p, and q and not q.) (p.~p.q.~q) Contradiction
(TFTF)(p,q) (p)
(TTFF)(p,q) (q)
(FTFT)(p,q) (Not p.) (~p)
(FFTT)(p,q) (Not q.) (~q)
(TFFF)(p,q) (p and q.) (p.q)
(FTTT)(p,q) (Not (p and q).) (~(p.q))
(FFFT)(p,q) (Neither p nor q.) (~p.~q or p|q)
(TTTF)(p,q) (p or q.) (p∨q)
(FTTF)(p,q) (p or q, but not both.) (p.~q:∨:q.~p)
(FFTF)(p,q) (p and not q.) (p.~q)
(FTFF)(p,q) (q and not p.) (q.~p)
(TFTT)(p,q) (If q, then p.) (q⊃p)
(TTFT)(p,q) (If p, then q.) (p⊃q)
(TFFT)(p,q) (If p, then q, and if q, then p.) (p≡q)

Wittgenstein calls those truth arguments that make a proposition true its verifiers.

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