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

The structures of propositions relate internally to one another.

Wittgenstein highlights internal relations in his notation by displaying a proposition as generated by an operation on other propositions, the bases of the operation. An operation makes the relation between the structures of its result and of its bases explicit because it expresses what has to be done to one proposition in order to generate another from it. That will naturally depend on the formal properties of both base and result, on the internal similarity of their forms. The internal relation that orders a series is equivalent to the operation that produces one term from another. It can only appear at the point at which one proposition is generated out of another in a logically meaningful way; where the logical construction of the proposition begins.

Truth functions of simple propositions are results of operations based on simple propositions. (Wittgenstein calls them truth operations.) The sense of a truth function of p is a function of the sense of p. Negation is an operation that reverses the sense of a proposition. Logical addition, logical multiplication, etc. etc. are also operations.

When an operation appears in a variable sentence, it shows how we can get from one form of proposition to the next. It expresses the difference between their forms. The common element between the bases of an operation and its result is just the bases themselves. An operation does not characterize any form, but only characterizes the difference between forms.

Let the operation that produces (q) from (p) also produces (r) from (q), and so on. This requires that 'p', 'q', 'r', etc. be variables that give general expression to particular formal relations. The occurrence of an operation, however, does not characterize the sense of a proposition. For an operation makes no statement, only its result does, which result depends on the bases of the operation.

Operations and functions must not be confused with each other. A function cannot be its own argument, whereas an operation can take one of its own results as its base. (A function can be defined recursively, an operation cannot.) An operation is the only way to proceed from one term of a series of forms to another.

Wittgenstein calls repeated applications of an operation to its own result its successive application. (OOOa) is the result of three successive applications of the operation (Oξ) to 'a'. In a similar sense, he speaks of successive applications of more than one operation to a number of propositions. Accordingly, the general term of the series of forms a, Oa, OOa, ... is written as: [a, x, O'x]. The bracketed expression is a variable where the first term is the beginning of the series of forms, the second is the form of a term (x) arbitrarily selected from the series, and the third is the form of the term that immediately follows (x) in the series.

The concept of successive application of operators is equivalent to 'and so on'. One operation can reverse the effect of another or cancel it. Operations can even vanish. For example, negation in (~~p) : (~~p = p).

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