Functions in Wittgenstein's Tractatus Logico-Philosophicus do not have exactly the same meaning as those in modern Mathematics. From its context in Tractatus, a function needs not have a well-defined domain (the set of arguments at which the function is well-defined) as in Mathematics, as long as it is logico-syntactical.
If all propositions form a set (the general form is given in 6 of Tractatus), then operations in Tractatus have the same meaning as those in Mathematics -- they are just functions (in the mathematical sense) of one or more arguments (propositions) to the set of all propositions. They are well-defined on all propositions.
As states in 5.251, a function cannot be its own argument because it is not logico-syntactical (3.333), whereas an operation can take one of its own results as its base because an operation is well-defined on all propositions as its bases and produces a proposition as its result.
If all propositions form a set, then what is the meaning of a function of Wittgenstein's with these propositions as arguments? Such a Wittgenstein's function is in fact the value of a mathematical function. In other words, if we denote the function with one argument (for example) by F in Mathematics, then Wittgenstein's function is just F(p) for some proposition p. Wittgenstein calls such F an operation. As states in 5.234, truth-functions of elementary propositions are results of operations with elementary propositions as bases. In Mathematics, we can have a composite function - a function of a function. That's why operations and functions must not be confused with each other (5.25); and a function cannot be its own argument, whereas an operation can take one of its own results as its base.
(Written on 22 September 2008; revised on 12 May 2009)
22 September 2008
Subscribe to:
Posts (Atom)
