Thursday, August 02, 2007

4.0411 Should we want to express, e.g., what we express with “(x) fx” by placing an affix before “fx” – something like “Gen. fx”, it would not suffice – we would not know what was being generalized. Should we want to indicate it by an affix “a” – something like “f(xa)” – it would still not suffice – we would not know the scope of the generality-sign.

Should we want to try it by the introduction of a mark in the argument place – something like “(A, A).F (A, A)” – it would not suffice – we could not fix the identity of the variables. Etc.

All these ways of symbolizing do not suffice, because they do not have the necessary mathematical multiplicity.

In other words, if we want to say “For all x, f is true of x” then the best way to do this is the standard way. Saying “Generally [or universally] f is true of x” does not say whether the generality applies to f or to x. Say “fx” means x is fierce. Then does “Gen. fx” say that fierce things are generally x, or that x’s are generally fierce? Saying “f(x-all)” doesn’t help either, because the “all” might apply only within the parentheses or more widely. And so on. The only way to do it is as we do.

It might be objected, since these days the more common way to represent “for all x” is “Ax” (with the A inverted) rather than “x”, that Wittgenstein, taken at face value here, is quite wrong. On its own “(x) fx” says nothing and has nothing to recommend it over any other possible notation. What matters is the use we make of the notation. In 4.0411 Wittgenstein points out possible ambiguities and misunderstandings that could arise if different notations were used. But any notation can be misapplied. Wittgenstein presumably realizes this and wants to minimize the chances of misapplication. On the other hand, perhaps these remarks should not be taken at face value.

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