In Principia Mathematica's Introduction, there is a definition for "incomplete" symbol:

By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts. -- Chapter III, Principia Mathematica, 1st edition, page 69.

I am not sure what "meaning" means in this definition. Judging by the passage that follows, I guess if a symbol has a "meaning," that symbol stands for an object, like "Socrates" stands for a certain man. As a matter of fact, in Russell's An Inquiry of Meaning and Truth, the meaning of an object-word is the thing it stands for. Please let me know if "meaning" in PM has the same definition as in An Inquiry of Meaning and Truth. Thanks.

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    $\begingroup$ Reminds me the classical ramble of Michael Palin, youtu.be/bDK2zdHBX3Q?t=1m37s $\endgroup$ – Asaf Karagila Jan 30 '14 at 10:22
  • $\begingroup$ hi George , you managed to do it completely the wrong way around, you ask the technical mathematical questions at philosophy and the philosophical here, better post them all at this forum i think. (and copy them to philforums if you cannot get your answr here) $\endgroup$ – Willemien Feb 4 '14 at 9:15
  • $\begingroup$ Hi @Willemien, Several people have told me that. The two asked in Phil were the first two. After that I moved everything to here. $\endgroup$ – George Chen Feb 4 '14 at 17:41

Russell is developing his theory of descriptions here. Roughly, he takes an "incomplete symbol" to be one that does not refer -- one that does not have a denotation in the way that proper names do. So, for example, the expression "the author of Waverley" -- according to Russell's theory -- does not denote; you cannot ask what its referent is. (EDIT 2: Russell does use the notation $\psi(\iota x\phi (x))$, but this doesn't mean that $\iota x\phi(x)$ denotes an object that can serve as the argument of $\psi(y)$, because $\psi(\iota x\phi(x))$ is really shorthand for $\exists x(\phi(x)\land\forall y(\phi(y)\rightarrow y=x)\land \psi(x))$.)

So noun phrases like "the author of Waverley," or "the more famous author of Principia Mathematica," cannot be thought of as proper names; and since Russell thinks of "meaning" in terms of reference, the expression cannot be said to have a meaning "in isolation." (One reason Russell makes this argument is to help explain how to formalize language about objects that don't exist, e.g. "the present King of France is bald" or "the greatest prime number.") Only in a sentence like "Scott is the author of Waverley" does the expression acquire a referent: that is what he means by the meaning only being "defined in certain contexts." In short, some expressions that look like object-words or proper names -- i.e., expressions that look like they refer (to an object) -- actually turn out not to be, if you're careful about the logical analysis.

Whether or not you are a proponent of the Russellian theory of definite descriptions, the mathematical examples Russell gives at the beginning of chapter 3 should not be taken too seriously. He says, for example, that the symbol $\frac{d}{dx}$ should be thought of as an incomplete symbol -- that it should not be thought of as having a denotation or referent by itself, when it isn't "completed" by a function symbol: e.g $\frac{d}{dx}(x^2)$.

To a contemporary mathematician, though, this might sound silly. A contemporary mathematician would likely say that $\frac{d}{dx}$ denotes a certain operator on a function space (e.g. the space of smooth functions from $\mathbb{R}$ to $\mathbb{R}$). Of course, there may be some ambiguity about which function space is meant, but that doesn't pose much of a problem: it is usually clear from context which space of functions one is working over.

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    $\begingroup$ This is a good answer; but I think that there are still "incomplete symbols" in modern mathematics, i.e., not all incomplete symbols are operators. For instance, the symbol "(". $\endgroup$ – mjqxxxx Jan 30 '14 at 3:17
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    $\begingroup$ I didn't say there weren't incomplete symbols in modern mathematics, or that every incomplete symbol is an operator. I said Russell's examples ($\int_a^b$ and $\nabla^2$) shouldn't be taken too seriously. $\endgroup$ – symplectomorphic Jan 30 '14 at 3:18
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    $\begingroup$ @GeorgeChen: yes, what Russell means by "meaning" is "the value of a variable" (to use a formulation Quine would later make more famous when he said "To be is to be the value of a variable"). And the value of a variable is an object. $\endgroup$ – symplectomorphic Jan 30 '14 at 14:29
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    $\begingroup$ Roughly, what "the author of Waverley" says is something like: "there exists an x such that x is the author of Waverley and, for all y, if y is the author of Waverley, then y=x. That is absolutely wrong. "The author of Waverley" expresses no proposition, your alleged equivalent does express a proposition (that there is one and only one author of Waverley). $\endgroup$ – Peter Smith Jan 30 '14 at 20:14
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    $\begingroup$ Yes, you're right; it does not express a proposition. I meant to give a flavor of what "the" contributes to the logical form of propositions containing definite descriptions, but you're right. $\endgroup$ – symplectomorphic Jan 30 '14 at 20:19

Following your own answer, recall that [page 66] :

By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts.

The issue is similar to that of terms [i.e.names] without denotation.

Basically, the "philosophy of meaning" of PM is based on propositions which "speak of" facts: the "ideal" language of PM's logic must express only true or false propositions.

Due to the paradoxes, R&W - in order to avoid the risk of propositions which are not true nor false - introduce the concept of "meaningless" expression. In their system, a menaingless proposition is detected by the fact that it violates syntactic constraints (see: types theory).

The paradigmatic case of "incomplete" symbol is that of definite descriptions, like :

the King of France is bald.

What is the meaning of the above proposition if there is no King of France ?

The solution Russell discovered was based on the well know paraphrase :

$(\exists x) (king_F(x) \land bald(x)) \land (\forall y)[(king_F(y) \land bald(y)) \rightarrow x = y]$.

Thus, in conclusion, if there is no King of France, the left conjunct is false, so that the complete sentence is false. We have found a way to give "meaning" to an expression also when it "seemingly" includes a non-denoting term.

Russell's solution is : circumvent the necessity of a term to safeguard "compositionalty" of meaning (i.e.the meaning or truth-value of the complete expression is based on the meaning of its components).

All this "machinery" is used in PM for classes [page 71] :

The symbols for classes, like those for descriptions, are, in our system, incomplete symbols: their uses are defined, but they themselves are not assumed to mean anything at all. That is to say, the uses of such symbols are so defined that, when the definiens is substituted for the definiendum, there no longer remains any symbol which could be supposed to represent a class. Thus classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects as their members are if they are individuals.

In other way, also class terms (i.e."modern" set terms : $\alpha = \{ x : \phi x \}$) like $\alpha = \hat{x} \phi x$, are "eliminated" in the same way, because non-denoting.

  • $\begingroup$ Excellent point, @Mauro. The more I consider this question, the more I get to grasp the philosophy aspect of this book. $\endgroup$ – George Chen Apr 16 '14 at 0:12
  • $\begingroup$ I just learnt that PM cannot prove that there are more than two individuals in the world but sees it unnecessary to postulate that there are more than two. This makes me wonder what meaningful symbols in PM really stand for. $\endgroup$ – George Chen Apr 16 '14 at 0:16
  • $\begingroup$ It seems that authors had "real things" in the back of their heads. $\endgroup$ – George Chen Apr 16 '14 at 0:21
  • $\begingroup$ Russell commented decades later that things are arbitrary and that counting, number, and mathematics like art and music are man-made beauties. But here and now I just want figure out what they were thinking in 1910. $\endgroup$ – George Chen Apr 16 '14 at 1:30
  • $\begingroup$ My mistake. One, not two. In the summary of *24, under*24.1 "Our primitive propositions do not require the existence of more than one individual $\endgroup$ – George Chen Apr 16 '14 at 5:21

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