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So far, to me, the greatest difficulty in studying philosophy is to recognize the importance of the problems: Exactly what make philosophers think these problems are worthy subject of study? Take Russell's The Principle of Mathematics §6 for example:

The notion of the variable is one of the most difficult with which Logic has to deal, and in the present work a satisfactory theory as to its nature, in spite of much discussion, will hardly be found.

I wonder, as of today, what questions regarding variables remain unanswered.

I'm aware of the prevailing prejudice in the area. I appreciate it if you don't automatically assume a dismissive tone.

George Chen
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  • @ClementC. Someone voted to close, was that you? – George Chen Aug 01 '15 at 10:27
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    @GeorgeChen No. I in general try to be consistent. – Clement C. Aug 01 '15 at 10:28
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    I'm voting to close this question as off-topic because this question belongs on philosophy.stackexchange.com. – A.P. Aug 01 '15 at 10:28
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    I didn't downvote: downvotes express a negative judgement on the quality of questions, while close votes simply say that a question doesn't belong here. I'm just saying that there is a more appropriate venue to ask about philosophical matters. While philosophy and mathematics often go hand in hand, they are two quite different subjects of study: in very broad, informal terms, the former studies the why of things, while the latter studies the how. – A.P. Aug 01 '15 at 10:40
  • @A.P. There is a foundations tag and this is a question about foundations. I agree that this is a borderline question. I appreciate it if you let me stay. – George Chen Aug 01 '15 at 10:51
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    This is a very serious question. In calculus, differential geometry, or mathematical physics we work with "variables" all the time. There are independent, dependent and even "derived" ($dx$ from $x$) ones. They have some personality of their own, but nobody cares what they actually are in the framework of logic, or of ZF set theory. – Christian Blatter Aug 01 '15 at 12:15
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    Nobody downvoted this question at this point of time. The comment about downvoters is meaningless. – Asaf Karagila Aug 01 '15 at 12:58
  • Three voted to close. If no one else sees it, I must be hallucinating. – George Chen Aug 01 '15 at 13:11
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    @GeorgeChen A downvote is when people vote down. Currently, your post has four upvotes (see the number on the left). Voting to close, as explained by A.P. above, is not the same as a downvote. – Clement C. Aug 01 '15 at 14:25
  • @ClementC. I know. Thanks. "Close votes" are similarly frustrating as downvotes. Pardon me for not distinguishing the two. – George Chen Aug 01 '15 at 14:53

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It seems to me that the modern approach to logic is "very far" from teh original view of W&R.

See Alfred North Whitehead & Bertrand Russell, Principia Mathematica, Introduction, Ch.I : PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS, page 4-on.

According to that view, what we today call "the connectives" are propositional functions; see page 6 :

The Logical Sum is a propositional function with two arguments $p$ and $q$, and is the proposition asserting $p$ or $q$ disjunctively, that is, asserting that at least one of the two $p$ and $q$ is true. This is denoted by $p \lor q$. Thus $p \lor q$ is the logical sum with $p$ and $q$ as arguments.

There is not the "modern" emphasis" on syntax : the initila list of symbols forming the alphabet, the definition of expression as a finite string of symbols, the recursive definition of formula as a specific type of expression, ...

Basically, W&R uses a "perfect" language where all the symbols denotes something : the symbols $\lor$ stays for the Logical Sum propositional function, and (presumibely) propositional functions are some sort of object in the world "out there" (recall Frege : the concept of function was basic and he "struggled" a lot with the issue of the denotation (Bedeutung) of such an "unsaturated" entity ...).

If so, for what kind of object the "variable symbols" stand for ?

See page 4 :

To sum up, the three salient facts connected with the use of the variable are: (1) that a variable is ambiguous in its denotation and accordingly undefined [...].

In a modern logic textbook we simply have symbols and interpretations, and some cunning device to assign a "temporary" denotation to variables in order to determine the meaning (an truth-value) of an expression with a variable inside.

Thus, a variable is like a pronoun of naural language; in "It is red", the pronoun does not denote outside the context where the sentence is uttered. If I'm uttering it now, it denotes the red book on my desk.

The device of "variable assignment" used by math logic in the recursive semantical clauses for a predicate logic language has exactly the same function : to give denotation to a variable in the context of an interpretation.

In conclusion : so what ? Have we solved the problem or only skipped it ?

We can consider the influence of Wittgenstein : he was absolutely crucial, with its move from the "perfect language" considered into the Tractatus to its second phase regarding "linguistic games" and so on, for leaving the idea of a language where every part of it must denote somethig ...

Mauro ALLEGRANZA
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  • At least I see where Russell's difficulties came from: the lingering belief of the Platonic eternal world. He found nothing up there and called it one of his failures, but I think it is one of his achievements. – George Chen Aug 01 '15 at 15:06
  • The theory of description probably put an end to their troubles. This leads to the next question: Exactly what were their headaches before the theory of description was invented? – George Chen Aug 02 '15 at 15:50
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    @GeorgeChen - denotation again ! See PM, Intro, CH.III : INCOMPLETE SYMBOLS (pag.66) : "By an 'incomplete' symbol we mean a symbol whichis not supposed to have any meaning in isolation, but is only defined in certain contexts. [...] Suppose we say: 'The round square does not exist.' It seems plain that this is a true proposition, yet we cannot regard it as denying the existence of a certain object called 'the round square.' For if there were such an object, it would exist: we cannot first assume that there is a certain object, and then proceed to deny that there is such an object." – Mauro ALLEGRANZA Aug 02 '15 at 17:27
  • That makes a lot of sense. In that case, one cannot simply replace "the round square" with a variable and create a propositional function "x does not exist." – George Chen Aug 02 '15 at 19:16
  • The question "what do variables stand for" was their major headache. Russell agrees with Parmenides that one cannot speak of something that is not. Parmenides believes that if one can speak of something that something must exist somewhere; Russell's difficulty, before the theory of description, was that he couldn't find solutions below the Platonic heaven. After the theory of description, if one speaks of something that is not, he is either speaking of a description or using bad grammar. Thanks, @Mauro. I think the modern view you presented is actually the formalist's view. I might be wrong. – George Chen Aug 03 '15 at 10:47
  • Thus, "$x$ does not exist" is bad grammar, because existence can only be asserted of a description. End of headache. – George Chen Aug 03 '15 at 10:59