The Logic of the Principia Mathematica

[xwolfhunter]

I had posted a question in the mathematics forum about some conceptual issues I was encountering while reading the book, but now I realize as I read on that I may need a dedicated thread for asking questions about the book. So I am asking two things in this thread. Firstly, I am asking someone who is quite comfortable with the field of logic (and generally familiar with the dot notation I think originating in Peano's book) to answer the following question with the knowledge that I may quote them in this thread asking another question in the context of the Principia Mathematica, more often in the near future and hopefully petering off as my mind becomes more accustomed to logical thought. That way I won't be making multiple threads asking many people many questions.

Secondly, here is my current question:

I'm trying to wrap my head around apparent (bound) and real (free) variables, using the same terminology as that in the book, in the context of propositions and propositional functions etc. Russel speaks of ambiguous assertion, without which, he says, the consideration of $\phi x$, which is an ambiguous member of $\phi \hat{x}$ (I think with $x$ in the former case being a real variable, and in the latter, an apparent one), would be meaningless. What I get from this is that we can't speak of $\phi x$ as referring to a real variable (in a useful way) without the assertion $$\vdash . \phi x$$

I think he basically says that $\vdash . \phi x \equiv \: \vdash . (x) \cdot \phi x$, where in the latter case $x$ is an apparent variable, and in the former, a real one, so the two $x$es are wholly distinct - so $\vdash . \phi x \equiv \: \vdash . (y) \cdot \phi y$, and even though $\vdash . \phi y \equiv \: \vdash . \phi z$, it is not the case that $y$ and $z$ are necessarily referring to the same ambiguous value of $\phi \hat{x}$, because they are real variables; and to top off the run-on, in all of the above cases, even though anything of the form $\vdash . \phi x \equiv \: (x) \cdot \phi x$ is correct, it is not the case that the two are identical, and are useful in different scenarios due to the use of a real variable in one, and an apparent in the other (respectively). My question is . . . is that correct? Do I have the use of apparent/free variables, and the concept of ambiguous assertion, down, or am I misunderstanding? Thanks.

 

[Ssnow]

I think the ambiguity is that you can consider $\phi x$ as a something that speak on $x$ or as something that speak about another something because the fact is that the variable $x$ is not fixed ( by $\exists$ or by $\forall$ ) so the ambiguity is to consider $\phi x$ a particular and in the same time a general assertion... I don't know if I gave a help...