# The Logic of the Principia Mathematica

1. Jan 4, 2016

### xwolfhunter

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.

2. Jan 4, 2016

### 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...