- #1
xwolfhunter
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I know that this book is both fundamentally flawed and an extremely important book. I got it for Christmas, and I kind of view it like a whetstone for my mind. I'm taking my time and I won't turn the page unless I understand everything beforehand. I've been on the same page for a little bit, though, and I want someone who is familiar with it to help me out a little. I got the rough draft version, so I think it's partly that, but anyway.
I'm in the introduction, at the point where he discusses propositional functions, and he says, "Let ##\phi x## be a statement containing ##x## and such that it becomes a proposition when ##x## is given any fixed determined meaning. Then ##\phi x## is called a "propositional function"; it is not a proposition, since . . . it really makes no assertion at all." (Due to the ambiguity of ##x##). Later on the same page he says "When we wish to speak of the propositional function corresponding to "##x## is hurt," [in other words, ##\phi x##, which is, according to him . . . a propositional function . . .] we shall write "##\hat{x}## is hurt." Thus "##\hat{x}## is hurt" is the propositional function and "##x## is hurt" is an ambiguous value of that function."
Ignoring the contradiction in definition, the structure as I see it is we have some variable ##x##, which can take the form of anything, potentially, but is never in and of itself a proposition. Then we have ##\phi##, the functional variable, which is essentially a proposition that describes nothing. When we apply this to ##x##, we get ##\phi x##, which is the "propositionization" of ##x##, and the completion of ##\phi##, giving it something to describe. But still at this point, ##\phi x## is not a proposition, since it speaks of nothing in particular, though a specific member of nothing in particular. So far everything is in terms of a variable ##x##. Now from the functional variable's side, saying nothing of ##x##, we can define ##\phi \hat{x}## as the collection of all possible values of ##x## applied to ##\phi##. ##x## itself has absolutely nothing to do with ##\phi \hat{x}##, because ##\hat{x}## is an apparent variable, while ##x## is a free variable - right? Or . . . do we say ##\phi \hat{x}## is a propositional function, and ##\phi x## is a propositional function of ##x##? Also, what exactly is ##\phi \hat{x}##? It's not a proposition, but I don't think it's a variable proposition either, since it . . . has . . . no . . . variable? We can't say ##\phi \hat{x}## is true, because ##(x) \cdot \phi x## is what we'd use to . . . or is ##\phi \hat{x}## always true regardless . . .
I think I'm like right on the verge of exact comprehension here, but the book is not written for the layperson, which I very much am, so a little elucidation would be very much appreciated.
I'm in the introduction, at the point where he discusses propositional functions, and he says, "Let ##\phi x## be a statement containing ##x## and such that it becomes a proposition when ##x## is given any fixed determined meaning. Then ##\phi x## is called a "propositional function"; it is not a proposition, since . . . it really makes no assertion at all." (Due to the ambiguity of ##x##). Later on the same page he says "When we wish to speak of the propositional function corresponding to "##x## is hurt," [in other words, ##\phi x##, which is, according to him . . . a propositional function . . .] we shall write "##\hat{x}## is hurt." Thus "##\hat{x}## is hurt" is the propositional function and "##x## is hurt" is an ambiguous value of that function."
Ignoring the contradiction in definition, the structure as I see it is we have some variable ##x##, which can take the form of anything, potentially, but is never in and of itself a proposition. Then we have ##\phi##, the functional variable, which is essentially a proposition that describes nothing. When we apply this to ##x##, we get ##\phi x##, which is the "propositionization" of ##x##, and the completion of ##\phi##, giving it something to describe. But still at this point, ##\phi x## is not a proposition, since it speaks of nothing in particular, though a specific member of nothing in particular. So far everything is in terms of a variable ##x##. Now from the functional variable's side, saying nothing of ##x##, we can define ##\phi \hat{x}## as the collection of all possible values of ##x## applied to ##\phi##. ##x## itself has absolutely nothing to do with ##\phi \hat{x}##, because ##\hat{x}## is an apparent variable, while ##x## is a free variable - right? Or . . . do we say ##\phi \hat{x}## is a propositional function, and ##\phi x## is a propositional function of ##x##? Also, what exactly is ##\phi \hat{x}##? It's not a proposition, but I don't think it's a variable proposition either, since it . . . has . . . no . . . variable? We can't say ##\phi \hat{x}## is true, because ##(x) \cdot \phi x## is what we'd use to . . . or is ##\phi \hat{x}## always true regardless . . .
I think I'm like right on the verge of exact comprehension here, but the book is not written for the layperson, which I very much am, so a little elucidation would be very much appreciated.