[PREV - LOGICAL_ATOM] [TOP]
LOGICAL_ATOM_CLASS
April 18, 2022
"Theory of Types and Symbolism: Classes",
Lecture 7 of "The Philosophy of Logical Atomism"
(1918) is pretty funny, if you've got an odd
enough sense of humor-- it starts out feeling
scattered, like endless preferences missing any
explanation of where he's going with it all--
It eventually becomes clear he's trying to find
some way out of Russell's Paradox with this material,
this is him trying in invent a Type Theory that
restricts the use of self-referential sets.
About a third of the way through, he suddenly says:
"I come now to the proper subject of my
lecture, but shall have to deal with it
rather hastily.
Then following Cantor, Russell points out that given
a set of Everything, you could have a new set of permutations
drawn from the set of Everything, which means you can
have a set bigger than the set of Everything--
from this Russell concludes that those permutations can't
possibly count as "things":
"You are met with the necessity, therefore,
of distinguishing between classes and
particulars. You are met with the necessity
of saying that a class consisting of two
particulars is not itself in turn a fresh
particular ..."
"You would say generally that you would not
expect a class to be a member of itself. For
instance, if you take the class of all the
teaspoons in the world, that is not in itself
a teaspoon."
"Normally you would say you cannot expect a
whole class of things to be itself a member of
that class. But there are apparent
exceptions. If you take, e.g., all the things
in the world that are not teaspoons and make
up a class of them, that class obviously (you
would say) will not be a teaspoon."
"Certainly you would have thought that it was
clear that the class consisting of all the
classes in the world is itself a class."
" ... certainly in all the cases of the
ordinary classes of everyday life you find
that a class is not a member of
itself. Accordingly, that being so, you could
go on to make up the class of all those
classes that are not members of themselves,
and you can ask yourself, when you have done
that, is that class a member of itself or is
it not?"
" ... either hypothesis, that it is or that
it is not a member of itself, leads to its
contradiction. If it is a member of itself,
it is not, and if it is not, it is."
"I think it is clear that you can only get
around it by observing that the whole
question whether a class is or is not a
member of itself is nonsense, i.e. that no
class either is or is not a member of itself,
and that it is not even true to say that,
because the whole form of words is just a
noise without meaning. "
"It is absolutely necessary, if a statement
about a class is to be significant and not
pure nonsense, that it should be capable of
being translated into a form in which it does
not mention the class at all."
When he finally gets down to trying to pin down
what's *wrong* with these self-referential assertions,
the argument threatens to dissolve into gibble-gabble--
the plain, ordinary meanings of words (e.g. "I am lying.")
get translated into another form that's *supposed* to
implicit in the first, but the reader lacks any feeling
for whether to agree or disagree because the constructions
feel so peculiar.
If the central goal of the Principia Mathematica
project is to ground all of math into simple,
incontrovertable points, that dissolves once
you're forced to make maneuvers like this: "You
see, what you think that means isn't exactly
what it does mean, it doesn't *really* mean
anything, unless you re-write it in this complex
form you've never seen before."
"The man who says 'I am lying' is really
asserting 'There is a proposition which I am
asserting and which is false'. That is presumably
what you mean by lying. In order to get out of
the contradiction you have to take that whole
assertion of his as one of the propositions to
which his assertion applies; i.e. when he says
'There is a proposition which I am asserting and
which is false', the word 'proposition' has to be
interpreted as to include among propositions his
statement to the effect that he is asserting a
false proposition."
"I have been talking, for brevity’s sake, as if
there really were all these different sorts of
things. Of course, that is nonsense. There are
particulars, but when one comes on to classes, and
classes of classes, and classes of classes of
classes, one is talking of logical fictions. When
I say there are no such things, that again is not
correct. "
Calling these distinctions "logical fictions" is a bit
much, but it is true that they're relatively arbitrary
intellectual constructs: we're free to divide up the
world into entities as we see fit--
Some divisions feel more natural to us (the wall of a
cell, the skin of a human, the surface of a planet, the
outer shells of atoms...) but even those are arguably
arbitrary in some respects, and whether they're
appropriate depends on the way you're using the
divisions-- e.g. human beings are arguably not really
independent entities, since they need social SELF
interaction to stay sane.)
Russell continually invokes the concept of
"particulars", which is to say stuff that can be
shoveled into a set without fear because they're the
real deal, and not some oddball "logical fiction".
I have my suspicions about how well that can
really be made to hold up.
The abstractions we get at with terminology such as:
sets
classes
categories
types
Can have many variations, and we might press
these words into use for a specific style of
logical concept we have in mind-- something
that's been done too often, I suspect-- or we
might contrive new terminology entirely.
Am I allowed to talk about sets of
different conceptions of sets?
"The theory of types is really a theory of
symbols, not of things. In a proper logical
language it would be perfectly obvious."
"All those statements are about symbols. They
are never about the things themselves, and
they have to do with 'types'. This is really
important and I ought not to have forgotten
to say it, that the relation of the symbol to
what it means is different in different
types."
--------
[NEXT - LOGICAL_ATOM_GOLF]