# [FOM] Solution for Buckner?

Lucas Wiman lrwiman at ilstu.edu
Thu May 8 19:00:08 EDT 2003

```Me:
>What about statements like ``all English sentences contain at least one
>word"?  On the surface, this seems to involve reference to an infinite
>totality--all English sentences.

Buckner
>No it doesn't.  Even if English contained infinitely many words, which
I don't
>think it does, your sentence just means " Every English sentence
contains
>at least one word" which means the same as " No English sentence
contains
>less than one word"

>Where's the "infinite totality"?

First of all, I did not mention the number of words in English, I
mentioned the number of sentences, which is plainly a different thing.
If you want to say that English only has a finite number of sentences,
then your finitism is more radical than most, as this would mean that
there are only a finite number of numbers (something you don't seem to
believe).  Also, you seem to have not read my post, where I said  "You
can, of course, interpret this to mean that there given an arbitrary
sentence in English,  it is the case that it contains at least one
word", which is approximately what you said.  This is based upon the
definition of English sentence, which stipulates that it have at least
one word.  I then gave a less trivial example:

``all English sentences can be generated by a single computational
procedure."

This sentence cannot be exactly interpreted in the way you ask, though
it could be recast as ``there is a computational procedure such that no
english sentence is not generated by it", but this still involves
quantification over an infinite domain, and the arbitrary sentence thing
doesn't work anymore, since the computational procedure has to apply to
all sentences.  Obviously a single computational procedure can be given
for any arbitrary sentence, namely one which just produces that sentence

How can you even understand your own sentence ``there is no sentence
which contains less than one word"?  Since there are infinitely many
sentences, this seems to make no sense to a finitist.  The first
sentence I gave was finitistically interpretable, I don't think my
second one is, and yours certainly isn't.

>in translating a Latin sentence to an English I aim
>to replace the Latin one with an English one that has the same meaning.
>That's what translation is.  So how can something that does not have a
>"finitistic" meaning be translated to one that does have a "finitistic"
>meaning?  If the meaning is not the same, it's not an accurate
translation
>in the first place.  If it is an accurate translation then both, or
neither
>of them, have a "finitistic" meaning.

In classical finitism, the statement ``there is no x such that P(x)"
has no meaning.  For nonfinists, this is seen to mean the same thing as
``For every x, P(x) is false" (which does have finitistic meaning) and
this can sometimes be proved using finistic methods.  So you've just
repeated my point.  If a finitist is asked the question ``is there no x
such that P(x)," then the finitist would respond ``I don't know how to
answer the question; the question is meaningless."  If you ask her  ``is
it the case that for every x, P(x) is false?", then she would respond
with a finitist proof (if there is one).  So to the finitist, these
statements are not equivalent, and to the nonfinitist, they are.  Thus
my question was, how could someone whose language only involves finite
totalities understand questions concerning infinite totalities as
equivalent to ones concerning only finite ones.

- Lucas

```