From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.philosophy.meta
Subject: Re: Excluded Middle (Long)
Date: 10 Nov 92 16:34:55 GMT
Organization: University of Helsinki

In <TORKEL.92Nov10130705@isis.sics.se> torkel@sics.se 
(Torkel Franzen) writes:

>You're ignoring the point at issue. Take a set S of English
>sentences, say the set of sentences found in the writings of Bertrand
>Russell. What subset of S is specified by "the set of true sentences
>in S"? If you hold that there is in fact such a well-defined subset,
>please say so, in which case further arguments can be brought to bear.
>If you do not, please take notice of the fact that nothing is gained
>by switching to "the set of true propositions expressed by sentences
>in S".

Here is a sketch of an argument for the existence of a set of all true 
propositions expressible in English:

Let's consider your example.  We start out with a set S of sentences in
English, say n of them (all the sentences in Russell's writings, I guess
at least several million).  We define a sentence to be any string of
words ending in a full stop that is syntactically correct (given the
clarity of Russell's style we needn't worry here).

Now, of these n sentences only some will be descriptive ie. they assert
something about the world (including mathematical sentences, and I guess
logical truths too).  Which sentences qualify would be determined by some
group of rational persons who were familiar with the subject matter of the
sentences, say, the readers of this newsgroup.  So, given enough time, we
could pick out from S all those sentences of which it would make sense to
ask whether they were true or false.  Call this subset SD (for descriptive
sentences).  Let us say there are m of them, with m < n.

Given this set SD, we could, again given enough time, go through each
SDi (i=1,2,...,m) and check whether it is true or not.  The method of
verification would vary from sentence to sentence and there is no
simple rule for how in general it would be done.  But, a group of
rational persons could in principle check at least most of the SDi.
Some might be left over for various reasons, but this ignorance does
not detract from the fact that in principle each SDi could be so checked.
We collect all the true sentences in SD into the set ST, the set of all
true sentences in Russell's writings.  The cardinality of this subset
would again be smaller than SD, say r < m.

Now, since we can take the relation between sentences and propositions
for granted, we say that each STi expresses some proposition.  The 
number of true propositions expressed would be smaller than the
number of sentences, since Russell presumably asserts the same thing 
several times in various places.  In any case, we get a further set,
call it SP, of all true propositions found in Russell's writings.
Let us say there are s of them, with s < r.  

We arrived at SP in stages from S and each stage seems legitimate.  So,
we are justified in saying that Russell's writings express some set SP
of true propositions.  We could in principle do this for any set of English
sentences, including the set of all possible sentences A, which would 
express the set of all true propositions T (expressible in English). 
There is nothing different about S and A, nor hence about SP and T.

Well, this is something to start from, anyway.  Where are my flaws?


-- 
Marko Amnell
amnell@klaava.helsinki.fi
Graduate Student in Philosophy