Further to the post on 25th Sept 2007 "Notes as I read..."...
Thus a system is either Eternal and XCI or it is not Eternal (mortal) and not XCI.
To be not XCI a system must be caused by something other than itself (even if it can also be caused by itself). Thus no mortal system can exist independent of other systems. Death implies Other.
Now the SRH (which states that referring to oneself is impossible) is based on similar logic. To refer to oneself we must also be in existence, for if we were not existing then how could we be referring. Equally if the statement is a self-referring statement, then it must be referring in order to exist (as a self-referring statement). So we have the chicken and the egg again, albeit rather loosely:
If the statement fails to refer to itself it is still an existing statement so we can say that the existence of the statement is prior. The fact of whether it is self-referential or not, is built upon the statement existence. So the SRH is complaining that the reference is apparently to something which somehow embodies "itself", as though failure to refer would destroy the "self". The reference to a statement is thus not strictly self-referential:
"This statement has five words." could refer to the existence of a statement alone which is independent of the further judgement that the statement referred to is itself. This wouldn't raise the chicken egg problem. Yes, there is a statement here which has five words, it happens to be itself, it could have been any statement on the page, it just happens to be the same one as we are reading. Very boring.
However to be self-referential it would have to be taken to mean that "This statement has five words." is really the statement we are reading. This exciting possibility is why these statements are so alluring. This is the feeling we have of ourselves, it is more than one person amongst billions who just happens to be us. This means that "this statement" is only true when there exists this statement which also refers to itself. This raises the chicken and egg problem. There is only a statement with 5 words, if there is a statement with five words.
Put another way if the statement is "really" and "essentially" and "necessarily" self-referential (EN-SR) then it must refer to a self-referential statement. It is not good enough to set up reference to any old statement and then notice that this statement is itself. That is only partial, secondary and non-essential, non-necessary self-reference. This point has been made before in this blog.
To further illustrate. Suppose this page forms an ordered set of statements, P. We now have a statement "Statement x of P" where x is a free variable. Now I could list on this page random versions of this general formula with random natural numbers substituted. Eventually one of these statement must be numbered with the same number as its position in the page P. Alternatively I'll just cut to the chase with the next statement. "Statement 37 of P." Now this happens to we statement 37 on the page because I just counted all the current statements and added 1. Now I added one because I knew that once the statement was formed it would be a new statement but I did this before it was formed, before there were 37 statements, done in anticipation of it being the 37th statement. Before I typed it out and put the full stop at the end it wasn't true. Thus I built a system by other means whose self-reference only became "switched on" when the full stop was put in place. Thus it was built without self-reference being in place. As above if self-reference was needed to build the statement then it would have to be immortal. But because it is created, there must be other ways to create the statement other than itself. So we know that the real nature of a self-referential statement is actually just a statement and this is what is referred to. The statement does not refer to a self-referential statement, so it is not really self-referential! This is the SRH.
To further illustrate. Suppose this page forms an ordered set of statements, P. We now have a statement "Statement x of P" where x is a free variable. Now I could list on this page random versions of this general formula with random natural numbers substituted. Eventually one of these statement must be numbered with the same number as its position in the page P. Alternatively I'll just cut to the chase with the next statement. "Statement 37 of P." Now this happens to we statement 37 on the page because I just counted all the current statements and added 1. Now I added one because I knew that once the statement was formed it would be a new statement but I did this before it was formed, before there were 37 statements, done in anticipation of it being the 37th statement. Before I typed it out and put the full stop at the end it wasn't true. Thus I built a system by other means whose self-reference only became "switched on" when the full stop was put in place. Thus it was built without self-reference being in place. As above if self-reference was needed to build the statement then it would have to be immortal. But because it is created, there must be other ways to create the statement other than itself. So we know that the real nature of a self-referential statement is actually just a statement and this is what is referred to. The statement does not refer to a self-referential statement, so it is not really self-referential! This is the SRH.
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