[FOM] A question about dialetheism and sorites

[Jesse Alama]

On Mon, Nov 18, 2002 at 10:12:24AM -0500, Axiomize at aol.com wrote:

> By "English", I mean both (the rules that define) the set of
> character strings that are syntactically correct English sentences,
> as well as the subset of these that are assigned a value of true.
> Would you agree that there is a (formal) deductive system behind the
> latter, and if so, what would you call it?  I'd be glad to consider
> alternate terminology.

Hi Charlie,

For our discussion on the liar paradox to proceed we must justify the
existence of what you call "English", which you define as a set.  An
argument for the existence of the set of syntactically correct English
sentences would be a great result of modern linguistics; to my
knowledge, this result has never been definitively established.  It is
unclear whether the question is even meaningful.

It is unclear what you mean by "English" as it appears in the
right-hand side of your definition, which also appears on the
left-hand side.

We must demonstrate the existence of the valuation you refer to.  A
sound argument for this conclusion would again be a fantastic result
in modern linguistics; I believe that it is currently unknown.

I do not agree that there is a formal deductive system behind the
subset you refer to.  It seems necessary to either describe the formal
deductive system or argue for its existence.  Neither of these tasks
has been carried out.  Carrying out even one of them would be a great
result in linguistics.

If we're going to honestly use set theory to help analyze the liar
paradox, then we must honestly adhere to the rule of demonstrating the
existence of the sets we use in our arguments.  If we don't adhere to
this rule, then we can be misled into accepting arguments whose
believability lies only in the mathematical language in which their
premises are stated.

> [snip]
 
> Yes, it is a correct refutation of the premise "Every sentence is
> true or false.", which is the English equivalent of "Every program
> halts yes or halts no."

This interesting equivalence has not been established.

> I would say, "is the formal analogue of".  Constructing an English sentence 
> that is neither true nor false is the analogue of constructing a program that 
> neither halts yes nor halts no, or a wff that is neither provable nor 
> refutable.
> 
> The syntax used in typical modern programming languages would be (where 
> "function" is a function declaration, "return" defines its value, "!" means 
> negation, "$a" is a variable, and ";" separates commands):
> 
>    "This is false."
>    function tif() { return !tif() }
> 
>    "This is true."
>    function tit() { return tit() }
> 
>    " 'It is false of itself.' is true of itself."
>    function iifoi($a) { return !$a($a) } ; iifoi("iifoi")
 
> That is: Function tif returns the negation of what it returns.
> Function tit returns whatever it returns.  Function iifoi returns
> the negation of what its argument applied to itself returns, so that
> iifoi applied to "iifoi" returns the negation of iifoi applied to
> "iifoi", which is the negation of itself.  (All three programs get
> into infinite loops.)

The relations between the English assertion and the "computer
programs" below them have not been spelled out.  What properties of
the English expression are also properties of the program?



Jesse