[FOM] A question about dialetheism and sorites

[Axiomize@aol.com]

On 16 Nov 2002, Charlie Silver wrote:

> Charlie Volkstorf writes:
>> The difference is, if "This is false." had a truth
>> value (i.e., its program halted), then we would have
>> an inconsistency, and English would be inconsistent....

> English _can't_ be inconsistent because it's only a language, not a
> formal deductive system.   I think this mistake lies at the bottom of the
> urgency many feel to resolve the so-called "Liar's Paradox."

> Charlie Silver

Do you consider English to be an informal deductive system?

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.

I believe there was justified urgency on the part of Gottlob Frege upon 
learning of Russell's paradox.  Current day interest in its English 
counterpart, the Liar, is more of an attempt to understand self reference in 
general and how it is utilized in such theories as computability and proof 
theory in particular, than an attempt to eliminate (resolve) it.

On 16 Nov 2002, Chris Menzel wrote:

> On Fri, Nov 15, 2002 at 08:54:49AM -0500, Axiomize at AOL.COM wrote:
>> Any system that concludes that a particular sentence is both true and
>> false is inconsistent, so that everything is provable ...

> That is exactly what does not follow in paraconsistent logics.

Does it follow in English (meaning the rules used to assign truth values to 
English sentences)?

>> The Liar paradox is simply the semantics of a program
>> that gets into an infinite loop, expressed in English.  

> The paradox surely is not the semantics of a program.  It is an
> argument from plausible premises to an inconsistent conclusion.

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."

> I guess what you say might be true if "is" means something like
> "bears certain structural similarities to".

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.)

> Chris Menzel

Charlie Volkstorf
Cambridge, MA