[FOM] A question about dialetheism and sorites

[Axiomize@aol.com]

On 18 Nov 2002, Jesse Alama wrote:

> Hi Charlie,

> For our discussion on the liar paradox to proceed we must
> justify the existence of "English".  This result has never
> been definitively established.  It is currently unknown.
> This would be a fantastic result in modern linguistics.

To require a task that has never been accomplished for discussion to ensue is 
to condemn everything written about the Liar paradox that mentions "English".

> We must demonstrate the existence of the sets we use
> in our arguments.

Yes, but we don't have to formalize all of English to show the existence of 
an inconsistency.  Every theory has its domain of interest, and we need only 
examine "semantic statements" to discover the paradoxes.  Consider the 
following 9 production rules, A-I (variables are UPPER CASE):

A.  BOOL = true
B.  BOOL = false
C.  BOOL = not BOOL
D.  OBJ = it
E.  OBJ = this
F.  OBJ = "SEN"
G.  SEN = OBJ is BOOL.
H.  SEN = OBJ is BOOL of OBJ.
I.  SEN = OBJ is BOOL of itself.

Using SEN as the start symbol, the language generated by this grammar is the 
set of English sentences of interest.  (Note that, for display purposes,  (1) 
the terminal symbol " alternately represents " and ' (2) The first letter of 
each SEN value is capitalized.  Also, (3) OBJ can more generally represent 
any recursive predicate.)

For example, to derive "This is true.":

1. G.  SEN = OBJ is BOOL.
2. E.  OBJ = this: SEN = This is BOOL.
3. A.  BOOL = true: SEN = This is true.

We can represent each sentence in SEN by the sequence of rules used in its 
derivation:

GEA = This is true.
GEB = This is false.
IFIDBA = "It is false of itself." is true of itself.

The first 10 semantic statements are:

  1. This is true.
  2. This is false.
  3. It is true.
  4. It is false.
  5. This is true of it.
  6. This is false of it.
  7. "This is true." is true.
  8. "This is true." is false.
  9. This is not true.
10. "This is false." is true.

The first 1,000 semantic statements are: http://groups.google.com/groups?dq=&
hl=en&lr=&ie=UTF-8&selm=af50c035.0211211856.6a6d9599%40posting.google.com

>>    "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")

> 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?

The above programs are written in a common subset of the programming 
languages Java and PHP.  In the sentence, the pronoun "it" (if any) 
represents input, corresponding to the value of $a that is input to the 
function.  The relationship is that the program returns true iff the sentence 
is true, and returns false iff the sentence is false.

> Jesse

Charlie Volkstorf
Cambridge, MA