FOM: Truth for sentence tokens

[Axiomize@aol.com]

In a message dated 8/30/02 10:38:01 AM Eastern Daylight Time, 
friedman at math.ohio-state.edu writes:

Subj:Re: FOM: Truth for sentence tokens 
Date:8/30/02 10:38:01 AM Eastern Daylight Time
From:friedman at math.ohio-state.edu
To:fom at math.psu.edu
Sent from the Internet (Details)

> When I see [the Liar] Paradox discussed, I always want to know what some 
criteria are for a "solution". I wish scholars would pay more attention to 
this issue.

Yes, very much so.  I would like to see:

1. A formal representation of "This is false.", preferably a single 
self-contained expression.

2. An infinite set of formal representations that includes # 1 and some 
(maybe all) of them are paradoxes - preferably an infinite number are 
paradoxes.

3. A formal generation of "This is false." - preferably as written here, as 
well as a shorter formal representation.

4. A formal generation of the other paradoxes described in # 2, also 
preferably in English plus a shorter formal representation.

5. A formal means of identifying and controlling (eliminating) 
contradictions.  (For example, in the Theory of Computation, we accept the 
fact that what is mathematically definable is a proper subset of what is 
programmable.)

6. Application of the principles to other domains e.g. Theory of 
Computation/Computability (Turing), Recursion Theory (Kleene) or 
Incompleteness in Logic (Godel, Rosser), which also use self-reference.  Any 
limitation imposed should enhance rather than interfere with any established 
results.

A. Are these 6 criteria useful?
B. What alternate criteria are there?
C. Who has done each of these?

> There seem to be very few cases where paradox "solutions" form the basis 
for further major developments.  (One is the) Zeno paradox, with its 
"solution" being the natural number and real number systems, and their 
connections with counting, space, and time.  A second is the Russell paradox 
for sets, with its "solution" being set theory, formal and informal.  (The) 
Liar Paradox has already spawned "self reference in formal systems" that 
Godel set up to prove his famous incompleteness theorem(s).

Only a few applications but in very fundamental, general areas.

> That is a very big deal, but is not the same as a "solution" to the Liar 
Paradox in the sense we are talking about.

First there is the question of the solution about which we are talking.

"It is not true of itself." is true of itself.

What is the truth value of the above sentence?

Charlie Volkstorf
Cambridge, MA