FOM: Truth for sentence tokens

[Sandy Hodges]

I have an idea that the question of plausibility (of accounts of the
Liar) would more profitably be delayed until we have at least two
accounts to consider.   An account being a mechanism of ascribing a
truth status to semantical entities, when those entities are themselves
ascribing truth status to each other.   Beck's model is one, and I have
a model which is sort of OK:
   http://www.powow.com/sandyhodges/logic/SelfAccH.htm
however, my model (and Beck's too I suppose) is overly sensitive to the
question of exactly when a sentence token refers to another.    Of these
sentence tokens:
  1: True(2)
  2: ~ True(1)
  3: ~ True(1)
my model calls 1 and 2 paradoxical (and thus not true), and it calls 3
true.    2 is treated as being inside the paradoxical "cell" while 3 is
outside of it, and thus 2 and 3, while they are the same formula, get
ascribed different status.    As far as I can see, I can't treat 2 as
outside the cell, or I get a contradiction, and if I don't treat 3 as
outside the cell, then my own description (whatever it is) of the Liar
will not be called true by my own model.

But my model treats 3 as outside based only on the fact that 1 and 2 do
not refer to 3.   Consider these:
  4:  True(5) & ( True(6) or ~ True(6) )
  5:  ~ True(4)
  6:  ~ True(4)
Now, my model thinks that 6 is inside the cell with 4 and 5, because 4
refers to 6.   But the reference to 6 in 4 is idle; the right conjunct
of 4 is a tautology and will always be true regardless of the status of
6.   But my model considers 6 inside the cell, and therefore calls 6,
like 5, paradoxical.    This over-sensitivity to token references that
are idle, becomes a serious problem as soon as you try to model
quantification over sentence tokens.    If a token begins:
  7:   For all tokens, ...
then my model will consider 7 as having referred to all tokens.   Thus
anything I say about 7 will be self-referential.

One way to disregard the idle reference from 4 to 6 would be to say that
if the status of 6 is not going to make any difference to 4, then 4 does
not really refer to 6.    But consider:
  8:  True(9) & ( ~ True(11) or ~ True(12) )
  9:  ~ True(8)
  10:  ~ True(8)
  11:  True(10)
  12:  ~ True(10)
Here the indirect reference from 8 to 10 is idle, because whatever the
status of 10, 11 and 12 will not both be true, so the right conjunct of
8 will never be true.   Also consider
   13: True(14) & ( 2+2=4 or True(15) )
   14: ~ True(13)
   15: ~ True(13)
Here again the reference of 13 to 15 is idle.    Examples like these
show you need to consider truth-status assignments in order to detect
idle reference, but my model assumes the references are all known before
the status assignment process can start.

My latest approach is what I call "two-level" models.    A "major" model
is tried, which assigns the status of "paradoxical", "inside but
bivalent", or "outside" to each sentence token.    Then while the major
model is held fixed, all possible "minor" models are tried: a minor
model being an assignment of T or F to every token.   For example, a
major model for tokens 8 - 12 is (Para, Para, Out, Out, Out).   Given
this major model, tokens 10, 11, and 12 must be T, T, and F.   The right
conjunct of 8 is T.    Then we try assignments of T and F to 8 and 9,
and we find that any assignment leads to a cycle:  (T,F) -> (F,F) ->
(F,T) -> (T,T) -> (T,F) -> etc.    The existence of this cycle justifies
the claim that 8 and 9 are paradoxical.   So the major model confirms
itself.

But suppose we started with a different major model for 8-12, namely
(Para, Outs, Para, Para, Para).    With this major model, 9 must be T.
We try assignments of T and F to 8, 10, 11, and 12.   But whatever the
minor model assigns to 10, in the next step 11 and 12 will have opposite
truth values, and in the step after that the right conjunct of 8 will be
T.   The left conjunct of 8 is T since 9 is T.   Thus, the only fixed
point minor model is T, T, T, T, F.    The existence of only a single
fixed point means there is no paradox, so the major model is wrong.    I
think the only major model that does not contradict itself is (Para,
Para, Out, Out, Out).

I'd be interested in any comments, especially any claim that the search
for models of these puzzles is pointless, if anyone thinks it is.   Also
any literature references to similar approaches.   - thanks
------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.