[FOM] Liar - Haim Gaifman's pointers

Sandy Hodges SandyHodges at attbi.com
Thu Sep 12 14:39:30 EDT 2002

Professor Gaifman's latest paper using his pointers to truth concept is

Gaifman, H. 2000. Pointers to propositions.  In: Circularity,
Definition, and Truth (editors: Andre Chapuis and Anil Gupta, Indian
Council of Philosophical Research) pp. 79–121.    A book which my
library does not own.    Fortunately it is online at:
 http://www.columbia.edu/~hg17/   (Click on the title)

Like Andreas Beck's algorithm, and my own, this is a "token-relative"
algorithm that can assign different truth values to different instances
of the same formula.   He  uses a third truth value (I do too.  He calls
it "GAP").

At any stage of the algorithm, each token either has one of the three
truth values, or it is undetermined (Undet.), having not yet been
assigned a value.   Semantic atoms Tr(x) and Fa(x) can thus be given
truth values from the truth value of x, using this table:
x |  Tr(x) |  Fa(x)
T |  T     |   F
F |   F     |  T
GAP | F   | F
Undet. | Undet. | Undet.
With this table, while sentences can have 4 values, the atoms that refer
to them have only 3 values, T, F, or Undet.    The values of sentences
containing the atoms can be worked out from the atoms using strong
Kleene truth tables (where the 3rd value is Undet, not GAP).   When this
process has reached a stopping point, Gaifman takes any closed cells of
mutually referring tokens (provided they refer to no Undet. token) and
gives them all the value GAP.

This is quite different from my algorithm - I am more sparing in calling
things GAP.    Gaifman does however suggest an alternative to this
wholesale assignment of GAP, namely to use supervaluations.    But
although I think I know what a supervaluation is, I am not quite clear
how he proposes to use them.   The algorithm proceeds until every token
gets determined, but then it goes back and re-assigns some things that
were GAP to be either T or F again.

Whatever the points of difference I am very excited to find another
algorithm of the same general kind; I've been having difficulty
explaining that the problem is even worth solving.
Here's an example which I think my algorithm treats differently than
1.  ~ Tr(2) or ~ Tr(3)
2.  Tr(4)
3.  ~ Tr(4)
4.  ~ Tr(1)
My thinking about this problem goes like this: whatever the status of 4,
2 and 3 can't both be true, so 1 must be true.   (This example is
related to Gupta's puzzle.)  Thus 4 is false, 3 is true, and 2 is
false.   My algorithm produces this output.   But as far as I can tell,
Gaifman's algorithm calls them all GAP, and his fix-it rule at the end
does not restore their values.

Here are two more examples:
5.  ~ Tr(5) & ( 1=1 or Tr(6) )
6.  ~ Tr(5) & ~ Fa(5)
7.  ~ Tr(5) & ~ Fa(5)

8.  ~ Tr(8) or ( ~ Tr(9) or ~ Tr(10) )
9.  ~ Tr(11)
10.  Tr(11)
11.  ~Tr(8) & ~ Fa(8)
In sentence token 5, the reference to token 6 is idle, because "( 1=1 or
Tr(6) )" is going to be true regardless of the status of 6.   Gaifman
has a rule to recognize this: token 5 does not "call" token 6, because
the clause "( 1=1 or Tr(6) )" can be evaluated without a value for
6.     But I think the indirect reference of 8 to 11 is equally idle,
since the right disjunct of 8 must be true.   But Gaifman's rule does
not recognize this.
It's worth pointing out that token-relativism is a quite radical
proposal.   All the logic I know is how to derive formulas from other
formulas.   Proving a token, rather than a formula, may require a
modification of the definition of proof.    Here is an example.

12.  Tr(16)
13.  ~ Tr(12)
14.   0=0 =>  ~ Tr(12)
15.   0=0
16.   ~ Tr(12)

12 and 16 are a classic split Liar: Gaifman, Beck, and I all call both
12 and 16 something other than true, and we all call 13, which states
our conclusion, true.    14 follows from 13, and 15 is a truth of
arithmetic.   All 3 algorithms agree that 14 and 15 are true.    But 16
follows from 14 and 15 by modus ponens.   And all three algorithms call
16 something other than true.    Thus, in the token-relative world, even
a conclusion by modus ponens from true conclusions, does not always
produce a true result.

16 is not true, even though provable by modus ponens, because it is
paradoxical.   But can there be a logic that requires a proof that
something is not paradoxical, before you are even allowed to use modus

------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.

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