[FOM] Truth value algorithm for tokens
Sandy Hodges
SandyHodges at attbi.com
Wed Oct 30 17:42:13 EST 2002
I've posted a write-up of my latest algorithm (which is simpler, and I
hope more elegant than my previous ones, and based on a different
principle). The write-up is at:
http://sandyhodges.topcities.com/logic/sybil/forhtm.htm
As always, comments and refutations are appreciated. I also
implemented the algorithm as a Perl program - here are its values for a
random set of sentence tokens:
Token a: ~ Fa(d) & (0=0 <=> Fa(c)) false
Token b: ~ ((Tr(c) => Tr(d)) or Fa(a)) gap
Token c: (Tr(g) or 1=0) & Fa(d) <=> (~ Tr(f) <=> 1=0) gap
Token d: 0=0 <=> Tr(d) or Tr(a) gap
Token e: 1=0 & Tr(c) <=> (Tr(e) <=> Fa(g)) gap
Token f: ~ ((0=0 <=> Tr(c)) or (Fa(b) <=> Tr(c))) gap
Token g: ~ Fa(h) & (Fa(d) => Fa(e)) true
Token h: ~ (Tr(b) & (Fa(a) or Tr(g))) true
The basic concept is that there is a definition of
"paradox-containing". A model (an assignment of truth values) to a
token set is called "justified" if the tokens it calls gap are a
paradox-containing set, and the tokens it doesn't call gap, are a
paradox-free set. But a justified algorithm is not necessarily
correct, because a model can meet the definition of "justified" even if
it calls tokens gap that are not really involved in the paradox. For
example for these tokens
1. Token 2 is not true and either 2+2=4 or token 3 is true.
2. Token 1 is true.
3. Token 1 is true or token 2 is true.
4. 5+7=13.
A model would still be "justified" if it called all four tokens gap
(that is, neither true nor false), since the ones it called gap would be
paradox-containing, and the ones it didn't call gap (the null set) are
paradox-free. The justified models for tokens 1 through 4 are
gap, gap, gap, gap
gap, gap, false, gap
gap, gap, gap, false
gap, gap, false, false
My claim is that we can reject model (gap, gap, gap, gap), simply
because (gap, gap, gap, false) is also justified, and it calls strictly
fewer things gap. This is a different principle for saying that token
4 is not involved in the paradox with tokens 1 and 2, than other
algorithms have used. Other algorithms have tried to determine which
tokens "call" which other tokens - the idea being that token 1 does not
really "call" token 3, because the right half of token 1 is true
regardless of the status of token 3. I think this notion of "call"
does not always work.
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Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.
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