Emma@SU-CSLI.ARPA (Emma Pease) (02/24/86)
From: Emma Pease <Emma@SU-CSLI.ARPA>
CSLI COLLOQUIUM
LOGIC OF POINTERS AND EVALUATIONS:
THE SOLUTION TO THE SELF-REFERENTIAL PARADOXES
Haim Gaifman
Mathematics Department The Hebrew University Jerusalem Israel
Visiting at SRI
February 27, 1986
Ventura Hall
Imagine the following exchange:
Max: What I am saying at this very moment is nonsense.
Moritz: Yes, what you have just said is nonsense.
Evidently Max spoke nonsense and Moritz spoke to the point. Yet Max
and Moritz appear to have asserted the same thing, namely: that Max
spoke nonsense. Or consider the following two lines:
line 1: The sentence written on line 1 is not true.
line 2: The sentence written on line 1 is not true.
Our natural intuition is that the self-referring sentence on line 1 is
not true (whatever sense could be made of it). Therefore the sentence
on line 2, which asserts this very fact, should be true. But what is
written on line 2 is exactly the same as what is written on line 1.
I shall argue that the unavoidable conclusion is that truth values
should be assigned here to sentence-tokens and that any system in
which truth is only type-dependent (e.g., Kripke's system and its
variants) is inadequate for treating the self-referntial situation.
Since the truth value of a token depends on the tokens to which it
points, whose values depend in their turn on the tokens to which they
point,and so on, the whole network of pointings (which might include
complicated loops) must be taken into account.
I shall present a simple formal way of representing such networks and
an algorithm for evaluating the truth values. On the input 'the
sentence on line 1' it returns GAP but on the input 'the sentence on
line 2' it returns TRUE. And it yields similarly intuitive results in
more complicated situations. For an overall treatment of
self-reference the tokens have to be replaced by the more general
pointers. A pointer is any obgect used to point to a sentence-type (a
token is a special case of pointer it points to the sentence of which
it is a token). Calling a pointer is like a procedural call in a
program, eventually a truth valye (TRUE, FALSE or GAP) is returned -
which is the output of the algorithm.
I shall discuss some more recent work (since my last SRI talk) -
variants of the system and its possible extensions to mathematical
powerful languages. Attempts to make such comprehensive systems throw
new light on the problem of constructing "universal languages".