Bruno, John, Russell
I am half-way through Smullyan's book. It is an entertaining book for someone motivated enough to do all these puzzles, but I think that what is missing is a metalevel discussion of what all this means.
Mathematical fireworks occur because we are dealing with self-referential systems. In the old days they may have called them "reflexive." Reflection is, I think, an essential component of conscious thought.
The type of reflection I have encountered so far in the book involves "infinite" reflections which lead to paradoxes.
For example if someone says " I am a knave" then obviously we have a paradox.
The human mind, however, does not have the capacity to deal with an infinite number of reflections. (I think that you think that I think that you think.....). If, however, self referential systems are limited to a finite number of reflections, such as the human mind is capable of, then these paradoxes may go away. With one reflection a knave would says: "I am a knight." With two reflections he would say "I am a knave." With three reflections, "I am a knight." With four "I am a knave." and so on. With an infinite number of reflections he would remain "Forever Undecided."
I am not sure if Physics is derived from an ideal infinite self-referential systems or from a more human and messy finite system and I cannot think of an obvious and clear-cut justification for either approach. What do you think?
Bruno Marchal wrote:
At 09:55 20/07/04 -0400, John Mikes wrote:
It all depends what do we deem: "POSSIBLE". According to what conditions,
belief, circumstances? If we accept the "here and now"
as "the world", Stathis #1 may be right.
This would mean Stathis first assumption was a first person assumption, but the
whole point of Stathis seems (to me) third person. Also what would be the
meaning of "physical" in a first person assertion.
Perhaps Stathis could comment.
Now you are right we should agree on what we deem "POSSIBLE". With the comp hyp I argued that POSSIBLE = arithmetically consistent, and then we can go back asking G and G* ....
Giving that logic is not so well known apparently I will soon or later invite you all to Smullyan's knight knaves Island. It is the gentlest path to G and G* which are the propositional psychologies from which UDA shows how to extract the quantum measure in case (comp is true). And from which I have extract some bits of von neuman's quantum logic (but I am just beginning opening a vast and heavy doors here).
Why not now? The native of that Island are all either knight or knaves and knight always tell the truth, and knaves always lie. You go there. Problem 1. A native tell you "I am a knight". Is it possible to deduce the native's type? Problem 2. You meet someone on the island, and he tells you "I am a knave". What can you deduce?
I would be please to get answers, or critics.
I think it will be useful if only by John Mike remark: we will not progress
if we make not clear the word "possible" in our everything context ...
Logic can help because it is the science of proofS, truthS, and possibilitieS
(note the s).