obnoxio@BRAHMS.BERKELEY.EDU (Obnoxious Math Grad Student) (05/20/87)
[Side comment before I get to my article: "tech" stands for "technical", not "technology". If you want to discuss the risks of technology in general, or the meaning of "high-tech", I propose posting to sci.misc. I don't really care, either--just MOVE it SOMEWHERE ELSE.] Raymond Smullyan is famous for his mathematical logic inspired puzzles. This newest book centers around Goedel's *second* incompleteness theorem: "Dedicated to All Consistent Reasoners Who Can Never Know That They Are Consistent." As usual, he frames things in terms of truth tellers and liars. This particular device makes self-reference very easy. Smullyan goes deeper this time, and makes explicit just how capable the logicians involved are. Puzzles and arguments this time center on "belief". Thus, what does one make of the logician who visits the island of Knights and Knaves, and who knows the rules of the island, and is told by a native, "You will never believe that I am a knight."? (Recall that in this setup, knights only utter truths, knaves only utter falsehoods, there are no other natives [unless explicity mentioned], and that logicians always deduce all correct logical conclusions instanter.) Is this a paradox? A quick analysis would say yes, arguing from the given that a knight cannot utter a falsehood, and a knave cannot utter a truth, etc. But the above is not a paradox. You see, the logician is not assumed to *know* that he is a logician. Smullyan touched on this notion briefly--but beautifully--in his earlier book _5000 BC_, in the chapter with the experimental epistemologist. Here, he stratifies reasoners into four levels, each level *believing* more and more about how logical their belief system is--as opposed to just *being* logical. Level 4 reasoners are the first level where reasoners actually believe that they are the type they in fact are. [For the record, type n reasoners believe all tautologies and the first n of the following types of statements: (a) q [whenever Bp and B(p=>q)], (b) (Bp&B(p=>q))=>Bq, (c) Bp [whenever Bp], (d) Bp=>BBp. No, there is no misprint in (c). There is an extra modality supra.] This arrangement allows him to reinterpret the above paradox. As expected, the resolution is that if an islander tells a logician of type 4 that he, the logician, will never believe that he, the islander, is a knight, then the logician, if consistent, will never believe that he is in fact con- sistent. An account on Goedel's second incompleteness theorem along the above lines can be found in Smorynski's article in _The Handbook of Mathematical Logic_. Smullyan naturally goes on and considers self-fulfilling beliefs and Loeb's theorem. I found his handling of this quite delightful. He starts off with the rheumatic type 4 reasoner whose doctor's cure, upon inquiry, is largely psychosomatic: it works if the patient *believes* it will work. Fortunately he ended up on the island, and a kindly native, upon hearing his story, said, "If you ever believe I'm a knight, then you will believe that the cure will work." [Loeb's theorem is the assertion that a self-referential sentence in Peano Arithmetic, which holds iff the sentence *is* provable, is true.] Smullyan's last part of the book is about modal logic. He imagines various universes of logicians, whose beliefs are all complete, ie for all propo- sitions p, each reasoner believes p or not-p, whose knowledge of others' beliefs is perfect, and further, ever reasoner has complete confidence in the beliefs of certain others. For "complete confidence", read "necessary", and for "certain others", read either "his/her parents" or "his/her ances- tors". The difference between these two universes is critical. The beliefs that are universal among all reasoners are a type 3 set in the first case, and a type 4 set in the second case. This is, suitably interpreted, nothing other than a baby version of Kripke semantics. This is then tied in with Goedel's theorems via provability interpretations. Amazing. Unfortunately, I have to temper my praise for the book. Towards the end I felt Smullyan rushed and faltered. He starts making various combinatorial assertions about the reasoners, in order to give a combinatory logic proof that self-reference is possible, as he had done in _To Mock a Mockingbird_, but by compressing it as briefly as he did, it will only make sense, I suspect, to somebody who already knew the background material. Similarly, he brings, somewhat clumsily I thought, the Fergusson machines from _The Lady or the Tiger?_. The complicated nature of the reasoning involved here is far beyond his earlier books. In particular, he gets more into blatant symbolic logic. I found this somewhat of a distraction. I feel if he wants to do sym- bolic logic, he should just write a twenty page summary of the book by going full hog with mathematical notation in the statement of his puzzles. There is a certain joy in turning a very strange puzzle into the under- lying mathematical notation--I once posted Eddington's notorious zoo problem (those on the ARPANET can snarf it from my .plan)--but I think it loses its force after awhile. In short, I think _Forever Undecided_ would have played better as two books. Still, it plays very well. Now, I wonder what he could do for an encore. Intuitionistic logic? The problem here is that modal logic follows actual standard interpretations of English, whereas intuitionistic logic pushes familiar predicates into twisted variants. I have confidence that Smullyan will find something. [And if any intuitionists out there are offended, well, then, good! Post something!] ucbvax!brahms!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720 Some billion years ago, an anonymous speck of protoplasm protruded the first primitive pseudopodium into the primeval slime, and perhaps the first state of uncertainty occurred. --I J Good
leimkuhl@uiucdcsp.cs.uiuc.edu (05/23/87)
My mistake: this group isn't for discussions of the import of technology for human thought, it's to provide a captive audience for obnoxio's diarrhea of the keyboard. Ah well, I should have realized that this new group was too good to be true; the thought-police are always watching. -Ben Leimkuhler
andrews@ubc-cs.UUCP (05/25/87)
In article <8705201819.AA05586@brahms.Berkeley.EDU> obnoxio@brahms.berkeley.edu (Obnoxious Math Grad Student) writes: >Now, I wonder what [Smullyan] could do for an encore. Intuitionistic logic? In his popular books, Smullyan seems to be trying to give general readers an intuitive understanding of logic, with (I assume) the goal of trying to show readers how logic really does relate to human reasoning. I would therefore think that he would approach intuitionism only if he thought it had some significance to practical reasoning. As far as I know, Smullyan has not displayed any interest in intuitionism in his technical writing, except as a mathematically interesting formal system. Perhaps he could work some "intuitionist" characters into his next book -- like the "monochrome chess players" he worked into _The Chess Puzzles of Sherlock Holmes_. >[And if any intuitionists out there are offended, well, then, good! Post > something!] I guess I approach intuitionism in much the same way as does Smullyan. It's interesting to explore, but I'm not too concerned about what kind of logic I "believe" in. --Jamie. ...!seismo!ubc-vision!ubc-cs!andrews "Oh, my Lolita, I have only words to play with"