Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
Chromatic Fantasy, And Feud Having had a splendid dip in the pond, the Tortoise is just crawling out and shaking himself dry, wh ...
Tortoise: I certainly am not. Tortoises treat words as sacred; Tortoises revere accuracy. Achilles: Well, then, why did you say ...
found a contradiction in a pair of sentences I uttered. Frankly, it's just as I said. Your own system of thought is so delusiona ...
to entrap a poor, innocent, bumbling Tortoise in a fatal contradiction. If it were so harmless, why would you be trying so blood ...
CHAPTER VII The Propositional Calculus Words and Symbols THE PRECEDING DIALOGUE is reminiscent of the Two-Part Invention by Lewi ...
welljormed strings. They will be defined in a recursive way. We begin with the ATOMS: P, Q, and R are called atoms. New atoms ar ...
(Answer: Those whose numbers are Fibonacci numbers are not well- formed. The rest are well-formed.) More Rules of Inference Now ...
square brackets '[' and ,]" respectively. Thus, whenever you see a left square bracket, you know you are "pushing" into a fantas ...
characters in the movie-there is no carry-over from the real world into the fantasy world, in movies. But in the Propositional C ...
RULE OF DETACHMENT: If X and < x:::> y> are both theorems, then y is a theorem. Incidentally, this rule is often called ...
If this mind is Buddha, then, if this flax weighs three pounds, then this mind is Buddha and this flax weighs three pounds. You ...
substituted, and the resulting string will also be a theorem. It must be kept in mind that the symbols 'x' and 'y' always stand ...
Either this mind is Buddha, or this mind is not Buddha. Once again, the interpreted theorem, though perhaps less than mind- bogg ...
(9) <-Q=>-P> carry-over of line 4 (10) -P detachment (11) <-Q=>--P> carry-over of line^6 (12) --P detachment ( ...
Propositional Calculus is not only r.e., but also recursive. It turns out that there is an interesting decision procedure-the me ...
Imprudence: Well, yes-provided you wish to interpret '-' as 'not'. But what would lead you to think that '-' should be interpret ...
one can give a proof of a proof, or a proof of a proof of a proof-but the validity of the outermost system always remains an unp ...
<-xv-y> and -< X/\y> are interchangeable. If this were a rule of the system, it could speed up many derivations cons ...
Reflections on the Strengths and Weaknesses of the System You have now seen one example of a system with a purpose-to represent ...
g-eneral method for synthesizing artificial proof-like structures. It does not, however, have much flexibility or generality. It ...
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