By Raymond M. Smullyan
Characters from Alice's Adventures in Wonderland and Through the Looking-Glass populate those 88 interesting puzzles. Mathematician Raymond Smullyan re-creates the spirit of Lewis Carroll's writings in puzzles related to note play, common sense and metalogic, and philosophical paradoxes. demanding situations diversity from effortless to tough and include strategies, plus 60 fascinating illustrations. "An inventive book." — Boston Globe.
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Extra resources for Alice in Puzzle-Land: A Carrollian Tale for Children Under Eighty
We could not let 3 serve as the 3 which appears in the axiom schemes A 1 to A 7. For if 3 (pl, p z ) denotes the truth function corresponding to Pl 3 Pz, then by rn 1 we get 3 (2, 3 (1, 2)) = 4, so that the truth function corresponding to A 1 takes an unde’signated value. However, a plausible axiomatic stipulation with 3 playing the role of 3 can be given by changing our set of axiom schemes. By reference to m 1, it can be shown that this latter axiomatic stipulation is plausible with respect to the truth-value stipulation defined by m 1 with M = 4 and S = 2 .
Xp,, PI, . ,Pyd). In order to define a partial normal form N , (1 r M ) which corresponds to a function F,(P,, . , Pa,), one can proceed as follows: Consider all the sets of values p,, . ,paj for which fi(p,, . . , pa,) = r. For each such set construct the two-valued logical product & P 2 , k , & . & p a i , k , where k; = pi (1 i C and c = a,). Then, our partial normal form N , will be the twovalued logical sum of all such logical products. 1 8Z P 2 . 3 . 3). 2) ( P 1 . 2) ( p 1 . 3 & P 2 , 3 ) * " A moment's reflection will indicate that N,, N,, and N3 of the present example are in effect the same as the partial normal forms for F,(P, Q ) which have already been described.
2. ( p ) Assume the theorem for p = k and prove it for p = k + 1. ) and lemma 3. 1. (Pr)fJ))* 3 ( J e k + l (Pk+l) 3 Also, lemma 3. 1. 5 and the definition of a chain symbol will give, (2) ( J e k + 1 (Pk+l)3 ((rt=l Jer(Pr)R)3 rt=1 Jer (pr)')) 3 3 ((I'&+: Jer (P,)R)3 rt2; Jer(Pr)fJ). -I But by the definition of a chain symbol and (1) we have, (3) t- (r,"=+,l Je,(Pr) (R3 f J )3 ) (Jq,+, ( P )3 ((r7k= Jer(Pr)R) 1 Jer(mW. 3 rt=1 3 If we now use (3), ( 2 ) , and A 3 our lemma follows. Lemma 3. 1. 7.
Alice in Puzzle-Land: A Carrollian Tale for Children Under Eighty by Raymond M. Smullyan