lianx4 - Notation fij denotes the i-th j-ary function...

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3 Notation: f j i denotes the i -th j -ary function symbol, and p j i the i -th j -ary predicate symbol. 1. Pick out the free and bound occurrences of variables in the following formulas: (a) x 3 ( x 1 p 2 1 ( x 1 , x 2 ) p 2 1 ( x 3 , a 1 )); (b) x 2 p 2 1 ( x 3 , x 2 ) → ∀ x 3 p 2 1 ( x 3 , x 2 ); (c) ( x 2 x 1 p 3 1 ( x 1 , x 2 , f 2 1 ( x 1 , x 2 )) ∨ ¬∃ x 1 p 2 1 ( x 2 , f 1 1 ( x 1 )) . 2. Translate the following sentences into formulas: (1) Anyone who is persistent can learn logic. (2) No politician is honest; (3) Not all birds can fly, and all birds cannot fly. (4) If everyone can solve the problem, Hilary can. (5) Nobody loves a loser. (6) Everyone loves somebody and no one loves everybody, or somebody loves everybody and someone loves nobody. (7) You can fool some of the people all of the time, and you can fool all the people some of the time, but you cannot fool all the people all the time. (8) John hates all people who do not hate themselves.
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