CM20019---Computation III:
Formal Logic and Semantics
2007
Exercise sheet 6---Solutions
Problem 1
=========
(1) -(FA X)((toothless(X) ^ feathered(X)) -> fly(X)) .
(2) (EX X)human-in-prison(X) -> (FA X)human-in-prison(X) .
(3) -(FA X)(EX Y)((tall(X) ^ -crazy(X)) ->
(likes(X,Y) ^ (-tall(Y) V -crazy(Y)) ^
-(tall(Y) V crazy(Y))
)) .
(4) (FA X)(FA Y)((gorilla(X) ^ ball(Y)) -> hairier(X,Y)) .
(5) (FA X)(EX Y)(-(X = Y) ^ (love(Y,X) V hate(Y,X)) ^
-(love(Y,X) ^ hate(Y,X))) .
(6) (EX X)(EX Y)(in(X,spain) ^ barber(Y) ^ in(Y,X) ^
(FA Z)(-shaves(Z,Z) -> shaves(Y,Z)) .
(7) (FA X)(-(X = jane) -> happy(X)) .
(8) (EX X)(has-wheel(X) ^ in-shed(X) ^
-(EX Y)(has-wheel(Y) ^ in-shed(Y) ^ -(X = Y)) ^
bicycle(X) ^ has-flat-tyre(X)) .
(9) (FA X)(FA Y)(EX Z)((X <= Z) ^ (Y <= Z) ^
(FA W)(((X <= W) ^ (Y <= W)) ->
(Z <= W))).
Problem 2
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For all formulae, we take I as the domain of natural numbers with all the canonical functions and predicates.
(1) F = even(X) (interpretation: X is even) ;
(2) F = successor(X,Y) (interpretation: Y = X+1) ;
(3) F = even(X), G = odd(X) ;
(4) F = even(X), G = odd(X) .