Exercise 4.2 - Answers

1

(i)

There's no need to do it all in one step. We might proceed thus:

First step: $x[x is an undergraduate who worked less hard than John Dx]

Next: $x[[UxHax]Dx]

(Altenatively: "x[[UxHax]Dx])

 
(ii)

Definitely worth taking slowly.

First step: "x[x is an undergraduate who passed all their exams Dx]

Next: "x[[Ux x passed all x's exams]Dx]

Finally: "x[[Ux"y[EyxPxy]]Dx]

 
(iii)

First step: "x[x is an undergraduate who did not pass all their exams Hax]

Next step: "x[[Ux"y[EyxPxy]]Hax] (or, "x[[Ux$y[EyxPxy]]Hax]

 
(iv)

(First step: "x[Dxx is an undergraduate who worked harder than John] and $x[x is an undergraduate who worked harder than John Dx] and $x[x is an undergraduate who worked harder than John Dx]

Next step: [["x[Dx[UxHxa]]$x[[UxHxa]Dx]]$x[[UxHxa]Dx]]

(Or: [[$x[Dx[UxHxa]]$x[[UxHxa]Dx]]$x[[UxHxa]Dx]])

Notice that, because none of the occurrences of "$x" were within the scope of any other quantifier containing "x" we could stick to the same variable throughout.

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