80-211 Spring 2003
Assignment #9
Due on Friday, March 21th
Problem 1 (4 pts): Prove the following sequents, using the basic rules for the quantifiers (of course you can use any primitive or derived rule for propositional logic too).
(a) "x(~Ax→Kx),
$y~Ky ├ $w(Aw v ~Lwf)
(b) "x"y((Ry
v Dx)→ ~Ky), "x$y(Ax → ~Ky), $x(Ax v Rx) ├ $x ~Kx
(c) "x$y"zFxyz ├ "x"z$yFxyz
(d) $x$y"zFxyz ├ "z$y$xFxyz
Problem 2 (3 pts): For the following proofs, you can use any
previously proved sequents from the predicate
and propositional calculus. You may find
it useful to call any rule where the negation operator is moved past a
quantifier a quantifier exchange (QE). Some examples of this would
include:
~$x~Fx ┤├ "xFx
$xFx ┤├ ~"x~Fx
~"xFx ┤├
$x~Fx
~$xFx ┤ ├
"x~Fx
(a) "z~Jz, ($y)(Hby v Ryy) → ($x)Jx
├ "y~(Hby
v Ryy)
(b) ~($y)(Ty
v ($x)~Hxy) ├ "x"yHxy & "x~Tx
(c) $xFx → $xGx ├ $x(Fx → Gx)
Problem 3: (3
pts) Show that each of the following sentences is not semantically true. Make
sure you give a quick explanation why (i.e., state briefly what your
interpretation is intended to show).
(a) $xFx→"xFx
(b) "x(Fx→Gx) → "xGx
(c) ~($x)Gx → "y(Fyy→Gy)