Symbolize
and provide derivations to show the equivalence for the following pairs
of sentences.
1. Not everyone who studies will pass. Someone who did not
pass had studied..
~(x)(Sx -> Px)
and (Ex)(~Px & Sx)
1.
~(x)(Sx -> Px) \show
(Ex)(~Px & Sx)
2.
(Ex)(Sx & ~Px) 1, QN
3.
(Ex)(~Px & Sx) 2, Com
1.
(Ex)(~Px & Sx) \show ~(x)(Sx -> Px)
2.
~(x)(~Px -> ~Sx) 1, QN
3.
~(x)(Sx -> Px) 2,
Trans
2. Unless everyone is ambitious there will be individuals who do
not
succeed. If everyone succeeds then everyone was ambitious.
(x)Ax v (Ex)~Sx
and (x)Sx -> (x)Ax
1.
(x)Ax
v (Ex)~Sx \show (x)Sx
-> (x)Ax
2.
(Ex)~Sx v (x)Ax
1, Com
3.
~(Ex)~Sx -> (x)Ax 2, Impl
4.
(x)Sx ->
(x)Ax
3, QN
1.
(x)Sx ->
(x)Ax
\show (x)Ax v (Ex)~Sx
2.
~(x)Sx v
(x)Ax
1, Impl
3.
(Ex)~Sx v
(x)Ax
2, QN
4.
(x)Ax v
(Ex)~Sx
3, Com
Do note that other derivations can be
presented in which the steps are taken in a different order. All
that matters is that any step is justified by looking back to a call
line earlier in the proof.
