p3.pdf

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P
∨ ¬Q
,
R
→ ¬P
Q
→ ¬R
We want to show that
P
∨ ¬Q,
R
→ ¬P
At rst I explain how to nd the proof.
Q
→ ¬R.
P
∨ ¬Q
,
R
→ ¬P
Q
→ ¬R
e rst premiss is
P
∨ ¬Q.
I try to
apply
∨Elim.
us I assume
P
and
¬Q
P
∨ ¬Q
P
¬Q
P
∨ ¬Q
,
R
→ ¬P
Q
→ ¬R
I need to show
Q
→ ¬R.
I hope to
get this by an application of
→Intro.
So I should try to prove
¬R
under
the assumption
Q.
First I look at the
case
P.
I hope to get
¬R
by
¬Intro;
so I assume
R.
R
P
∨ ¬Q
P
¬Q
P
∨ ¬Q
,
R
→ ¬P
Q
→ ¬R
R
is can be combined with the pre-
miss
R
→ ¬P
to obtain
¬P.
R
P
∨ ¬Q
P
R
→ ¬P
¬P
¬Q
P
∨ ¬Q
,
R
→ ¬P
Q
→ ¬R
¬P
yields a contradiction with the
assumption
P.
So I can conclude
¬R
and discharge
R
by applying
¬Intro.
[R]
P
∨ ¬Q
P
¬R
R
→ ¬P
¬P
¬Q

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