Section 1.3
which we check with a truth table.
p
q
8
T
T
T
T
F
T
F
T
T
F
F
T
T
T
F
T
F
F
F
T
F
F
F
F
pVq
p'>8
T
T
T
T
T
T
F
T
T
F
T
F
T
T
F
T
qV8
T
T
T
T
T
F
T
F
*
*
*
*
23
There are four rows when the premises are true and, in each case, the conclusion is also true. The
argument is valid.
(d) Since
«
,q)
"
r)
¢:::=}
,
(q
V
(,r))
and ,
(p
"
8)
¢:::=}
[(
,p)
V
('8)]
¢:::=}
(p
'>
('8)),
the
premises become
(q
V
(,r))
'>
p
P
'>
('8)
so the chain rule gives
(q
V
(,r))
'>
('8),
which is logically equivalent to
(,(
q
V
(,r)))
V
('8),
which is the desired conclusion by a law
of
De
Morgan.
4. (a) [BB] Since
[(,r)
V
(,q)]
¢:::=}
[q
'>
(,r)],
the first two premises give
p
'>
(,r)
by the chain
rule. Now
,p
follows by modus tollens.
(b) This argument is not valid.
If
q
and
r
are false,
p
and
8
are true, and
t
takes on any truth value,
then all premises are true, yet the conclusion is false.
(c) [BB] This argument is valid. Since
p
+t
(t
V 8), we can replace
t
V 8 with
p
so that the premises
become
p
V
(,q), p
'>
(p
V
r), (,r)
V
p.
The first
of
these is logically equivalent to
q
'>
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 Summer '10
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 Logic, Graph Theory, Modus ponens, Modus tollens, Argument form

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