Methods to Prove Logical Implications
The standard form of an argument (or theorems) can be
represented using the logical symbols we have learned so far:
(p
1
∧
p
2
∧
p
3
∧
... p
n
)
⇒
q
Essentially, to show that this statement is always true (a
tautology), we must show that if all of p
1
through p
n
are true,
then q must be true as well.
Another way to look at this is that we must show that if q is
ever false, then at least ONE of p
1
through p
n
must be false as
well. (This is the contrapositive of the original assertion.)
Let’s look at an example of how you might go about proving a
statement in a general form, given some extra information.
Let p, q, are r be the following statements:
p: Rudy loses the immunity challenge.
q: Kelli lets go of the pole during the immunity challenge.
r: Rich will win Survivor.
Let the premises be the following:
p
1
: If Kelli does not let go of the pole during the immunity
challenge, then Rudy will lose the immunity challenge.
p
2
: If Rudy loses the immunity challenge, then Rich will win
Survivor.
p
3
: Kelli did not let go of the pole during the immunity
challenge.
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Now, I want to show that
(p
1
∧
p
2
∧
p
3
)
⇒
r
First, I must express p
1
, p
2
, and p
3
in terms of p, q, and r.
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 Fall '09
 Logic, Rudy, Kelli, immunity challenge

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