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Section 15:
Formal Deductions
, Barker, pp. 9498.
***Read the section, then the following additional comments.
Try not to let the process of formal deductions intimidate you too much.
The key to it is not so
much one of skill as perception; viz. to learn to look at a line and see what potential it might
carry for a simple argument or an equivalence.
Ultimately, you may have to go through a bit of
trial and error before being able to complete some of the deductions.
There is no substitute for being familiar with the simple arguments and equivalences.
If you get
stuck, take time out to review the material on the fly leafs of the inside front cover or on pp.
9996.
Remember, they all work.
Keep in mind that you can always insert a tautology.
Every step needs to come with a
justification.
You can’t just add a line just because it seems “obviously” true.
If so, you should
be able to prove it with a justification.
You can’t perform simple arguments on parts of lines.
If you have two lines, such as
G e (C > F)
and
C
you can’t just do a modus ponens and get
G e F.
The reason is that, even though “C” is considered true, you don’t know whether “C > F” is true
since it only appears as a part of a disjunction.
Remember, all of the exercises in this section are valid.
Let’s now look at an example.
We want to prove that the following argument is valid:
If either Johnny is very quiet or he took a good nap, then he will not be sent to bed on time.
He
will be sent to bed on time.
Therefore, he is not very quiet.
We symbolize this argument as:
1.
(Q e N) > B
premise
[
∴
Q]
2.
B
premise
1
LESSON 8
Truthfunctional Logic, Part 3
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View Full DocumentLooking at the premises, we can see that B is the consequent of the first line, and B is given in
the second line.
So we can perform a modus tollens.
3.
(Q e N)
from 1, 2 by modus tollens
Now, if we have our mind geared to recognizing equivalences, we should see that here we have
potential for application of DeMorgan’s law.
So we get:
4.
from 3 by DeMorgan’s law
Now we can simplify this expression:
5.
Q
from 4 by conjunctive simplification.
There’s our conclusion already.
Here’s another example:
It is not the case that we will go to the zoo if and only if we drive to Ft. Wayne.
We do go to the
zoo.
Therefore, we do not drive to Ft. Wayne.
1.
(Z = D)
premise
[
∴
D]
2.
Z
premise
Clearly, there’s not much we can do until we break up premise 1.
Then we get by equivalence:
3.
from 1 by equivalence
Now we can apply the reverse of DeMorgan’s law to this expression.
4.
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 Spring '11
 BrentKelly

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