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Examples 7.1
Natural Deduction
, the topic we’ll be concerned with for the next bit of the course, is
the most efficient method for establishing validity (the thing we’re most concerned with
in logic).
Here, we derive a conclusion of an argument from the premises via a few steps
which appeal to
rules of inference
.
Truth tables were another way, you’ll recall, of
analyzing the validity of arguments—it was also really clumsy and tedious.
So think of
deduction as your friend.
There are 18 rules of inference and replacement, plus some
rules of conditional proofs, that will suffice to derive any conclusion of any valid
argument that can be symbolized in symbolic logic.
For the first section, however, we’ll
only be concerned with 4 of the rules.
A proof in natural deduction has the following form: there is a sequence of propositions
which is either a premise of the argument, a premise derived from other premises, or
finally the conclusion.
So, a question you’ll be asked to answer will look like the
following:
1.
A
⊃
B
2.
~A
⊃
(C v D)
3.
~B
4.
~C
/
D
The conclusion (here ‘D’) will then be derived with appropriate justification (don’t worry
about the justification just now—when I give the rules of inference you can look back
and it will make sense).
1.
A
⊃
B
2.
~A
⊃
(C v D)
3.
~B
4.
~C
5.
~A
1,3 MT
6.
(C v D)
2,5 MP
7.
D
4,6 DS
So, you number the premises and the conclusion and then justify, by the rules we’ll
introduce, the steps you take to produce the conclusion.
Thus, what we’ve shown here is
that, given the four premises above, we can validly argue the conclusion symbolized by
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 Spring '08
 Barrett

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