PHI 108.04
Logical and Critical Reasoning
Spring 2008
Hurley,
Logic,
and Munson and Black,
The Elements of Reasoning
were used as sources for the above information.
1
The Basics of Arguments III
1.
NECESSARY AND SUFFICIENT CONDITIONS
A
necessary condition
is a condition which
must
be present in order for a given result to follow.
For
example, there are a number of necessary conditions that govern whether or not our car will start moving
when we press the accelerator.
There must be gas in the car, the car must be turned on, the engine must
be working, etc.
A necessary condition is
necessary
to bringing about a result, but does not guarantee that
result.
A
sufficient condition
is one which, whenever it is present,
always
produces a given result. If my Saturn SL2
lacks an engine, this is a sufficient condition for knowing that it will not work properly.
A sufficient
condition is just that
–
it is
sufficient
, it is all that is needed, to bring about a certain result.
These terms can be applied to some of the statement forms we
’
ve looked at so far.
For example, in the
statement
“
P
→ Q”
, we know that if P is true then Q must also be true.
Therefore, P is sufficient to
guarantee Q; in other words, P is a sufficient condition for Q.
On the other hand, Q is a necessary condition
for P: Q must be true, in order for P to be true.
Q is necessary for P.
2.
EQUIVALENT FORMS
In evaluating arguments symbolically, we must use the following rules establishing equivalent (and
therefore substitutable) argument forms.
An equivalent form is another way of stating the same thing.
Many of these will look like common sense, but it is necessary to know them in their standard forms.
a.
Double Negation (DN)
--P
↔
P
This is certainly common sense: a thing is the opposite of its opposite; P is not the opposite of P; P
is not not-P.
b.
Commutation (Com)
(P
• Q
)
↔
(Q
• P)
This allows us to switch the two terms of a conjunction:
“
P and Q
”
is the same as
“
Q and P
”
.
It also
applies to disjunctions,
“
either/or
”
statements.
We can switch their terms because the order of
terms in
“
and
”
and
“
either/or
”
statements doesn
’
t matter.
(On the other hand, this is not true in
the case of implication:
“
If Q then P
”
is
not
the same as
“
If P then Q.
”
)
For example,
Albert has a pet lizard and a pet monkey.
…
is equivalent to
…
Albert has a pet monkey and a pet lizard.
or
(P v Q)
↔ (Q v P
)
Either James will win the game or Marie will.
…
is equivalent to
…
Either Marie will win the game or James will.
c.