Consider
the statement
A
prisoner is paroled only if the prisoner obeys the rules.
Notice
that if the prisoner does not obey the rules, he will not be paroled. This is the contrapositive
of the statement
If
the prisoner is paroled then he obeyed the rules.
So
a statement P only if Q is the same as P Þ Q.
The
following are several other ways of saying P Þ Q.
q follows from p.
Not
p unless q.
Whenever
p, q.
q whenever p.
p
is sufficient for q.
q
is necessary for p.
p
is a sufficient condition for q.
q is a necessary condition for p.
The main application of a
truth table is to determine whether an argument is valid. An argument consists of two parts:
1. Given statements called the premises.
2. The conclusion.
An argument is said to be valid
if the conclusion follows logically from the premises. In other words, if the
premises imply the conclusion. So
we take the premises and connect them using conjunction and then use this
compound statement as the antecedent of a conditional statement of which the
conclusion is the consequent. If this
conditional is a tautology, then the argument is valid.
Here is an example:
If a given figure is a square, then
it is a rectangle.
The figure is a square.
These are the premises. The first is called major premise and
the second the minor premise.
Therefore, the figure is a
rectangle.
This is the conclusion.
The argument may be
symbolized using P = A given figure is a square, Q = It is a rectangle. Then the argument is
( P Þ Q ) Ù P Þ Q
The argument in #32 is one of
the basic laws of logic: the law of
detachment. If the statement P
Þ Q is true and P is known to be true, the
statement Q must be true.
Notice that a valid argument
may have either a true or a false conclusion. The truth or falseness of the conclusion does
not determine the validity of the argument.
Also the validity of an argument does not guarantee the truth of its
conclusion. However, if the premises are
true, a valid argument has to have a true conclusion. If you have an argument in which all the
premises are true and the conclusion is false, then the argument is invalid.
The argument in #34 is one of
the basic laws of logic: the law of
contraposition.
If the statement P Þ Q is true
and ~ Q is known to be true, then ~ P must be true.
The argument in #36 is a
common form of incorrect reasoning called the fallacy of affirming the
consequent.
There are two other common
valid forms of argument. The law of
syllogism says that if the statement P Þ Q is true
and Q Þ R is true, then the
statement P Þ R is true.
The last common valid
argument is the disjunctive syllogism.
If the statement P Ú Q is true and ~ P is true, then Q
is true.
There is one other invalid
argument that is commonly made. It is
the fallacy of the inverse. In
this argument, P Þ Q is true and ~ P is
true. It is then concluded that ~ Q is true. Exercise #40 shows that this is an invalid
argument.
Below
are the symbolic forms of the four valid arguments and the two invalid
arguments.
Valid Arguments
Law of Contraposition (P Þ Q ) Ù (~Q) Þ ~ P
Law of Detachment (P Þ Q ) Ù P Þ Q
Law of Syllogism (P Þ Q ) Ù (Q Þ R) Þ (P Þ R)
Disjunctive Syllogism (P Ú Q)
Ù
(~ P) Þ Q
Invalid Arguments
Affirming the Consequent (P Þ Q) Ù Q Þ P