The Inverse Error
Jeffrey Martinez
Math 170 Dr. Lipika Deka
10/15/13
Inverse Error
Also known as the fallacy of denying the antecedent (Epp p.
58). The inverse error is a reasoning error that an inexperienced
logician may fall into, since in at least one case the conclusion can be
true given the premises, but not in all cases.
The premises present a given implication, and then state that
the hypotheses, or antecedent is false. This step is called denying the
antecedent. The conclusion then states that since the antecedent is not
true, the consequent must not be true.
Although at first glance the argument looks valid, but can be
soon shown invalid by using truth tables, or by counterexample.
We see an example of this error in exercise #29 in section 2.3
of our Discrete Mathematics and Applications textbook. We can see
the problem on the next slide.
Section 2.3 #29
Use symbols to write the logical form of each argument. If the argument is
valid, identify the rule of inference that guarantees its validity. Otherwise, state
whether the converse or the inverse error is made.
29. Premise #1: If at least one of these two numbers is divisible by 6, then the
product of these two numbers is divisible by 6.
Premise #2: Neither of these two numbers is divisible by 6
∴The product of these two numbers is not divisible by 6.
Solution: Symbolically the initial implication (Premise #1) is of the basic form If
R, then S, or R→S, letting
R = At least one of these two numbers is divisible by 6.
S = The product of these two numbers is divisible by 6.
Premise #2 states that R is not true or ~R is true.
The conclusion states that S is not true, or ~S is true.
The simple answer is that the argument is invalid, because the inverse
error is made. But lets look at truth table to prove this answer.
Looking at the truth table below, we identify our premises. We look
for instances when both our premises R → S and ~R are true. In the fourth row,
our premises are true and our conclusion ~S is true. But in the third row we see
that even though R → S and ~R are true, ~S is not true.
R
S
~R
~S
R→S
T
T
F
F
T
T
F
F
T
F*
F
T
T
F
T
F
F
T
T
T
*R → S is only false when R is true, and S
is false. This means that S could be true
even when R is false, which is important
to our solution.
To state this as a counterexample, given the original implication “If at
least one of these two numbers is divisible by 6, then the product of these two
numbers is divisible by 6.” We let the two numbers be 2 and 3. Both 2 and 3
are NOT divisible by 6 (~R), but their product(2x3=6) IS divisible by 6 (S).
Therefore the argument is invalid by counterexample.
This is known as the inverse error, because ~R →~S is the inverse of
our original implication. But we cannot use the inverse, because an implication
and it’s inverse are not logically equivalent.
Therefore, the conclusion is false, and the argument is invalid due to
the inverse error.