Lecture 3: Formal Fallacies
Necessary and Sufficient
Conditions. First, a little vocabulary. We have already seen a number of argument
forms which involve conditional statements―if
p, then q. First we will look a little bit
at the structure of conditional statements.
A conditional is composed of two clauses, p and q,
related by the connective ‘if, then’.
The first clause (i.e., the clause that follows the ‘if’) is known as
the antecedent of the conditional; the second clause (i.e., the
clause that follows the ‘then’) is known as the consequent.
Now consider the meaning of a conditional. Suppose I say, ‘If we go backpacking, then it
will rain.’ What I am saying is that the
antecedent (i.e., that we go backpacking) is enough to make it true that
it will rain (but be careful here, it doesn’t assert that there is a causal
connection). In general, we will say
that the antecedent of a conditional is a sufficient condition
for its consequent. Thus, our going
backpacking is a sufficient condition for its raining (at least, if my
statement is true). This is why modus ponens is deductively valid.
Notice also that, if my statement about backpacking
is true, then if we do go backpacking, it must rain. In general, the consequent of a conditional
is a necessary condition for its antecedent. Thus, its raining is a necessary condition
for our going backpacking. Without the rain, no backpacking. This is why modus tollens
is deductively valid.
Finally, there are other ways of saying ‘if p, then
q’ in English. For instance, I could say
‘q, if p’ (e.g., ‘It will rain, if we go backpacking). Logically speaking, ‘q, if p’ means exactly
the same thing as ‘if p, then q.’ Similarly, I could say ‘p only if q’
(e.g., ‘We will go backpacking only if it rains’). Again, logically this just
means ‘if p, then q’.
Given these alternative ways of saying ‘if p, then
q’, what does it mean if I say ‘p if and only if q’? (Try it before continuing.) Well, this claim breaks down into two claims:
‘p, if q’ and ‘p only if q’. The first
of these (i.e., ‘p, if q’) means the same as ‘if q, then p’ (check this). In this case, q is a sufficient condition for
p. The second part of
the claim (i.e., ‘p only if q’) just means ‘if p, then q’. In this case, q is a necessary condition for
p. Thus, ‘p if and only if q’ means that
q is both necessary and sufficient for p.
If q is both necessary and sufficient for p, then p
and q are logically equivalent―one is true if
the other is, and conversely.
One final, but important,
point.
Suppose that I give you a biconditional one side of which is a complex
conjunctive statement (e.g., x is a bachelor v x is unmarried, x is an adult, and x is a male). In
this case, only the entire
conjunction
is sufficient for the left hand side. In such cases, we say that the complex
conditions are jointly
sufficient for
the left-hand side (e.g., being unmarried AND being an adult AND being
unmarried are jointly sufficient for being a bachelor). Taken one by one, the
conditions are not, however, sufficient (e.g., being a male isn’t sufficient
for being a bachelor). But notice that they are individually necessary (e.g.,
being unmarried is necessary for being a bachelor).
Formally: p v (q Ù r Ù
s). Here q, r and s are jointly necessary and sufficient for p; they are not
individually sufficient (thought they are individually necessary).
It is absolutely essential that you get necessary
and sufficient conditions straight when arguing. Failure to do so, gives rise to the following
two logically fallacious arguments.
Affirming the Consequent. A fallacy is a type of argument
that may seem to be correct, but that proves, on examination, to involve an
error of reasoning.
The fallacy of affirming the consequent is exactly
what the name says: it is an argument in which one asserts a conditional,
asserts the consequent of the conditional, and then concludes that the antecedent
of the conditional is true. So the
argument has the following form:
1.
If p, then q.
2.
q
–––––
3.
\ p.
Here is an example:
1.
If George Bush is president, then he lives in the White
House.
2.
George Bush lives in the White House.
––––––
3.
\ George Bush is president.
Even though this argument form is superficially
similar to the deductively valid argument forms modus ponens
and modus tollens, it is not itself valid. In order to show this, we need to describe a
counterexample. A counterexample
is a situation in which both of the premises are true while the conclusion is
false. Do this.
We can see why affirming the consequent is a fallacy
by thinking in terms of necessary and sufficient conditions. The first premise of the argument (i.e., ‘if
p, then q’) asserts that p is a sufficient condition for q. But no claim is made to the effect that p is
a necessary condition for q―there might well be
other conditions that are sufficient for q.
The fallacy of affirming the consequent, therefore, goes wrong because
it mistakenly treats p as a necessary condition, rather than merely a
sufficient condition.
Denying the Antecedent. A related fallacy, the fallacy of denying
the antecedent, occurs when one asserts a conditional, denies the
antecedent of the conditional, and concludes by denying the consequent. Thus:
1.
If p, then q.
2.
Not-p.
––––––
3.
\ Not-q.
Here is an example:
1.
If Al Gore is president, then he lives in the White House.
2.
Al Gore is not president.
–––––––
3.
\ He doesn’t live in the White House.
Again, you should be able to show why this argument
is fallacious by producing a counterexample.
Like the fallacy of affirming the consequent, the
fallacy of denying the antecedent confuses necessary and sufficient
conditions. When someone commits this
fallacy they are mistakenly treating p as a necessary condition for q. But the first premise asserts only that p is
a sufficient condition for q.