THINKING WELL AND THINKING LOGICALLY
by Mark Hodes
I. The Thesis
The literature of skepticism has generated its share of cliches.
Skeptics claim to use scientific reasoning, to think logically,
to be guided by evidence in forming beliefs as though these are
clear-cut descriptions of well-understood activities. Thinking
logically, by implication, is taken as an explication of thinking well
or clearly. Those who think logically are regarded as less likely to
err by adopting false beliefs or rejecting true ones. I will argue
that thinking logically is not all it's cracked up to be, and that
thinking logically is not the same as thinking well.
In the propositional calculus of elementary symbolic logic, p -> q,
material implication, has the following truth table:
p q p->q
1. T T T
2. T F F
3. F T T
4. F F T
If most live birds fly, then 8 - 3 = 5 (true).
If most live birds fly, then 8 - 3 = 4 (false).
If pigs fly, then 8 - 3 = 5 (true).
If pigs fly, then 8 - 3 = 4 (true).
Line 3 is true because the sentence makes no claim if in fact pigs do
not fly. If in line 4 it seems strange that F -> F is T, consider the
sentence "If I am president of France, then I have a chauffeur." I
assure my readers that both the hypothesis and conclusion are false,
though intuitively (as well as logically) the sentence is true.
The next step in my argument is to consider the CONTRAPOSITIVE of
p -> q. This is the sentence (not q) -> (not p). For example, if an
exemplar of p -> q is "If something is a dog, then it is an animal",
then the exemplar of (not q) -> (not p) would be "If something is not
an animal, then it is not a dog." A conditional sentence and its
contrapositive are logically equivalent. Their truth tables are:
1 2 3 4 5 6
p q p->q (not q) (not p) (not q)->(not p)
T T T F F T
T F F T F F
F T T F T T
F F T T T T
The entries in columns 4 and 5 are derived as the opposites of the
entries in 2 and 1, respectively. The matching entries in columns 3
and 6 demonstrate the logical equivalence of the sentences p -> q and
(not q) -> (not p).
In case you still are not convinced of the logical equivalence of a
sentence and its contrapositive, select a conditional sentence and try
to imagine a world in which it is true, but its contrapositive is
false (or vice versa). Your inability to do this should convince you
of their synonymity.
A primary epistemological activity of science is the generation of
probably true or well-confirmed generalizations from evidence. The
more extensive and diverse the evidence, the more soundly based are
the generalizations. A single apparently disconfirming instance can,
however, jeopardize any general statement.
The following hackneyed example is often given as a model for
inductive generalization. You observe a crow and notice that it is
black. You seek out many other crows and find (surprise) that they,
too, are black. You conclude, tentatively, that all crows are black.
As the years roll by, you have occasion to observe other black crows,
and never observe a crow that is not black. Your observations of black
crows occur under widely varying conditions. Each new observation in
the absence of disconfirming instances strengthens your belief that
all crows are black.
Originally, of course, being black was not a defining characteristic
of crows. After years of study, you reformulate the concept of
crowness by including melanism amongst the theoretical baggage of
being a crow. You are awarded the Nobel Prize for Avian Trivia and
retire to Woodshole, where you spend your declining years wiring
flowers to the grave of Burt Lancaster to commemorate his portrayal of
the Bird Man of Alcatraz.
IV. Is Science Logical?
The general statement involved in our example of generalization is "If
something is a crow, then it is black." We have seen that this
statement is logically equivalent to its contrapositive, "If something
is black, then it is not a crow", or, more simply, "All non-blacks are
non-crows." Now, if science really proceeds logically, any evidence
that tends to confirm a statement should tend to confirm any logically
equivalent statement to exactly the same extent. Similarly,
disconfirming evidence should be equipotent in respect to a statement
and its contrapositive. We have arrived at the end of my garden path.
So you see the paradox?
V. The Paradox
Our Nobel Laureate could have confirmed his hypothesis by observing
non-blacks and noticing that they never turn out to be crows. He
could, for example, have gone to Sears, inventoried all non-black
items, noticed that none was a crow, and proclaimed his tentative
conclusion, "All crows are black." Notice that if Sears does not sell
parrots, the generalization "All parrots are black" is equally well
Receiving a grant from William Proxmire, our savant could then have
toured the shopping malls of Europe, sampling non-black merchandise,
finding no crows, and proclaiming, "All non-blacks are non-crows", or,
equivalently, "All crows are black."
Now, you may feel that I have cheated in some way, because non-blacks
are so much more numerous than crows. I am not sure how this affects
the prinicple involved, but it does create an apparent asymmetry
between the original condition and its contrapositive. Consider,
however, an astronomer attempting to support her thesis, "All type G
stars have planetary systems."
Lacking the funding for adequate telescope time, she peers from the
roof of her observatory onto the used car lot below. She proceeds by
examining all Chevrolets failing to have planetary systems, and notes
that none of them is a type G star. Given the frequency-of-repair
records of GM cars, there are surely more type G stars than
Chevrolets. So, here the numerical asymmetry is in the opposite
There is some rationale for our stargazer's procedure. The more
diverse the confirmatory evidence, the more effective the
confirmation. Chevrolets are more diverse (remember the '59 Impala?),
though less numerous, than type G stars. Nevertheless, we would be a
bit uncomfortable accepting our astronomer's conclusion on the basis
of such evidence. She would be laughed off the roof of the
VI. The Lineus Bottomus
Our conclusion is that sentences that are logically equivalent are not
necessarily epistemologically equivalent. This would come as no
surprise to serious students of science, for the philosophical and
psychological literature is replete with such examples. Ours is merely
a cautionary tale for those who believe that thinking logically is an
adequate explication of thinking well. Sorry, Mr. Spock.