A TRIVIU M
The medieval curriculum comprised seven liberal arts, divided into the
lower-level trivium (grammar, logic, and rhetoric) and the upper-level
quadrivium (geometry, astronomy, arithmetic, and music).
meant three roads, then it meant crossroads, then commonplace
(since common people hang around crossroads), and finally trifling or
The etymology is, in a sense, apt: with the exception of
astronomy, none of the liberal arts is about anything.
They don't explain
plants or animals or rocks or people; rather, they are intellectual tools
that can be applied in any realm.
Like the students who complain that
algebra will never help them in the real world, one can wonder whether
these abstract tools are useful enough in nature for natural selection to
have inculcated them in the brain. Let's look at a modified trivium: logic,
arithmetic, and probability.
"Contrariwise," continued Tweedledee, "if it was so, it might be, and if it were
so, it would be; but as it isn't, it ain't. That's logic!"
Logic, in the technical sense, refers not to RATIONALITY in general but to
inferring the truth of one statement from the truth of other statements
based only on their form, not their content.
I am using logic when I reason
If P, Q.
P and Q.
Therefore, P is true.
P or Q.
If P Q,
I can derive all these truths not
knowing whether P means
i. There is a unicorn in the garden.
iii. My car has been eaten by rats.
Does the brain do logic?
College students' performance on logic problems
is not a pretty sight.
There are some archeologists, biologists, and
chess players in a room.
None of the archeologists are biologists.
the biologists are chess players.
What, if anything, follows?
A majority of
students conclude that none of the archeologists are chess players,
which is not valid.
None of them conclude that some of the chess players
are not archeologists, which is valid.
In fact, a fifth claim that the
premises allow no valid inferences.
Spock always did say that humans are illogical.
But as the psychologist
John MacNamara has argued, that idea itself is barely logical.
rules of logic were originally seen as a formalization of the laws of
That went a bit overboard.
Logical truths are true regardless of
how people think.
But it is hard to imagine a species discovering logic if
its brain did not give it a feeling of certitude when it found a logical
There is something peculiarly compelling, even irresistible, about
If P, Q.
With enough time and patience, we discover
why our own logical errors are erroneous.
We come to agree with one
another on which truths are necessary.
And we teach others not by force
of authority but Socratically, by causing the pupils to recognize truths by
their own standards.
People surely do use some kind of logic.
All languages have logical
terms (or "formal devices", as Grice called them at Harvard) like "not", "and", "if", "same", "equivalent", and "opposite".
Children use "not", "and", "or," and "if" appropriately before they turn three, not only in English but in
half a dozen other languages that have been studied.
are ubiquitous in human thought, particularly when we understand language.
Here is a simple example from the psychologist Martin Braine:
John went in for lunch.
The menu showed a soup-and-salad special, with
free beer or coffee.
Also, with the steak you got a free glass of red wine.
John chose the soup-and-salad special with coffee, along with something
else to drink.
(a) Did John get a free beer?
(Yes, No, Can't Tell) ---- NO.
(b) Did John get a free glass of wine?
(Yes, No, Can't Tell) ----- NO.
Virtually everyone deduces that the answer to (a) is no.
of restaurant menus tells us that the or in free beer or coffee implies "not both"—you get only one of them free.
If you want the other, you have to
pay for it.
Farther along, we learn that John chose coffee.
premises "not both free beer and free coffee" and "free coffee," we
derive "not free beer" by a logical inference.
The answer to question (b)
is also no.
Our knowledge of restaurants reminds us that food and beverages
are not free unless explicitly offered as such by the menu.
therefore add the conditional
"if not steak, no free red wine."
chose the soup and salad, which suggests he did not choose steak.
conclude, using a logical inference, that he did not get a free glass of
Logic is indispensable in inferring true things about the world from
piecemeal facts acquired from other people via language or from one's
Why, then, do people seem to flout logic in stories
about archeologists, biologists, and chess players?
One reason is that logical words in everyday languages like English
are ambiguous, often denoting several formal logical concepts.
word "or" can sometimes mean the logical connective OR (A or B or
both) and can sometimes mean the logical connective XOR (exclusive or:
A or B but not both).
Wood notes this in his review in "Mind". "OR" implicates "XOR".
The context often makes it clear which one the
speaker intended, but in bare puzzles coming out of the blue, readers
can make the wrong guess.
Another reason is that logical inferences cannot be drawn out willynilly.
Any true statement can spawn an infinite number of true but useless
ii. Iowa grows soybeans.
we can derive
iv. Iowa grows
soybeans or the cow jumped over the moon.
v. Iowa grows soybeans and
either the cow jumped over the moon or it didn't.
ad infinitum. (This is
an example of the "frame problem").
has all the time in the world, even the best logical inferencer has to guess
which implicatures to explore and which are likely to be blind alleys.
Some rules have to be inhibited, so valid inferences will inevitably be
The guessing can't itself come from logic.
Generally, it comes
from assuming that the utterer is a Griceian cooperative conversational partner
conveying relevant information and not, say, a hostile lawyer or a tough-grading
logic professor trying to trip one up. (Grice taught logic, but he never tried to trip Strawson up).
Perhaps the most important impediment is that mental logic is not a
hand-held calculator ready to accept any As and B's and C's as input.
is enmeshed with our system of knowledge about the world.
step of mental logic, once set into motion, does not depend on world
knowledge, but its inputs and outputs are piped directly into that knowledge.
In the restaurant story, for example, the links of inference alternate
between knowledge of menus and applications of logic.
Some areas of knowledge have their own inference rules that can
either reinforce or work at cross-purposes with the rules of logic.
famous example comes from the psychologist Peter Cathcart Wason.
inspired by the philosopher Sir Karl Popper's ideal of scientific reasoning: a
hypothesis is accepted if attempts to falsify it fail.
Wason wanted to see
how ordinary people do at falsifying hypotheses.
In his A3D7 'card selection task', Wason told ordinary people that a set
of cards had letters on one side and numbers on the other, and asked
them to test the 'if' utterance
vi. If a card has a vowel on one side, it has an odd number on the
-- a simple If P, Q statement.
The Wason subjects were shown 4 cards and were asked which ones they would have to turn over to see if
the rule was true.
Most people choose either the D card or the D card and the 3 card.
correct answer is D and 7.
~(p ⊃ q) iff p & ~q.
The 3 card is irrelevant.
The 'if' utterance iplicates that Ds have 3s, not that 3s
The 7 card is crucial.
If it had a D on the other side, the 'if' utterance would be false.
Only about five to ten percent of the people who are
given the test select the right cards.
Even people who have taken logic
courses get it wrong
Incidentally, it's not that people interpret "If D, 3" as "If D, 3 and vice versa."
If they did interpret it that way
but otherwise behaved like logicians, they would turn overall four cards.
Dire implications were seen.
John Q. Public was irrational, unscientific,
prone to confirming his prejudices rather than seeking evidence that
could falsify them.
But when the arid numbers and letters are replaced with real-world
events, sometimes — though only sometimes — people turn into logicians.
You are a bouncer in a bar, and are enforcing the rule "If a person is
drinking beer, he must be eighteen or older."
You may check what people
are drinking or how old they are.
Which do you have to check: a beer
drinker, a Coke drinker, a twenty-five-year-old, a sixteen-year-old?
people correctly select the beer drinker and the sixteen-year-old.
mere concreteness is not enough.
vii. If a person eats hot chili
peppers, he drinks cold beer.
is no easier to falsify than the D's and
L. Cosmides discovers that people get the answer right when the 'if' utterance is a contract, an exchange of benefits.
In those circumstances,
showing that the rule is false is equivalent to finding cheaters.
is an implication of the form "
viii. If you take a benefit, you must meet a
Cheaters take the benefit without meeting the requirement.
Beer in a bar is a benefit that one earns by proof of maturity, and
cheaters are underage drinkers.
Beer after chili peppers is mere cause
and effect, so Coke drinking (which logically must be checked) doesn't
Cosmides shows that people do the logical thing whenever
they construe the Ps and Qs as benefits and costs, even when the
events are exotic, like eating duiker meat and finding ostrich eggshells.
It's not that a logic module is being switched on, but that people are
using a different set of rules.
These rules, appropriate to detecting
cheaters, sometimes coincide with logical rules and sometimes don't.
When the cost and benefit terms are flipped, as in
ix. If a person pays $20,
he receives a watch.
people still choose the cheater card (he receives
the watch, he doesn't pay $20) — a choice that is neither logically correct
nor the typical error made with meaningless cards.
In fact, the very same
story can draw out logical or non-logical choices depending on the
reader's interpretation of who, if anyone, is a cheater.
x. If an employee
gets a pension, he has worked for ten years.
Who is violating the rule?
people take the employee's point of view, they seek the twelve-year workers
If they take the employer's point of view, they seek
the eight-year workers who hold them.
The basic findings have been
replicated among the Shiwiar, a foraging people in Ecuador.
The mind seems to have a cheater-detector with a logic of its own.
When standard logic and cheater-detector logic coincide, people act like
When they part company, people still look for cheaters.
gave Cosmides the idea to look for this mental mechanism?
It was the
evolutionary analysis of altruism.
does not select public-mindedness.
A selfish mutant would quickly
outreproduce its altruistic competitors.
Any selfless behavior in the natural
world needs a special explanation.
One explanation is reciprocation.
A creature can extend help in return for help expected in the future.
favour-trading is always vulnerable to cheaters.
For it to have evolved, it
must be accompanied by a cognitive apparatus that remembers who has
taken and that ensures that they give in return.
The evolutionary biologist
Robert Trivers had predicted that humans, the most conspicuous
altruists in the animal kingdom, should have evolved a hypertrophied
Cosmides appears to have found it.
Cosmides, L. Deduction or Darwinian algorithms? An explanation of the "elusive"
content effect on the Wason selection task. Ph.D. dissertation, Harvard University.
So is the mind logical in the logician's sense?
Sometimes yes, sometimes
A better question is, Is the mind well-designed in the biologist's
Here the "yes" can be a bit stronger.
Logic by itself can spin
off trivial truths and miss consequential ones.
The mind does seem to
use logical rules, but they are recruited by the processes of language
understanding, mixed with world knowledge, and supplemented or
superseded by special inference rules appropriate to the content.