Rules of Inference

Before examining how people reason deductively, two rules of inference
must be considered: modus ponens (the “method of putting,” which involves
affirming a premise) and modus tollens (the “method of taking,” which involves
negating a premise). Considering P and Q as content-free abstract
variables (much like algebraic variables), modus ponens states that given “P
implies Q” and given P, one can infer Q. In the following example, applying
modus ponens to 1 and 2 (in which P is “it rained last night” and Q is “the game
was canceled”), one can infer 3.
1. If it rained last night, then the game was canceled.
2. It rained last night.
3. The game was canceled.
Modus tollens states that given “P implies Q” and ~Q (read “not Q”; “~” is a
symbol for negation), one can infer “~P.” Applying modus tollens to 1 and 4,
one can infer 5.
4. The game was not canceled.
5. It did not rain last night.
In general, people apply modus ponens properly but do not apply modus
tollens properly. In one experiment, four cards showing the following letters
or numbers were placed in front of subjects:
E K 4 7
Subjects saw only one side of each card but were told that a letter appeared
on one side and a number on the other side. Subjects were asked to judge
the validity of the following rule by turning over only those cards that provided
a valid test of the following statement: If a card has a vowel on one side,
then it has an even number on the other side. Turning over E is a correct application
of modus ponens, and turning over 7 is a correct application of modus
tollens (consider P as “vowel on one side” and Q as “even number on the
other side”). Almost 80 percent of subjects turned over E only or E and 4,
while only 4 percent of subjects chose the correct answer, turning over E and
7. While many subjects correctly applied modus ponens, far fewer correctly applied
modus tollens. Additionally, many subjects turned over 4, an error called
affirmation of the consequent.
When stimuli are concrete, reasoning improves. In an analogous experiment,
four cards with the following information were placed before subjects:

beer Coke 16 22
One side of each card showed a person’s drink; the other side showed a person’s
age. Subjects evaluated this rule: If a person is drinking beer, that person
must be at least nineteen. In this experiment, nearly 75 percent of the
subjects made the correct selections, showing that in some contexts people
are more likely to apply modus tollens properly.
When quantifiers such as “all,” “some,” and “none” are used within syllogisms,
additional errors in reasoning occur. People are more likely to accept
positive conclusions to positive premises and negative conclusions to negative
premises, negative conclusions if premises are mixed, a universal conclusion
if premises are universal (all or none), a particular conclusion if premises
are particular (some), and a particular conclusion if one premise is
general and the other is particular. These observations led to the atmosphere
hypothesis, which suggests that the quantifiers within the premises
create an “atmosphere” predisposing subjects to accept as valid conclusions
that use the same quantifiers.