that the problem of fi nding the shortest in this sentence, “John is taller
than Fred and Fred is shorter than Peter” is the same as fi nding the darkest
in this sentence, “Mary is fairer than Ann and Ann is darker than Mandy.”
Concrete operational thinking deals with objects. Formal operational
thinking deals with truth values. This allows the logic used in concrete
operational thinking, the logic of classes and relations, to be replaced by
propositional logic. Second, the formal operational thinker is freed from
the immediate situation to consider a wide range of possibilities, and it
is the consideration of what might be rather than what is which explains
adolescent predilection for idealism. 5
Propositional logic is expressed by the connections between proposi-
tions. Take the propositions “it is raining” and “I shall put up an umbrella.”
When we try to work out the relationship between raining and whether
or not the umbrella is up, we notice that each proposition can exist in two
forms (raining or not raining and putting up an umbrella or not putting
up an umbrella). Connections between propositions can also exist in two
forms, conjunction (“and,” that is, both are jointly true) and disjunction
(“or,” that is, one is true). Altogether there are sixteen possible combina-
tions of A, B, C, and D, including that none of them is true and that all
of them are true. 6
The concrete operational child does not think of all the combinations
of variables in any particular system. He or she typically focuses on only
one variable. The formal operational adolescent, however, appreciates that
several variables may be relevant to a particular problem and can appreci-
ate how they might combine and all the possible combinations that might
occur. The evidence for this change is given by a series of classic Piagetian
experiments. One involves fi ve liquids which have to be mixed in a par-
ticular way to obtain a particular color. The formal thinker is able to work
systematically through the possibilities until arriving at the answer. He or
she “undertakes the coordination of concrete groups into a single system
(of the second degree) because it deals with possible combinations and no
longer with objects directly.” 7
The shift to formal operational thinking is as a result of a series of
transformations which take place on the structures of concrete operations.
Concrete operational thinking is the necessary foundation for formal
operational thinking and, for this reason, the sequence of mental stages
cannot be varied. Each stage is characterized by a corresponding set of
structures, but the structures of formal operational thinking are the most
PHILOSOPHY AND DEVELOPMENTAL PSYCHOLOGY: RELEVANCE... 271