1 The Birth of Science - MSRI

7.4 Propositional Logic 191
Regarding the dream of incest between son and mother, Artemidorus
talks as if he knew of and agreed with Freud’s ideas about
the Oedipus complex. 49
7.4 Propositional Logic
Aristotle devoted much attention to logic, and syllogisms in particular.
But in his analysis of the various forms of syllogisms their validity was
justified only through the evidence provided by examples. In other words,
he described the use of logic, but he did not formulate a theory thereof, in
our sense of the word.
The first steps toward a scientific theory of logic seem to have been taken
around 300 B.C. by Diodorus Cronus and Philo the Dialectician. The latter
is known to have defined the conditional proposition “if p then q” as
“not (p and not q)”. 50 But although his awareness of the need for a crisp,
unambiguous definition of the conditional seems to herald proofs of theo- page 242
rems in logic, on the whole the existing testimonia point to Chrysippus as
the founder of the scientific theory of propositional logic. While Aristotle
adopted variables to represent generic terms in propositions, Chrysippus
used them to stand for the propositions themselves, and constructed a
theory of logical inference based on five postulates. 51 If we use p and q to
denote generic propositions, the five inference rules assumed by Chrysippus
as undemonstrated correspond to the five lines below; the first two entries
of each line are premises and the last says what can be deduced from
them.
1. if p then q p q
2. if p then q not q not p
3. not (p and q) p not q
4. p or 52 q p not q
5. p or 52 q not p q
From these five postulates an unlimited number of inference schemes
could be deduced as theorems. But none of Chrysippus’ theorems in logic
has been preserved.
49 [Artemidorus/Musatti], p. 17.
50 Sextus Empiricus, Adversus dogmaticos, II, 113; Pyrrhoneae hypotyposes, II, xi, 110.
51 These five postulates are listed by Diogenes Laertius (Vitae philosophorum, VII, 79–81) and by
Sextus Empiricus (Pyrrhoneae hypotyposes, II, xiii, 157–158). Another passage of Sextus (Adversus
dogmaticos, II, 224–227) reports only the first three postulates but gives more information about
theorems in logic.
52 Here “or” is understood in the exclusive sense: either p or q is true but not both.
Revision: 1.9 Date: 2002/10/11 23:59:33