**Syllogistic**

Aristotle’s claim to be the founder of logic rests primarily on the
Categories, the De interpretatione, and the Prior Analytics, which deal
respectively with words, propositions, and syllogisms. These works, along with
the Topics, the Sophistical Refutations, and a treatise on scientific method,
the Posterior Analytics, were grouped together in a collection known as the
Organon, or “tool” of thought.

The Prior Analytics is devoted to the theory of the syllogism, a central
method of inference that can be illustrated by familiar examples such as the
following:

Every Greek is human. Every
human is mortal. Therefore, every Greek is mortal.

Aristotle discusses the various forms that syllogisms can take and
identifies which forms constitute reliable inferences. The example above
contains three propositions in the indicative mood, which Aristotle calls “propositions.”
(Roughly speaking, a proposition is a proposition considered solely with
respect to its logical features.) The third proposition, the one beginning with
“therefore,” Aristotle calls the conclusion of the syllogism. The other two
propositions may be called premises, though Aristotle does not consistently use
any particular technical term to distinguish them.

The propositions in the example above begin with the word every;
Aristotle calls such propositions “universal.” (In English, universal propositions
can be expressed by using all rather than every; thus, Every Greek is human is
equivalent to All Greeks are human.) Universal propositions may be affirmative,
as in this example, or negative, as in No Greek is a horse. Universal
propositions differ from “particular” propositions, such as Some Greek is
bearded (a particular affirmative) and Some Greek is not bearded (a particular
negative). In the Middle Ages it became customary to call the difference
between universal and particular propositions a difference of “quantity” and
the difference between affirmative and negative propositions a difference of
“quality.”

In propositions of all these kinds, Aristotle says, something is
predicated of something else. The items that enter into predications Aristotle
calls “terms.” It is a feature of terms, as conceived by Aristotle, that they
can figure either as predicates or as subjects of predication. This means that
they can play three distinct roles in a syllogism. The term that is the
predicate of the conclusion is the “major” term; the term of which the major
term is predicated in the conclusion is the “minor” term; and the term that
appears in each of the premises is the “middle” term.

In addition to inventing this technical vocabulary, Aristotle introduced
the practice of using schematic letters to identify particular patterns of
argument, a device that is essential for the systematic study of inference and
that is ubiquitous in modern mathematical logic. Thus, the pattern of argument
exhibited in the example above can be represented in the schematic proposition:

If A belongs to every B, and B
belongs to every C, A belongs to every C.

Because propositions may differ in quantity and quality, and because the
middle term may occupy several different places in the premises, many different
patterns of syllogistic inference are possible. Additional examples are the
following:

Every Greek is human. No human
is immortal. Therefore, no Greek is immortal.

Some animal is a dog. Some dog
is white. Therefore, every animal is white.

From late antiquity, triads of these different kinds were called “moods”
of the syllogism. The two moods illustrated above exhibit an important
difference: the first is a valid argument, and the second is an invalid
argument, having true premises and a false conclusion. An argument is valid
only if its form is such that it will never lead from true premises to a false
conclusion. Aristotle sought to determine which forms result in valid
inferences. He set out a number of rules giving necessary conditions for the
validity of a syllogism, such as the following:

At least one premise must be
universal.

At least one premise must be
affirmative.

If either premise is negative,
the conclusion must be negative.

Aristotle’s syllogistic is a remarkable achievement: it is a systematic
formulation of an important part of logic. From roughly the Renaissance until
the early 19th century, it was widely believed that syllogistic was the whole
of logic. But in fact it is only a fragment. It does not deal, for example,
with inferences that depend on words such as and, or, and if…then, which,
instead of attaching to nouns, link whole propositions together.

## No comments:

## Post a Comment