Aristotle’s Logic
Syllogism- an instance of a form of reasoning in which a
conclusion is drawn (whether validly or not) from two given or
assumed propositions
Inference- a conclusion reached on the basis of evidence and
reasoning.
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.