Aristotle - Foundations of Logic

Logic - 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.

Propositions and Categories

Aristotle’s writings show that even he realized that there is more to logic than syllogistic. The De interpretatione, like the Prior Analytics, deals mainly with general propositions beginning with Every, No, or Some. But its main concern is not to link these propositions to each other in syllogisms but to explore the relations of compatibility and incompatibility between them. Every swan is white and No swan is white clearly cannot both be true; Aristotle calls such pairs of propositions “contraries.” They can, however, both be false, if—as is the case—some swans are white and some are not. Every swan is white and Some swan is not white, like the former pair, cannot both be true, but—on the assumption that there are such things as swans—they cannot both be false either. If one of them is true, the other is false; and if one of them is false, the other is true. Aristotle calls such pairs of propositions “contradictories.”

The propositions that enter into syllogisms are all general propositions, whether universal or particular; that is to say, none of them is a proposition about an individual, containing a proper name, such as the proposition Socrates is wise. To find a systematic treatment of singular propositions, one must turn to the Categories. This treatise begins by dividing the “things that are said” (the expressions of speech) into those that are simple and those that are complex. Examples of complex sayings are A man runs, A woman speaks, and An ox drinks; simple sayings are the particular words that enter into such complexes: man, runs, woman, speaks, and so on. Only complex sayings can be statements, true or false; simple sayings are neither true nor false. The Categories identifies 10 different ways in which simple expressions may signify; these are the categories that give the treatise its name. To introduce the categories, Aristotle employs a heterogeneous set of expressions, including nouns (e.g., substance), verbs (e.g., wearing), and interrogatives (e.g., where? or how big?). By the Middle Ages it had become customary to refer to each category by a more or less abstract noun: substance, quantity, quality, relation, place, time, posture, vesture, activity, and passivity.

The categories are intended as a classification of both the kinds of expression that may function as a predicate in a proposition and of the kinds of extralinguistic entity such expressions may signify. One might say of Socrates, for example, that he was human (substance), that he was five feet tall (quantity), that he was wise (quality), that he was older than Plato (relation), and that he lived in Athens (place) in the 5th century BCE (time). On a particular occasion, his friends might have said of him that he was sitting (posture), wearing a cloak (vesture), cutting a piece of cloth (activity), or being warmed by the sun (passivity).

If one follows Aristotle’s lead, one will easily be able to classify the predicates in propositions such as Socrates is potbellied and Socrates is wiser than Meletus. But what about the term Socrates in propositions such as Socrates is human? What category does it belong to? Aristotle answers the question by making a distinction between “first substance” and “second substance.” In Socrates is human, Socrates refers to a first substance—an individual—and human to a second substance—a species or kind. Thus, the proposition predicates the species human of an individual, Socrates.(See below Physics and metaphysics: Form.)

Aristotle’s logical writings contain two different conceptions of the structure of a proposition and the nature of its parts. One conception can trace its ancestry to Plato’s dialogue the Sophist. In that work Plato introduces a distinction between nouns and verbs, a verb being the sign of an action and a noun being the sign of an agent of an action. A proposition, he claims, must consist of at least one noun and at least one verb; two nouns in succession or two verbs in succession—as in lion stag and walks runs—will never make a proposition. The simplest kind of proposition is something like A man learns or Theaetetus flies, and only something with this kind of structure can be true or false. It is this conception of a proposition as constructed from two quite heterogeneous elements that is to the fore in the Categories and the De interpretatione, and it is also paramount in modern logic.

In the syllogistic of the Prior Analytics, in contrast, the proposition is conceived in quite a different way. The basic elements out of which it is constructed are terms, which are not heterogeneous like nouns and verbs but can occur indifferently, without change of meaning, as either subjects or predicates. One flaw in the doctrine of terms is that it fosters confusion between signs and what they signify. In the proposition Every human is mortal, for example, is mortal predicated of humans or of human? It is important to distinguish between use and mention—between the use of a word to talk about what it signifies and the mention of a word to talk about the word itself. This distinction was not always easy to make in ancient Greek, because the language lacked quotation marks. There is no doubt that Aristotle sometimes fell into confusion between use and mention; the wonder is that, given his dysfunctional doctrine of terms, he did not do so more often.