Aristotle

Aristotle’s most famous contribution to logic is the syllogism, which he discusses primarily in the Prior Analytics. A syllogism is a three-step argument containing three different terms. A simple example is “All men are mortal; Socrates is a man; therefore, Socrates is mortal.” This three-step argument contains three assertions consisting of the three terms Socrates, man, and mortal. The first two assertions are called premises and the last assertion is called the conclusion; in a logically valid syllogism, such as the one just presented, the conclusion follows necessarily from the premises. That is, if you know that both of the premises are true, you know that the conclusion must also be true.

Aristotle uses the following terminology to label the different parts of the syllogism: the premise whose subject features in the conclusion is called the minor premise and the premise whose predicate features in the conclusion is called the major premise. In the example, “All men are mortal” is the major premise, and since mortal is also the predicate of the conclusion, it is called the major term. Socrates” is called the minor term because it is the subject of both the minor premise and the conclusion, and man, which features in both premises but not in the conclusion, is called the middle term.

In analyzing the syllogism, Aristotle registers the important distinction between particulars and universals. Socrates is a particular term, meaning that the word Socrates names a particular person. By contrast, man and mortal are universal terms, meaning that they name general categories or qualities that might be true of many particulars. Socrates is one of billions of particular terms that falls under the universal man. Universals can be either the subject or the predicate of a sentence, whereas particulars can only be subjects.

Aristotle identifies four kinds of “categorical sentences” that can be constructed from sentences that have universals for their subjects. When universals are subjects, they must be preceded by every, some, or no. To return to the example of a syllogism, the first of the three terms was not just “men are mortal,” but rather “all men are mortal.” The contrary of “all men are mortal” is “some men are not mortal,” because one and only one of these claims is true: they cannot both be true or both be false. Similarly, the contrary of “no men are mortal” is “some men are mortal.” Aristotle identifies sentences of these four forms—“All X is Y,” “Some X is not Y,” “No X is Y,” and “Some X is Y”—as the four categorical sentences and claims that all assertions can be analyzed into categorical sentences. That means that all assertions we make can be reinterpreted as categorical sentences and so can be fit into syllogisms. If all our assertions can be read as premises or conclusions to various syllogisms, it follows that the syllogism is the framework of all reasoning. Any valid argument must take the form of a syllogism, so Aristotle’s work in analyzing syllogisms provides a basis for analyzing all arguments. Aristotle analyzes all forty-eight possible kinds of syllogisms that can be constructed from categorical sentences and shows that fourteen of them are valid.

In On Interpretation, Aristotle extends his analysis of the syllogism to examine modal logic, that is, sentences containing the words possibly or necessarily. He is not as successful in his analysis, but the analysis does bring to light at least one important problem. It would seem that all past events necessarily either happened or did not happen, meaning that there are no events in the past that possibly happened and possibly did not happen. By contrast, we tend to think of many future events as possible and not necessary. But if someone had made a prediction yesterday about what would happen tomorrow, that prediction, because it is in the past, must already be necessarily true or necessarily false, meaning that what will happen tomorrow is already fixed by necessity and not just possibility. Aristotle’s answer to this problem is unclear, but he seems to reject the fatalist idea that the future is already fixed, suggesting instead that statements about the future cannot be either true or false.

Aristotle’s logic is one of the most mind-boggling achievements of the human intellect, especially when we bear in mind that he invented the entire field of logic from scratch. His work was not significantly improved upon until the invention of modern mathematical logic in the late nineteenth century. Obviously, Aristotle is not the first person to make use of a syllogism in an argument, and he is not even the first person to reason abstractly about how arguments are put together. However, he is the first person to make a systematic attempt to sort out what kinds of arguments can be made, what their structure is, and how we can prove rigorously whether they are true or false, valid or invalid. His analysis of the syllogism lays bare the mechanics of rational argumentation so that we can see the truth plainly through the many layers of rhetoric, ambiguity, and obscurity. With the proper analysis, Aristotle tells us, any argument can be laid out as a series of simple and straightforward statements, and its validity or invalidity will be obvious.

Aristotle’s logic rests on two central assumptions: the fundamental analysis of a sentence divides it into a subject and a predicate, and every sentence can be analyzed into one or more categorical sentences. Aristotle identifies four kinds of categorical sentences and distinguishes each by the way the subject relates to the predicate. In other words, the way in which subject and predicate are connected is what allows us to distinguish one kind of sentence from another. Furthermore, Aristotle argues that, at heart, there are only four kinds of sentences. Every variation that we see in ordinary human speech is just one categorical sentence, or a combination of several, with window dressing to make it look less plain. With these twin assumptions, Aristotle can show that there are only forty-eight possible kinds of arguments that can be made—fourteen of them are valid and thirty-four of them are invalid. In theory, he has given us a foolproof map: with sufficient analytical skill, we can reduce any argument to a series of simple subject–predicate sentences of four different kinds and then quickly determine whether the combination of these sentences produces a valid or an invalid inference.

Modern mathematical logic departs from Aristotle primarily by recognizing that the subject–predicate form of grammar is not the fundamental unit of logical analysis. Bertrand Russell famously uses the example of the sentence, “the present king of France is bald” to show that, on Aristotle’s logic, we are committed to accepting that the phrase “the present king of France” has a clear meaning, which leads to all sorts of difficulties. A modern logician would analyze that same sentence as being a combination of three smaller sentences: “there is a person who is the present king of France,” “there is only one person who is the present king of France,” and “that person is bald.” We know that there is no king of France, so we can immediately see that the first of these three sentences is false and don’t need to worry about the complications of accepting “the present king of France” as a subject in a syllogism.

The fundamental insight that there is more to logic than subject–predicate analysis opens the way for several other important blows to Aristotle’s logic, primarily that the categorical sentence is not the only kind of sentence and that the syllogism is not the only form of argument. There are a number of kinds of sentence that cannot be analyzed into one or more categorical sentences, most notably sentences that contain other sentences (“If you are over forty or have false teeth then you will not enjoy candy as much as a ten-year-old unless you have recently undergone surgery”), sentences that express relations (“My left foot is bigger than my right foot”), and sentences that involve more than one quantifier (“No people love all people who hate some people”). These sentences can be easily analyzed with the technical machinery of modern logic but only by accepting that they can fit into nonsyllogistic arguments. The first and the third examples of noncategorical sentences just given contain more than two terms and so cannot fit into a syllogism. Logical deductions can be made from them in combination with other premises, but the conclusion may take many more than two steps to reach.