The Five Most Common Arguments

Five Common Arguments

One of the things a logic student learns is that, of the 64 possible kinds of arguments (also called syllogisms), only 19 of them are valid. Let’s take the most common argument form of all:
PREMISE #1: All flowers are plants (A)
PREMISE #2: All roses are flowers (A)
CONCLUSION: Therefore, all roses are plants (A)

This argument form is called a “BARBARA” (for reasons we will explain in a minute). If we converted each of the terms into a letter (S=roses, P=plants, and M=flowers), the argument above would look like this:

PREMISE #1: All M is P
PREMISE #2: All S is M
CONCLUSION: Therefore, all S is P

When we put an argument into symbols like this, we call it the “form” of the argument. It is just an easy way to see how the argument is structured.

In the BARBARA syllogism above, all the statements are A statements, which just means they are constructed in the form “All S is P.” There are three other kinds of statements: E statements use the form “No S is P,” I statements use the form “Some S is P,” and O statements use the form “Some S is not P.” If we take into account all four of these kinds of statements, we could come up with the 64 possible kinds of arguments.

But of these 64 kinds of arguments, only 19 are valid. Validity just means that the argument is structured so that, if the premises are true, the conclusion must also necessarily be true. Of the 64 possible arguments, only 19 of them are structured this way. In the other forms—the invalid forms—there is no guarantee that the conclusion is true if the premises are true.
In the Middle Ages, a philosopher named William of Sherwood gave names to these 19 valid syllogisms and put them in a four-line verse (the most common ones are in bold):


The names are not accidental. The vowels and some of the consonants in each name tell us something about each syllogism. The advanced logic student learns what these letters mean. But the most basic thing they tell you is what kind of statements make up each argument. A BARBARA, for example (like the one above), is made up of three A statements.

Note that all three statements in our first argument—the two premises and the conclusion—are A statements: BARBARA. We could also call it an AAA argument, or, simply, an AA argument, since it is according to the kind and arrangement of the premises by which we categorize arguments.
You will see both A and E statements (but not I and O statements, curiously) in the five most common argument forms. In addition to BARBARA, there are:

No flowers are trees (E)
All roses are flowers (A)
Therefore, no roses are trees (E)

No trees are flowers (E)
All roses are flowers (A)
Therefore, no roses are trees (E)

All roses are flowers (A)
No trees are flowers (E)
Therefore, no trees are roses (E)

All roses are flowers (A)
No flowers are trees (E)
Therefore, no trees are roses (E)

Notice that the middle term (the term in both of the premises but not the conclusion) is in bold in the arguments above. It is the position of the two middle terms that dictated on what line William of Sherwood put these syllogisms in his verse. The first line of the verse includes only syllogisms in which the middle term is the subject of the first premise and the predicate of the second premise—like the CELARENT above and the BARBARA we showed in the beginning of this article. Syllogisms in which the middle terms are in the predicates of both premises—like the CESARE and the CAMESTRES above—are in the second line. The third line of Sherwood’s verse includes only those in which the middle term is in the subjects of both premises, and the fourth only those with the middle terms in the predicate of the first premise and the subject of the second (like the CAMENES above).

Of these five common argument forms, the first, BARBARA, is the most common. Most of the arguments in Euclid’s Elements, for example, the great classical work on analytic geometry, are BARBARAs.

There is no obvious reason why these five arguments should be the most commonly used, although, if we analyzed them for long enough, we would see that they are conceptually the most simple and straightforward.

Many logic students only learn enough of the subject to test arguments for validity. They would see one of the 64 forms, perform seven tests, and see if the argument passes all of these tests.

But imagine how much simpler it would be for the student if he had simply memorized the 19 valid forms (the ones in William of Sherwood’s verse). He wouldn’t have to worry about all those validity tests: He would simply look at the argument, and see if it was one of the 19 forms he had memorized. If it was, then it would be valid; if not, then it would be invalid.

This is what the logic student who sticks with the subject will learn. He will have mastered a difficult subject, and will end up with a simple procedure for analyzing arguments.

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