Syllogisms

A syllogism is a logical argument in which a conclusion is reached based on some known facts which are usually stated in if-then or all form. This kind of logic played a central role in ancient Greek mathematics and is the essence of proof in mathematics today. Aristotle (around 350 BC) wrote extensively about logic and syllogisms, and this forms the basis of western culture.

Here is an example of a syllogism. If we know the following facts:

1. All teachers are in the library.

2. Everyone in the library is watching Dr. T’s lecture.

3. Ms. Jones is a teacher.

Then we can reach any of the following three conclusions:

(a) Ms. Jones is in the library.

(b) Ms. Jones is watching Dr. T’s lecture.

In other words, we just follow the string of statements in a common-sense way. However, we must be cautious in some cases. Many people confuse a statement with its converse. For example, if we also know this statement is true:

4. Mr. Smith is in the library.

then we can not conclude that Mr. Smith is a teacher. The first statement says all teachers are in the library, but it does not say that everyone in the library is a teacher.

Sometimes you have to use the contrapositive to reach a conclusion. The contrapositive is important to understand, because it is the basis of the logic used in certain proofs. Such proofs are called “indirect” since they rely not on the given form of a statement, but on its contrapositive.

The logic of an indirect proof is essentially this: If you want to prove B follows from A, try assuming B is not true and show that this leads to the conclusion that A is not true. Thus to prove “If A then B” you instead prove “If not B then not A,” which is equivalent.

Here is an example of the use of the contrapositive (or an indirect proof) to arrive at a conclusion. Suppose the following two facts are true:

1. Everyone in the library is watching Dr. T’s lecture.

2. Murphy is not watching Dr. T’s lecture.

We can conclude that Murphy is not in the library, because the first statement is equivalent to, “If someone is in the library then that person is watching Dr. T’s lecture,” and this in turn is equivalent to its contrapositive, “If someone is not watching Dr. T’s lecture then that person is not in the library.”

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