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1. A syllogism is an argument made up of three statements. Here is an example:

All Greeks are philosophers.

All wise men are Greeks.

All wise men are philosophers.

The first two statements are called premises and the third is called the conclusion. The conclusion either does or does not follow logically from the premises. If it does, the conclusion is valid. If it does not, the conclusion is invalid. Whether or not the premises are true has nothing to do with the validity of the conclusion. But if the premises are true and the conclusion is valid, then the syllogism is said to be sound.

2. Any sentence generally has a subject and a predicate, and these descriptions are applied to syllogisms in a particular way. The subject (S) of a syllogism is the first element of the conclusion, ‘all wise men’ in our example. The second element of the conclusion, ‘philosophers’, is the predicate (P).

3. The major term is the one that forms the predicate of the conclusion, 'philosophers' in the example. The minor term is the subject of the conclusion, 'all wise men' in the example. The term which does not appear in the conclusion, but appears in both premises, is the middle term, 'Greeks' in the example.

4. The three statements in the example are of a similar pattern. They say that all of one thing is something else. Such a pattern is assigned the letter A. Since both the premises and the conclusion of our example are in this form, we can say that the syllogism as a whole is AAA.

5. However, premises and conclusions don’t have to be on the pattern of ‘All Greeks are philosophers’. They can also appear as ‘No Greeks are philosophers ’, ‘Some Greeks are philosophers’ and ‘Some Greeks are not philosophers’. A premise or conclusion like ‘No Greeks are philosophers’ is assigned the letter E, one that takes the form ‘Some Greeks are philosophers’ is assigned the letter I and one that takes the form ‘Some Greeks are not philosophers’ is assigned the letter O. So a syllogism such as the following, showing in brackets :

Some Greeks are not philosophers.

No wise men are Greeks.

Some philosophers are wise men.

can be described as being OEI. Premises and conclusions that begin with ‘All’ and ‘No’ are called universal affirmatives. Premises that begin with ‘Some’ are called particular affirmatives.

6. The patterns AAA, OEI, and so on are called moods. The variations possible for each syllogism produce 64 different moods. Each of the 64 moods is found in the four figures described below. There are thus 256 possible syllogisms altogether.

7. With our knowledge of subjects, predicates and middles, we can label our AAA example as follows:

All Greeks (M) are philosophers (P).

All wise men (S) are Greeks (M).

All wise men (S) are philosophers (P).

This is a syllogism of the form

All M is P.

All S is M.

All S is P.

This syllogism is valid because the conclusion follows from the two premises. We know from the second premise that all wise men are in the same category as Greeks and we know from the first premise that all Greeks are in the same category as philosophers. All wise men must therefore also be in the same category as philosophers. Any other syllogism having of this mood (AAA) in the First Figure (M-P, S-M, S-P) will be valid.

8. If we change the terms around we can produce:

All philosophers (S) are Greeks (M).

All wise men (P) are Greeks (M).

All philosophers (S) are wise men (P).

This is a syllogism of the form

All S is M.

All P is M.

All S is P.

This is a syllogism of the Second Figure. Is it valid? We know that all philosophers and all wise men are Greeks, but does that tell us that all philosophers are wise men? It does not, because, in spite of their both being Greeks, all philosophers could be stupid men. Any other syllogism having this pattern (AAA) in the Second Figure (S-M, P-M, S-P) will be invalid.

9. Here’s a third way of arranging the sentences:

All Greeks (M) are philosophers (P).

All Greeks (M) are wise men (S).

Therefore all wise men (S) are philosophers (P).

Here we have reversed the order of the second premise as well as the first, so we now have:

All M is P.

All M is S.

All S is P.

This is said to be of the Third Figure Let’s see if it’s valid. The second premise tells us that all Greeks are in the same category as wise men. Does the first premise give us any grounds to conclude that those same wise men in turn be in the same category as philosophers? It doesn’t. It tells us only that all Greeks are philosophers, not that all wise men are.

10. Finally, we can order the sentences in a fourth way:

All philosophers (P) are Greeks (M).

All Greeks (M) are wise men (S).

Therefore all wise men (S) are philosophers (P).

Here, we have reversed the order of both premises and created a premised of the form:

All P is M.

All M is P.

All S is P.

This is of the Fourth Figure and it, too, is invalid. The second premise tells us that all Greeks are wise men. Does the first premise entitle us to conclude that all wise men are philosophers because all Greeks are wise men? No, because there could be some Greeks who are not philosophers.

11. On this site:

• Premises and conclusions of type A (All S is P) are in this colour.

• Premises and conclusions of type E (No S is P) are in this colour.

• Premises and conclusions of type I (Some S is P) are in this colour.

• Premises and conclusions of type O (Some S is not P) are in this colour.

The 15 valid syllogisms are marked thus: VALID. All the others are invalid.

WARNING

I have made this site for my own amusement, so please don't rely on it for any academic or other purpose. It is quite possible that there remain inaccuracies, in spite of repeated checking. If you find any I'd be most grateful if you'd let me know.