## 1  An example and some terminology

A (categorical) syllogism is a basic form of reasoning consisting of three statements. The first two statements are the premises, the third statement is the conclusion. Here is a first example, showing the way we are going to write syllogisms in these notes.

 Some North Americans are tall. All Canadians are North Americans. Some Canadians are tall.

At this point, do not worry wether the reasoning is right or wrong. Notice:

• The conclusion is the final statement, the one below the line. The other two statements are the premises

• The statements refer to three terms, namely Canadians, North Americans, and tall (i.e., tall people). Of these three, one (namely North Americans) appears in both premises and not in the conclusion. That is the middle term.
• Of the remaining two terms, the term tall is in the predicate of the conclusion. The term which is in the predicate of the conclusion is known as the major term and the premise containing it is the major premise.
• The term Canadians is in the subject of the conclusion; such a term is known as the minor term and the premise in which it appears is the minor premise.
• It is customary to write the major premise on top, followed by the minor premise; I'll follow this in the vast majority of cases but, perhaps, not always.
• In the given example, the major premise is a syllogism of type I, the minor premise is of type A, the conclusion is of type I; we say it is an IAI syllogism, or that the mood of the syllogism is IAI.

Notions discussed so far: Syllogism, major term, major premise, minor term, minor premise, conclusion, mood of a syllogism. Do you understand what they all mean? More examples will follow.

## 2  The stripped down syllogism

The basic ingredients of a syllogism are thus the middle term, which we'll denote by M (for Middle), the major term, which we'll denote by P (for Predicate of the conclusion) and the minor term, which we'll denote by S (for Subject of the conclusion). In the example we gave, M is North Americans, P is tall and S is Canadians. Stripped down, our syllogism is

 Some M are P All S are M Some S are P

Syllogisms can refer to cats, birds, Canadians, green cheese on the moon, whatever; but once one removes the icky layers of emotional or affective content, which can obscure the processes of thought, one sees that there are only a few different types of syllogisms. In the first place, there are exactly 4×4×4 = 64 different ways in which we can arrange the letters A, I, E, O as a triple; that is there are precisely 64 different moods. Let us look at one of these moods, say AAA, in which all three statements are universal affirmatives. Consider the major premise. It involves the terms M and P. In how many different ways can you make an A out of these two terms? There are just two ways; you can say All P are M or you can say All M are P. That is it. The same holds for the minor premise; you can say All S are M or you can say All M are S. On the other hand, for the conclusion you don't have a choice at all. If the conclusion is to be an A statement, it must be All S are P. This gives us a total of 2×2×1 = 4 possible syllogisms in the AAA mood. The same holds for each one of the other 64 possible moods, giving a grand total of 64×4 = 256 possible syllogisms. As we shall see, very few of these possibilities are valid syllogisms; i.e., valid forms of reasoning.

## 3  The classification

Since the middle ages, syllogisms have been classified by dividing them into four groups or figures, depending on the position of the terms in the major and minor premise. Consider the following syllogisms.

1.
 All children are cute. All brats are children. All brats are cute.

2.
 All professors are clowns. Some wise people are professors. Some wise people are clowns.

3.
 Some Americans are rich. Some poor people are Americans. Some poor people are rich.

4.
 No Americans are French. All New Yorkers are American. No New Yorkers are French.

5.
 No politician is dishonest. Some liars are politicians. Some liars are not dishonest.

6.
 All cows are green. Some dogs are cows. No green object is a dog.

All of these syllogisms have something in common. Ignoring qualifiers such as all, some, not, no, the scheme for all these syllogisms is:

 M-P S-M S-P

which is known as the first figure. For example, in the first syllogism, the term appearing in both premises (and not in the conclusion) is children. That means that children is the middle term; M = children. Looking at the conclusion, we see that S = brats and P = cute. The syllogism can be abbreviated to

 All M are P All S are M All S are P

Now get rid of the qualifier all, replace the verb by -, and you are left with the first figure scheme given a few lines above.

Exercise. For each one of the six syllogisms:

1. Identify the terms; i.e., identify M, P, and S.
2. Verify that it does indeed belong to the first figure.
3. Determine its mood.
4. The only (classically) valid first figure syllogisms are those of moods

AAA,  EAEAII, and EIO.

(known, respectively, as Barbara, Celarent, Darii, and Ferio). With this information, determine the validity of the syllogism at hand.

Enough about the first figure! Here are the schemes for all four figures.

 M-P S-M S-P

 P-M S-M S-P

 M-P M-S S-P

 P-M M-S S-P
1st     2nd     3rd     4th

## 4  The valid moods

Classically, of all 256 possible syllogisms, only 19 were considered valid. In general, it is better to analyze syllogisms according to well established rules of reasoning to determine their validity (rather than blindly using the classification we are about to see). However, this being a very introductory overview, we'll just blindly classify and limit our more serious analysis to the use of the modified Venn diagrams (see below). The valid syllogisms per figure are given on page 143 of your textbook, using their cutesy names (Barbara, Celarent, etc.-only the vowels matter). Here they are once more.

FigureMood and Cute Name
First Figure
 AAA Barbara EAE Celarent AII Darii EIO Ferio
Second Figure
 AOO Baroco EAE Cesare AEE Camestres EIO Festino
Third Figure
 OAO Bocardo AAI Darapti AII Datisi IAI Disamis EAO Felapton EIO Ferison
Fourth Figure
 AAI Bramantip AEE Camenes IAI Dimaris EAO Fesapo EIO Fresison

THAT A SYLLOGISM IS VALID MEANS THAT IF BOTH PREMISES ARE TRUE, THEN THE CONCLUSION MUST ALSO BE TRUE. The scholastics, as the people who worked on these and other intellectual matter during the middle ages were called, believed that the table above listed ALL valid forms, and ONLY valid forms. However, logic has shifted a little bit and we do not quite consider all listed cases as being valid. We'll discuss this a little bit later. For now, keep in mind that verufying whether a syllogism is valid using the table consisits in figuring out its mood (AAA, AAI, etc.), its figure, and then checking whether it is listed. A computer program could do it (and a link to one that does it is provided here and in the links section of our main web page). So can you.

Exercise Classify each of the following syllogisms by figure and mood, and decide whether it is valid or not (according to the table of valid moods).

1.  Some evergreens are objects of adoration. All evergreens are trees. Some trees are objects of adoration.
2.  Some impractical people are intellectuals. All poets are impractical. Some intellectuals are poets.

3.  All students are bright. No bright person is a litterer. No litterer is a student.

4.  No bright person is a student. All litterers are bright. No litterer is a student.

5.  All well paid people are educated. All teachers are educated. All teachers are well paid.

6.  Some snakes are not venomous. All snakes are reptiles. Some reptiles are not venomous.

7.  No fish is a mammal. Some mammals are aquatic. Some (aquatic) animals are not fish.

8.  No man is an island. All islands are rocky. No man is Rocky.

9.  All horses are equines. All equines are vertebrates. Some vertebrates are horses.

10.  All dogs are mammals. No cat is a dog. No cat is a mammal.

11.  All ants are insects. Some ants have wings. Some winged animals are insects.

12.  Some birds of prey are eagles. All eagles have excellent eye-sight. Some animals with excellent eyesight are birds of prey.

13.  All textbooks are worthy of careful study. No textbook is a work of Shakespeare. No work of Shakespeare is worthy of careful study.

14.  All Toyotas are cars. Some cars are not made by General Motors. Some Toyotas are not made by General Motors.

15.  No motorcycle is a car. Some Hondas are cars. Some Hondas are not motorcycles.

16.  Every honest person is worthy of trust. No liar is worthy of trust. No liar is an honest person.

17.  All Athenians were philosophers. All Athenians were Greek. Some Greeks were philosophers.

18.  All of John's statements are true. Some statements I heard yesterday were not true. Some statements I heard yesterday were not made by John.

19.  No litterer is a bright person. All students are bright. No student is a litterer.

20.  No reptile is a mammal. Some reptiles are carnivorous. Some carnivorous animals are not mammals.

## 5  Some remarks about the power of nonsense

In the middle ages, when people were perhaps more credulous, perhaps more in touch with their inner child, more platonic, perhaps smarter (who knows?), words seemed to have a power of their own. To give something a name was, in a way, to create it. If I said All unicorns are equines, people might say, look at him, he believes in unicorns. And some people believed in them, others had some doubts. We have a somewhat different point of view today. Today any universal statement about nothing is considered to be true. For example All unicorns are green is absolutely true, and so is All unicorns are blue, because there are no unicorns (of course, if it turns out that there are unicorns after all, the situation changes). If something doesn't exist, you can say anything you wish about it, and what you say will be true. The funny thing is that this applies to the universal statements; we are more particular with the particular ones. We interpret nowadays a statement such as Some unicorns are blue as meaning There exists at least one blue unicorn; if there are NO unicorns the statement is automatically false. We won't dig to deep into these matters, only enough to point out that some syllogism moods considered valid in scholastic times, are not considered completely valid anymore. The main examples are the third and fourth figure EAO and AAI syllogisms (Felapton, Fesapo, Darapti, and Bramantip). Here is an example of a fourth figure EAO (Fesapo) syllogism:

 No dog is a bird. All birds are winged. Some winged animals are not dogs.

It seems like a perfectly valid argument. It is valid, because there are birds. However, suppose we replace birds by purple three winged mountain goats and winged by border collie. We get

 No dog is a purple three winged mountain goat. All purple three winged mountain goats are border collies. Some border collies are not dogs.

In mediaeval times, people might have said that the second premise is false because there are no purple three winged mountain goats. Today we see both premises as true, the conclusion as false. We don't discard poor Fesapo altogether; we just say that Fesapo, which follows the scheme

 No P is M All M are S Some S are not P

is valid as long as M does not describe some non-existent class of objects. To put it in the form of an answer to exercise 5 of Section 5.5 of the textbook, if the (set of objects corresponding to the) middle term is not empty.

Exercise. What about Felapton, Darapti, Bramantip?

## 6  Venn Diagrams

In these diagrams, the class corresponding to each term is represented as a circle. In an improvement (maybe) our textbook represents the minor term (S) as a circle, the major term (P) as a square, the middle term as a triangle. A statement All S are M can then be represented by a circle inside of a square. But we would start usually with the major premise, which tells you how to draw the square (P) and the triangle (M). The minor premise tells you how to place the circle (S). To decide now if the syllogism is valid, look at the picture, ignoring the triangle. Is the picture consistent with the conclusion of the syllogism? You have some pictures for Barbara, Celarent and Darii in your text.
Solutions to the exercises in these notes can be found at http://www.math.fau.edu/schonbek/mfla/mfla1f01syl_s.html

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On 20 Apr 2000, 14:06.