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:
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.
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.
All children are cute. |
All brats are children. |
All brats are cute. |
All professors are clowns. |
Some wise people are professors. |
Some wise people are clowns. |
Some Americans are rich. |
Some poor people are Americans. |
Some poor people are rich. |
No Americans are French. |
All New Yorkers are American. |
No New Yorkers are French. |
No politician is dishonest. |
Some liars are politicians. |
Some liars are not dishonest. |
All cows are green. |
Some dogs are cows. |
No green object is a dog. |
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:
AAA, EAE, AII, 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.
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1st | 2nd | 3rd | 4th |
Figure | Mood and Cute Name | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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First Figure |
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Second Figure |
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Third Figure |
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Fourth Figure |
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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).
Some evergreens are objects of adoration. |
All evergreens are trees. |
Some trees are objects of adoration. |
Some impractical people are intellectuals. |
All poets are impractical. |
Some intellectuals are poets. |
All students are bright. |
No bright person is a litterer. |
No litterer is a student. |
No bright person is a student. |
All litterers are bright. |
No litterer is a student. |
All well paid people are educated. |
All teachers are educated. |
All teachers are well paid. |
Some snakes are not venomous. |
All snakes are reptiles. |
Some reptiles are not venomous. |
No fish is a mammal. |
Some mammals are aquatic. |
Some (aquatic) animals are not fish. |
No man is an island. |
All islands are rocky. |
No man is Rocky. |
All horses are equines. |
All equines are vertebrates. |
Some vertebrates are horses. |
All dogs are mammals. |
No cat is a dog. |
No cat is a mammal. |
All ants are insects. |
Some ants have wings. |
Some winged animals are insects. |
Some birds of prey are eagles. |
All eagles have excellent eye-sight. |
Some animals with excellent eyesight are birds of prey. |
All textbooks are worthy of careful study. |
No textbook is a work of Shakespeare. |
No work of Shakespeare is worthy of careful study. |
All Toyotas are cars. |
Some cars are not made by General Motors. |
Some Toyotas are not made by General Motors. |
No motorcycle is a car. |
Some Hondas are cars. |
Some Hondas are not motorcycles. |
Every honest person is worthy of trust. |
No liar is worthy of trust. |
No liar is an honest person. |
All Athenians were philosophers. |
All Athenians were Greek. |
Some Greeks were philosophers. |
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. |
No litterer is a bright person. |
All students are bright. |
No student is a litterer. |
No reptile is a mammal. |
Some reptiles are carnivorous. |
Some carnivorous animals are not mammals. |
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?