Categorical syllogisms are sets of three categorical propositions. The first two are given and presumed to be true. These first two categorical propositions are called premises. The third categorical proposition is the conclusion. The third categorical proposition is in the form _____ S is (___) P.

The first categorical proposition is called the major premise and introduces P and a term that will be called M, the middle term. The second categorical proposition is called the minor premise and includes S and M.

A categorical proposition is termed "valid" if the premises are sufficient support to prove the conclusion true. The premises are always presumed to be true. To avoid confusing oneself, the use of factually true premises is useful when examining a syllogism.

There are four types of categorical proposition, the universal affirmative A, the universal negative E, the particular affirmative I, and the particular negative O.

A: All rocks are hard things.

E: No rocks are hard things

I: Some rocks are hard things

O: Some rocks are not hard things [Not every rock is a hard thing]

Syllogisms consist of three of these, any three. The order in which the three occur specifies the mood of the syllogism. Consider the following EAO syllogism

E: No women named Sepe are outer island Yapese women

A: All outer island Yapese women are weavers of the *gulfoy*

O: Some weavers of the *gulfoy* are not women named Sepe.

In the above syllogism the subject S is *weavers of the gulfoy*, the predicate P is *women named Sepe* and the middle term M is *outer island Yapese women*.

When analyzing a syllogism, always remember that the premises are taken to be true, whether or not they are factually true.

The above syllogism is term an EAO syllogism. Note the the middle term is the predicate of the major premise and the subject of the minor term. This could be abstracted in the following notation.

```
Major premise: No P is M
Minor premise: All M is S
Conclusion: Some S is not P
```

As long as there are women named Sepe, outer island women, and *gulfoy* weavers, the above syllogism is indeed valid. In modern logic, the particular proposition imply the existence of actual members of the class, while the universal propositions do not imply the existence of members of the class. In simplified terms, the issue is what happens when no one knows how to make the *gulfoy*. When that knowledge is lost, then there the conclusion, which demands existing members, is false because there are no weavers.

Categorical syllogism have four possible figures depending on the position of the middle term. The "flying brick" is a good way to remember the four figures. The flying brick refers to the possible positions of the middle term without regard to quantity. The following is a picture of the flying brick.

Seen more abstractly:

M M M M \ / M M M M

Now with the predicate and subject terms introduced.

```
M-P P-M M-P P-M
\ /
S-M S-M M-S M-S
S-P S-P S-P S-P
First Second Third Fourth
Figure Figure Figure Figure
```

Mood is AAA, figure I.

```
Major premise: All M is P
Minor premise: All S is M
Conclusion: All S is P
```

All Micronesians are Pacific Islanders

All Kosraens are Micronesians

All Kosraens are Pacific Islanders

When reading a syllogism, think of the statements this way:

Given that "All Micronesians are Pacific Islanders" is true and

Given that "All Kosraens are Micronesians" is true,

Therefore "All Kosraens are Pacific Islanders" is true.

In the above example the middle term M, the term in common to the major and minor premise, is Micronesians. The predicate P is Pacific Islanders. The subject S is Kosraens.

To diagram this syllogism start by laying out the major premise All Micronesians are Pacific Islanders or All M is P.

Since all M is P, there is no M that is not P. Thus the part of M outside of P can be "erased" or marked out.

The third circle added is the subject circle S. In the diagram below the S circle has been temporarily laid "on top" to show the full circle.

"All S is M" is diagrammed by "erasing" or blacking out all areas of S that are not inside the M circle.

Note that the areas of M blacked out by the major premise remain blacked out. This results in a section of S also being blacked out - the lower left "triangle" where S and M intersect outside of the P circle.

Look at the Venn diagram and consider whether the diagram supports the conclusion that All S is P. The only area of S that remains is the center triangle of the three circles. And this triangle is wholly inside of P. All S is indeed P.

Note that the three categorical syllogisms can be a mixture from any of the four types. There are three categorical propositions in each syllogism and four types or 4^{3} = 64 possible combinations (moods). With four figures possible for each of 64 moods there are 256 total possible arrangements of mood and figure. For no particular reason these examples first consider triplets of identical types of categorical propositions in figure 1.

```
Major premise: No M is P
Minor premise: No S is M
Conclusion: No S is P
```

[Minor pre] No bats are cats

[Major pre] No rats are bats

[Therefore] No rats are cats

This sounds reasonable. Yet consider the EEE-1 syllogism;

[Minor pre] No fish are birds

[Major pre] No golden plovers are fish

[Therefore] No golden plovers are birds

Both premises are factually true, yet the conclusion is not true. Golden plovers are birds is the true statement. The syllogism at first appeared to be valid, but a second example shows that EEE-1 is not valid for all possible S, P, and M. This failure means that EEE-1 is invalid.

Thinking of all the possible examples is difficult, this is where Venn diagrams help.

There are no fish (M) that are birds (P).

No S is M

For the first example involving bats (M), cats (S), and rats (P) the area where the k is located above is empty: there are no rats that are cats. For the second example involving fish (M), birds (S), and golden plovers (P), golden plovers are birds, so there are elements at k. The result is that either j or k might be empty of elements. When there are elements at k, then the conclusion that No S is P is false. This means the whole EEE-1 categorical syllogism is considered invalid.

Although for the purposes of this discussion the English language common sense presumption of existential import is being retained, in modern logic there is the possibility that S and P lack existential import and thus both j and k could be empty.

Note that if No S is P then No P is S. That is, the universal negative categorical proposition is valid when the subject and predicate are interchanged. The interchange of the subject and predicate is called conversion. Only the universal negative E and the particular affirmative I convert. This conversion for E means that all possible positions of S, P, and M reach the same result for EEE. In all four figures the syllogism is invalid.

This will later be given as a general and broader rule that includes the particular negative O: no syllogism with two negative premises is valid.

```
Major premise: Some M is P
Minor premise: Some S is M
Conclusion: Some S is P
```

[Minor pre] Some rocks are in the water

[Major pre] Some coral are rocks

[Therefore] Some coral are in the water.

Note that the particular affirmative I proposition converts. That is, if some S is P, then it is also true that some P is S. If *some rocks are in the water*, then it is also the case that *some in the water [things] are rocks*. Thus whether III-1 is valid or invalid will also hold for III-2, III-3, and III-4.

Note that where a predicate is an adjective or adjective phrase, this course uses the approach of attaching the implicit [things] to convert the proposition. In modern logic there are problems of existential import and the empty class to be considered.

```
Some in the water [things] are rocks
Some coral are rocks
[Therefore] Some coral are in the water
```

The other two figures are left to the reader.

This sounds reasonable. Yet consider the III-1 syllogism:

```
Some people who wear blue jeans are men
Some women are people who wear blue jeans
[Therefore] Some women are men.
```

One appears to be valid, the other appears to be invalid. A Venn diagram helps sort out the cause of this contradiction visually. The major premise, *Some M is P* is diagrammed below.

The minor premise,* Some S is M* is diagrammed below.

Note that the X's are placed on the line. This is intentional. The X being on the line signifies that the "some" things could be on either side of the line. In one arrangement the result appears to be a valid syllogism, in the other arrangement the result is an invalid syllogism. That we cannot decide between the two possibilities will mean that that syllogism is considered invalid.

In the diagram on the left below, coral is a rock in the water, hence the "X" is placed in the intersection of all three circles. On the right below the diagram has been altered with additional information. The additional proposition is the reality that *No women are men*. This removes the possibility of "some" elements in the common intersection, thus the minor and major premise have to marked in the lower left intersection of S and M but not P, and in the lower right intersection of P and M but not S.

When diagramming I and O propositions, the X's are placed on the appropriate line to remind us that the X could be on either side of the line. If placing the X on one of the sides leaves the conclusion unsupported, then the syllogism is invalid.

```
Major premise: Some M is not P
Minor premise: Some S is not M
Conclusion: Some S is not P
```

[Minor pre] Some rocks are not in the water

[Major pre] Some wood are not rocks

[Therefore] Some wood are not in the water.

Diagram the major premise

Diagram the minor premise. Be very careful. *Some wood are not rocks* means placing the X outside of the rocks circle but inside of the wood circle. This is not easy or obvious to either the beginner nor sometimes the more experienced Venn diagrammer. Think about where the X should go if there was no predicate P circle. Imagine only the S and M circles being present.

Then combine the two preceding Venn diagrams into a single Venn diagram.

For some wood to not be in the water there would have to be an X in the lower left triangle at the intersection of wood and rocks. The X on the line at the bottom of the triangle, could be either in the wood circle or outside the wood circle in the rocks circle. In the abstract where *Some M is not P* and *Some S is not M*, there is no way to assure validity. OOO-1 is not valid.

The following is a reproduction at twice the scale of the above. This proved useful in determining that the version of Amaya for Fedora Core 5 in late December 2006 was not supporting scale and rotate transforms, only translate transformations. Note that Amaya 9.52 for windows does support these transforms.

Amaya on Fedora Core 5 supports MathML despite the apparent lack of the Mathematica fonts. How Amaya accomplishes this while FireFox 1.5.0.9 fails to do so is a puzzle as of 07 January 2007.

Compound document comments • Geometry • Lee Ling • COMFSM