Categorical Syllogisms

Now you know the basics about soundness, validity, counterexample, and categorical syllogisms. You learned that, if an argument is valid, there will be no case in which all the premises are true while the conclusion is false.  You also learned a basic method (counterexample) for showing that an argument is invalid. Here you’ll learn another method for proving whether a given categorical syllogism is valid or invalid. First, though, we have to learn a little more about categorical syllogisms themselves.

All categorical syllogisms consist of exactly three statements (or propositions): there are two premises and one conclusion. Also, in a categorical syllogism, there are exactly three terms (such as the terms A, B and C, below; or the terms “cats”, “trees” and “dogs”, in the example below that):

All A are B
All B are C
All A are C   

All oaks are trees*

All trees are plants

All oaks are plants

*This is not in the standard form that we’ll discuss below.

You already know that arguments are made up of statements. In categorical syllogisms, there are only four kinds of statements. These are called standard form categorical statements or propositions, as follows:

1) All S is/are P

2) No S is/are P

3) Some S is/are P

4) Some S is not/are not P

(Notice that a statement such as “All S is not P” is not a possibility, but it is equivalent to “No S is P”.)

For logical purposes, “Some” means “at least one”. Thus, the statement, “Some cats are felines” is true. You might think: “But wait – ALL cats are felines! So how can it be true to say that ‘Some cats are felines’? The answer is, “some” means at least one. So, both of the following statements are true: “Some cats are felines”, and “All cats are felines”.

Similar considerations apply to the statement, “Some cats are not dogs”. If at least one cat is not a dog, the statement is true; so, of course, it is true. It’s also true that “No cats are dogs”.

Statements have parts: you will need to learn them in order to apply the rules for categorical syllogisms. The parts are as follows:

1) A quantifier (All, No or Some)

2) A subject term (“S” in the four statements given above)

3) A predicate term (“P” in the four statements given above)

4) A copula (is/are and is not/are not). The copula joins the subject term to the predicate term.

So, for example, let’s look at the parts of the following statement:

“No oaks are maples.”

- The quantifier is “No”

- The subject term is “oaks”

- The predicate term is “maples”

- The copula is “are”

Here’s another example:

“Some mammals are not canines.”

- The quantifier is “Some”

- The subject term is “mammals”

- The predicate term is “canines”

- The copula is “are not”

Additionally, the four kinds of statements that make up categorical syllogisms are said to have both quantity and quality. There are two kinds of quantity: universal and particular. Also, there are two kinds of quality: affirmative (it affirms something) and negative (it denies something). Let’s see how these apply to the four kinds of statements, below.

1) All S is/are P = Universal (because “all” is universal), Affirmative (because it affirms something – that All S is/are P).

2) No S is/are P = Universal (because “no” is universal), Negative (because it denies something).

3) Some S is/are P = Particular (because “some” is particular), Affirmative (because it affirms something).

4) Some S is not/are not P = Particular, Negative.

Finally, for the sake of abbreviating these statements, the following naming convention has been in effect for several hundreds of years:

A = All S is/are P

E = No S is/are P

I = Some S is/are P

O = Some S is not/are not P

(For the future, I will drop the is/are convention and use one or the other – but remember that either is fine.)

So, here is the complete table of information for the four kinds of standard form statements that make up categorical syllogisms:

Statement	Letter Name	Quantity	Quality
All S are P	A	universal	affirmative
No S are P	E	universal	negative
Some S are P	I	particular	affirmative
Some S are not P	O	particular	negative

 OK, now for a somewhat trickier concept: distribution. For the moment, just accept and memorize the following, then I will attempt to explain it:

“A term is said to be distributed if it is either the subject of a universal or the predicate of a negative.”

Here is another table that reflects that additional piece of information:

Statement	Letter Name	Quantity	Quality	Terms Distributed
All S are P	A	universal	affirmative	S
No S are P	E	universal	negative	S and P
Some S are P	I	particular	affirmative	none
Some S are not P	O	particular	negative	P

So, you may be thinking: “Wonderful. What exactly is distribution, and why should I care?” Good questions. You need to know how to determine if a term is distributed when it comes to applying the rules for categorical syllogisms. What does “distribution” mean? This is a little confusing, but bear with me. I’m not making it confusing – it just is, and has been, for countless generations of logic students. But here goes: When a term is distributed in a statement, it means that the statement says something about every member of the class that the term denotes.

So, in the “A” statement, “All S is P”, the term “S” is distributed, because the statement “All S is P” says something about every single S – namely, that they are all P.

In the “E” statement, “No S are P”, both the S term and the P term are distributed, since you are making a claim about all S’s, and all P’s.

Moving on to the “I” statement, “Some S are P”, neither term is distributed, and that’s pretty easy to see. You are not making a claim about all of S, nor are you making a claim about all of P.

OK – so the last one is the toughest, and to many, it never makes intuitive sense, you just have to accept that it is so, and if you want to know more, take an advanced logic class. In the “O” statement, “Some S are are not P”, the P term is distributed. Why? Because you have asserted something about all of P – namely, that all of P falls outside at least one S. You don’t like that, do you? No one does. 

Next: Every syllogism has a major term, a minor term, and a middle term, each of which appears exactly twice. The major term, by definition, is the predicate of the conclusion. The minor term is the subject of the conclusion. The middle term occurs once in each premise, but not in the conclusion. Here is an illustration:

All felines are cats

No dogs are cats

No dogs are felines

- In this syllogism, the major term, or the predicate of the conclusion, is felines.

- The minor term, or the subject of the conclusion, is dogs.

- The middle term is cats.

(- By the way, it is valid. )

Here is another illustration:

All horses are animals

Some dogs are not horses

Some dogs are not animals

- The major term is animals.

- The minor term is dogs.

- The middle term is horses.

(- By the way, it is invalid.)

There are 1-letter abbreviations for the major term, minor term, and middle term. They are:

P = major term (because it is the predicate of the conclusion)

S = minor term (because it is the subject of the conclusion)

M = middle term (“M” for middle term)

Also, we distinguish the two different premises in a categorical syllogism according to whether they contain the major term or the minor term. Quite simply, the major premise contains the major term, the minor premise contains the minor term.

Finally, just as a categorical statement or proposition (A, E, I or O) is said to be in standard form, a categorical syllogism has a standard form, too. In order for a categorical syllogism to be in standard form, the following requirements must be met:

1) All three statements are standard form categorical propositions (A, E, I or O).

2) The two occurrences of each term (major, minor and middle) are identical, and are used in the same sense throughout the argument.

3) The major premise is listed first, the minor premise second, and the conclusion last.

Here, for example, are three categorical syllogisms in standard form:

1)

All P are M

Some S are not M

No S are P

2)

All M are P

All S are M

All S are P

3)

All felines are mammals

All cats are felines

All cats are mammals

Here are two that are NOT in standard form, because the minor premise is listed first:

4)

All S are M

All M are P

All S are P

(It is still valid, but it is not in standard form.)

All cats are felines

All felines are mammals

All cats are mammals

(Again, still valid, but not in standard form.)

It may surprise you to know that categorical syllogisms have both moods and figures. To determine these, you first need to be sure that your categorical syllogism is in standard form. Then, the mood is simply the letter names of the categorical statements, in order, from the first premise, to the second premise, then the conclusion. It’s easier to see this by an example:

1) The mood below is AEO:

All P are M (A)

No S are M (E)

Some S are not P (O)

2) The mood below is IAE:

Some M are P (I)

All M are S (A)

No S are P (E)

3) The mood below is AAO:

All P are M

All M are S

Some S are not P

So much for mood. Now, you may be wondering, what determines the order in which the middle term occurs in each premise? Does it come first or second? The placement of the middle term is determined by the figure of the categorical syllogism. Since there are only four possible placements, there are only four possible figures. What are the four possible placements? Think about it – in each premise the middle term could be either first or second. So, the four possibilities are:

Figure 1	Figure 2	Figure 3	Figure 4
First premise: Middle term first	First premise: Middle term second	First premise: Middle term first	First premise: Middle term second
Second premise: Middle term second	Second premise: Middle term second	Second premise: Middle term first	Second premise: Middle term first

These four figures were originally devised by Aristotle, and have been memorized in exactly this order ever since. So, join two thousand years’ worth of logic students in remembering your four figures! Here is a mnemonic that many have found useful in trying to remember which figure is which:

The first figure sort of looks like an “S”, the last a “Z”, with two back-to-back “C’s” in the middle. How does it work? The horizontal lines represent the premises, and the middle term falls where the lines intersect. So, in the first figure, which looks like an “S”, the middle term comes first in the first premise (because that’s where the lines intersect), then it comes second in the second premise (again, because that’s where the lines intersect). In the second figure, which looks like a backward “C”, the middle term comes second in both premises, and so forth, as per the table above.

By the way, it may have occurred to you by now that there are a finite number of standard form categorical syllogisms. How many do you think there are?

It may also have occurred to you that of this finite number of standard form categorical syllogisms, only a certain number of them are valid. You’ll soon be able to tell for yourself.

Next – some practice on mood and figure. Remember – you must make sure the syllogism is in standard categorical form, or the mood and figure may be wrong! Be sure, for example, that the major premise (the one containing the major term) is listed first. Here are some sample moods and figures for you to study:

1) Mood AAA, Figure 1:

All M are P

All S are M

All S are P

- or –

All monkeys are primates

All spider monkeys are monkeys

All spider monkeys are primates

2) Mood AAA, Figure 2:

All P are M

All S are M

All S are P

- or –

All pandas are mammals

All sloths are mammals

All sloths are pandas

3) Mood AAA, Figure 3:

All M are P

All M are S

All S are P

- or -

All mammals are pandas

All mammals are sloths

All sloths are pandas

4) Mood AAA, Figure 4:

All P are M

All M are S

All S are P

- or –

All poodles are mammals

All mammals are sentient beings

All sentient beings are poodles

Now, let’s practice identifying the mood and figure of an existing standard form categorical syllogism.

1) Identify the mood and figure:

No P are M

Some S are not M

All S are P

2) Identify the mood and figure:

Some M are P

Some S are not M

Some S are not P

3) Identify the mood and figure:

No M are P

No M are S

All S are P

4) Identify the mood and figure:

All P are M

No M are S

Some S are not P

5) Identify the mood and figure:

All S are M

No M are P

All S are P

(Hint: Be careful – remember standard form!)

Answers

1) Mood: EOA Figure 2 (Usually just written together as EOA-2)

No P are M

Some S are not M

All S are P

2) IOO - 1

Some M are P

Some S are not M

Some S are not P

3) EEA - 3

No M are P

No M are S

All S are P

4) AEO - 4

All P are M

No M are S

Some S are not P

5) Be careful – this syllogism is not written in standard form:

All S are M

No M are P

All S are P

Before you determine the mood and figure, you have to switch the order of the premises so that the major premise is listed first:

No M are P

All S are M

All S are P

The mood is EAA, and the figure is 1

OK, now we can finally move on to the much-anticipated rules for categorical syllogisms. First, I’ll list them, then discuss them further.

Rules for Categorical Syllogisms!

Rule 1: The middle term must be distributed at least once.

Rule 2: If a term is distributed in the conclusion, then it must be distributed in the premise.

Rule 3: Two negative premises are not allowed.

Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.

Rule 5: If both premises are universal, the conclusion cannot be particular.

Note: If only rule 5 is broken, the syllogism is said to be conditionally valid – that is, valid on the condition that certain terms actually denote things that are real, or that exist.

If no rules are broken, the syllogism is valid. Otherwise, it is invalid. (I’ll address the exception concerning rule 5, below).

Let’s look at some examples. 

Here is the table again from above:

Statement	Letter Name	Quantity	Quality	Terms Distributed
All S are P	A	universal	affirmative	S
No S are P	E	universal	negative	S and P
Some S are P	I	particular	affirmative	none
Some S are not P	O	particular	negative	P

Rule 1: The middle term must be distributed at least once. (Now you understand why it was so important to learn about distribution!) Let’s see how this rule works with the following syllogism:

All sharks are fish

All salmon are fish

All salmon are sharks

First, identify the middle term, which is “fish”. Then, check to see if it is distributed in either premise. You can see that it is not distributed in the first premise, because it is neither the subject of a universal, nor the predicate of a negative (in fact, it is the predicate of a universal affirmative). So, now check the second premise. The term “fish” is not distributed here either, for the same reason. So, this rule is violated, and the syllogism is invalid.

Let’s see another example that violates Rule 1:

Some cats are pets

Some pets are dogs

Some dogs are cats

First, identify the middle term, which is “pets”. It is not distributed in the first premise, because the first premise is an “I” proposition that distributes no terms. The same is true for the second premise, so the middle term, “pets”, is not distributed at all. The rule is violated, and the syllogism is invalid. By the way, you can see that no categorical syllogism that has two “I” statements for premises could ever be valid, because the middle term could never be distributed. In other words, IIA, IIE, IIO and III could never be valid, no matter what figure they’re in.

Rule 2: If a term is distributed in the conclusion, then it must be distributed in the premise. Here is a syllogism that violates this rule:

All squirrels are animals

Some wombats are not squirrels

Some wombats are not animals

First, check to see if a term (either the subject term or the predicate term) is distributed in the conclusion. The subject term, “wombats”, is not distributed, but the predicate term, “animals”, is (because it is the predicate of a negative). Next, check to see if the term “animals” is distributed in the premise in which it occurs, which is the statement, “All squirrels are animals”. In this statement, the term “animals” is not distributed, because it is neither the subject of a universal nor the predicate of a negative. So, this rule is violated, and the syllogism is invalid.

Let’s look at a second example that violates Rule 2:

All tigers are mammals

All mammals are animals

All animals are tigers

First, check to see if a term is distributed in the conclusion. In this case, the term “animals” is the subject of a universal, so it is distributed, but the term “tigers” is not distributed. OK, so now check to see if the term “animals” is distributed in its premise, which reads, “All mammals are animals”. The A proposition, as you know, does not distribute the predicate term, so the term “animals” is not distributed in the premise, and the rule is violated. This syllogism is invalid.

Let’s do one more example of a syllogism that violates Rule 2:

Some creatures are camels

Some creatures are not leopards

No leopards are camels

First, check the conclusion to see if any terms are distributed. In this case, the conclusion is an “E” statement, so both terms, “leopards” and “camels” are distributed. (The conclusion is a universal negative, so “Leopards” is the subject of a universal, and “camels” is the predicate of a negative). Now, we have to check to see if they are both distributed. Is the term “leopards” distributed in the premise that reads, “Some creatures are not leopards”? Yes, it is, because it is the predicate of a negative. So far, so good. Now, let’s check the term “camels”. Is it distributed in the premise that reads, “Some creatures are camels”? No, an “I” statement does not distribute any terms. So, the rule is violated, because “camels” is distributed in the conclusion, but not in the premise. The syllogism is invalid.

Rule 3: Two negative premises are not allowed. This rule is probably the easiest to check. What is a negative premise? One that is either an “E” statement (No S are P) or an “O” statement (Some S are not P). So, if you have EE, OO, EO, or OE as your two premises the syllogism is invalid. Let’s look at some examples:

No fish are mammals

Some dogs are not fish

Some dogs are mammals

The first premise is negative (an “E” statement), and so is the second (an “O” statement), so both premises are negative, and the rule is violated. The syllogism is invalid.

Here’s a second example:

No bricks are marshmallows

No marshmallow are thimbles

Some thimbles are bricks

The first and second premises are both “E” statements, so they are both negative, and the rule is violated. The syllogism is invalid.

Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise. Let’s look at an example:

All ducks are birds

Some elephants are not ducks

Some elephants are birds

First, check to see if you have a negative premise. In this case, we do, “Some elephants are not ducks”. Then, check to see if the conclusion is negative. In this case, “Some elephants are birds” is not negative, so the first part of this rule is violated, and we know the syllogism is invalid.

Here’s another example:

All circles are shapes

All squares are shapes

Some squares are not circles

First, check to see if you have a negative premise. In this case, we do not. Next, for the second part of the rule, see if the conclusion is negative. In this case, it is negative. So, since the conclusion is negative, we need to see if one of the premises is negative, too. As we’ve already seen, neither premise is negative, so the second part of the rule is violated, and the syllogism is invalid.

Rule 5: If both premises are universal, the conclusion cannot be particular.

Note: If only rule 5 is broken, the syllogism is said to be conditionally valid – that is, valid on the condition that certain terms actually denote things that are real, or that exist.

First, let’s just see this rule works, and then I’ll explain the point about conditional validity. Here is an example of a syllogism that violates Rule 5:

All mammals are animals

All unicorns are mammals

Some unicorns are animals

Both premises are universal, but the conclusion is particular, so the rule is violated. The syllogism is invalid. But only Rule 5 is violated, so doesn’t that mean it’s conditionally valid? Yes. But what does that mean? The condition it refers to is the condition that certain terms actually exist. When Aristotle invented this logic, he wanted to use it to talk about things in the world – things that really existed – so that assumption was built in. More modern logics, however, need to be able to talk about things that may or may not exist, and it can make a difference. In the argument above, the argument is conditionally valid – it is valid on the condition that unicorns actually exist. But, of course, they do not. So, the argument is invalid.

A fuller treatment of this problem, called “the problem of existential import”, is beyond the scope of this course. I just wanted you to have a basic explanation for the sake of completing the rules.

Now for the good news: I won’t require you to learn the difference between syllogisms that are conditionally valid and ones that are unconditionally valid. For testing purposes, the syllogisms will either be valid (unconditionally) or invalid. I won’t give you test questions in which only Rule 5 is violated, though I may give you some which violate Rule 5 along with one or more other rules.

Practice Problems

Reconstruct the following syllogisms and indicate any and all rules for categorical syllogisms that it violates. Indicate if it is valid or invalid.

1) I I I – 4

2) IAO – 3

3) EAA - 1

4) AEE – 4

5) AOO – 2

Answers:

1) I I I - 4

Some P are M

Some M are S

Some S are P

Invalid, violates Rule 1 (middle term is not distributed).

2) IAO - 3

Some M are P

All M are S

Some S are not P

Invalid, violates Rule 2 and Rule 4.

3) EAA - 1

No M are P

All S are M

All S are P

Invalid, violates Rule 4

4) AEE – 4

All P are M

No M are S

No S are P

Valid, no rules broken.

5) AOO – 2

All P are M

Some S are not M

Some S are not P

Valid, no rules broken.