7.3 Categorical Logic: Syllogisms

As you already know, Categorical syllogisms are arguments made up of categorical propositions. Until the 20th century, virtually all formal logical analysis was done in terms of categorical logic, and learning how to construct and evaluate categorical syllogisms was central to the study of Logic. Today, in the aftermath of the mathematization of our techniques for representing relations amongst statements and amongst categories, the study of categorical syllogisms is often left aside. But I believe that it has some value for beginning students of logic for two reasons: it is good to know the foundations that more advanced approaches build from, and categorical logic is certainly the foundation. Also, categorical logic and categorical syllogisms are “closer” to common everyday modes of expression than are the approaches of symbolic logic. For this reason, you may well find the techniques of syllogism analysis more useful than the more powerful techniques of symbolic logic.

I’ll begin with an example we can talk about:

All men are mortal, and Socrates is a man, so Socrates is mortal.

From the previous reading, you know that to be strict about it, we’d restate these in standard form A statements, like this:

All men are mortal beings.

All things identical to Socrates are men.

So All things identical to Socrates are mortal beings.

 

Now let’s look at the parts of this syllogism:

All men are mortal beings.

All things identical to Socrates are men.

So All things identical to Socrates are mortal beings.

Thanks to the bolding and italics, you can spot that there are three distinct terms in the syllogism, and that they each occur two times. This is one defining feature of categorical syllogisms; when you go to construct one, you must make sure each term occurs twice and that there are exactly three terms.

Another thing to note is the order of the statements; it is not arbitrary. Whichever premise contains the same term that occurs as the predicate of the conclusion, is the premise that must come first. The predicate of the conclusion is “mortal beings,” so the premise containing “mortal beings” must be listed first. (It does not matter whether this term occurs as the subject or the predicate of the first premise, just that it be in that premise.) The term that is the predicate of the conclusion is called the MAJOR term, and the premise containing the major term is called the Major Premise. The first premise in a properly arranged syllogism is the Major Premise.

The subject of the conclusion is called the MINOR term, and the statement that contains it and comes second in order is called the Minor Premise.

Each term occurs twice, and so far we have accounted for the two occurences of the Major term and the Minor term. The other term also occurs twice, and it occurs only in the premises. It is known as the MIDDLE term. The idea behind this name is undoubtedly that it stands between (in the middle of) the Major and Minor terms, and makes (or fails to make) a connection between them.

That addresses the question of what order to present our statements in. Once we have them in the right order, there are two important things to observe, and these go by the names MOOD and FIGURE.

The Mood of the syllogism above is AAA. (Of course it would be AAA no matter what order you put the statements in, wouldn’t it?) So the mood is very easy to report: it’s just the letter names of the three propositions, listed in order.

The Figure has to do with the location of the Middle term. Perhaps you can tell at a glance that there are four and only four possible arrangments of the locations of the middle term:

M __

__ M

is the arrangement called figure 1.

 

__ M

__ M

is the arrangement called figure 2.

 

M __

M __
is the arrangement called figure 3.

 

And

__M

M__

is the arrangement called figure 4.

 

 

The four figures are known simply as “1,” “2,” “3,” and “4.”

 

If we use “S” and “P” to refer to the Minor and Major terms (i.e., to the Subject and Predicate of the conclusion), we could fill this in:

 

Figure 1

M P

S M

S P

 

Figure 2

P M

S M

S P

 

Figure 3

M P

M S

S P

 

Figure 4

P M

M S

S P

 

You can now see that our Socrates example from above

All men are mortal beings.

All things identical to Socrates are men.

So All things identical to Socrates are mortal beings.

is what’s called AAA-1, because it reduces down to

A M P

A S M                                      AAA-1 is known as the FORM of the syllogism

A S P

 

In medieval schools, you were expected to memorize a little poem in Latin that told you the valid syllogisms. Just as they used the “A” and “I” from “Affirmo” to give names to the affirmative categorical propositions, they used the vowels of names to convey the moods of syllogisms. The AAA-1 was known as Barbara. Here’s the first line of the poem:

Barbara, Celarent, Darii, Ferioque prioris

Most logicians today will know what you are talking about if you mention “Barbara” to them.   This line also tells you that EAE1, AII1 and EIO1 are valid.

 

Anyway, the Form is simply the Mood and the Figure stated together. The Form provides an exhaustive account of the syllogism. Take a minute and fill out the syllogisms (using S, P, and M) on the basis of these Forms:

1. AII-2

2. EAA-3

3. OIE- 1

4. OAO-4

EIEIO !

 

Now take a few minutes to identify the Forms of these syllogisms. You won’t be able to just do it at a glance, because they are not in standard form categorical propositions yet, and they are not in the right order. Do you know what to do?

1. No republicans are Democrats, so none of them are big spenders, since all big spenders are Democrats.

2. All pranksters are exasperating so some leprechauns are exasperating, since they’re all pranksters.

3. Any corporation that overcharges is unethical and some investor-owned utilities overcharge, so some unethical businesses are investor-owned utilities.

4. Some intelligible statements are true, because all meaningful ones are intelligible and some meaningful statements are not true.

5. “All our ideas derive from our experiences. But we have no experience of God. I need not conclude my syllogism.”   –David Hume,Dialogues Concerning Natural Religion

 

 

Answers to these exercises are below:

1. No republicans are Democrats, so none of them are big spenders, since all big spenders are Democrats.

All big spenders are Democrats

No Republicans are Democrats

No Republicans are big spenders

 

All B are D

No R are D

No R are B

AEE-2

2. All pranksters are exasperating so some leprechauns are exasperating, since they’re all pranksters.

All pranksters are exasperating individuals

All leprauchans are pranksters

Some leprauchans are exasperating individuals

 

All P are E

All L are P

Some L are E

AAI-1

 

3. Any corporation that overcharges is unethical and some investor-owned utilities overcharge, so some unethical businesses are investor-owned utilities.

Some investor-owned utilities are corporations that overcharge

All corporations that overcharge are unethical businesses

Some unethical businesses are investor-owned utilities

 

Some I are C

All C are U

Some U are I

IAI -4

4. Some intelligible statements are not false, because all meaningful ones are intelligible and some meaningful statements are not true.

Some M are not T

All M are I

Some I are not F (“false” is the same as “non-true” –nonT)  –has to be obverted to:    Some I are T

OAI -3

So this one looked like OAO at first, but it had four terms, since “true statement” occurred once and “false statement” occurred once.  When reduced to three, that O conclusion turned into an I.  The obversion could have been done to the major premise rather than the conclusion, which would have made it IAO3.

 

5. “All our ideas derive from our experiences. But we have no experience of God. I need not conclude my syllogism.” David Hume, Dialogues Concerning Natural Religion

All real ideas are ideas derived from experience.

No idea of God is an idea derived from experience

No idea of God is a real idea

A R E

E G E

E G R

 

AEE -2

 

 

 

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