Logic Class # 5

I have struggled with how to present this particular set of information. (Note 1) There is some drudgery involved, but the outcome is beautiful and worth the trip. So I finally have decided to present the outcome first, at least rough idea of where this is heading.

The objective here is to find a set of argument forms that are consistently valid. We can show that there are 256 different forms for Categorical Syllogisms, and we can show further that only 15 of those forms are valid arguments (!) That leaves 241 non-valid forms of Categorical Syllogisms. Is there any wonder that there are so many false arguments?

The fun part is that the 15 valid forms of Categorical Syllogisms have been given names. For example, one need not say to a logician any more than “Barbara”, and the logician will immediately know the exact form. I think that is a hoot, and it reminds me of the old joke about the comedian’s convention, where one comedian yells out, “number 47”, and the whole audience laughs…

In order to get to that point we have some things to understand first. Here is a map of where we are headed in this lesson:

1. Standard Form for Categorical Syllogisms.
2. The Concept of “Distribution”.
3. Affirmation / Negation.
4. Contraries and Contradictories.
5. Rules and Fallacies.
6. Moods and Figures.
7. A List of the Valid Categorical Syllogisms (of Standard Form)

We will need to define some things as we go along.

1. Standard Form for Categorical Syllogisms

The Structure of Categorical Propositions has a standard form. The form consists of this:

[subject] [some form of the verb “to be”] [ predicate]

Example:

[The king of France] [ is ] [bald].

This proposition is in a valid form, but is false: France has no king.

2. The Concept of Distribution

Distribution is defined as how much of the population of a set is covered by the proposition. If an element of the proposition refers to an entire population then it is said to be distributed (or “universal”); otherwise it is not distributed (also called “particular”).

Examples:

The modifier, “some”, does not cover the entire population and is thus not distributed.

The modifiers, “all” and “none”, refer to the entire population of the set, and are called distributed.

[All of P] [ is ] [some of Q]: Here P is distributed, and Q is not distributed.

3. Affirmation and Negation

Now, recalling the types of propositions, we can further refine them:

A. All Q is P;

This means that all of the Q category is a subset of the P category.
Q is distributed; P is not distributed.
Q is universal: P is particular.
The statement is affirmative.

E. No Q is P;

Q is completely outside and independent of the p category.
Q is distributed; P is distributed.
Both Q and P are universal.
The statement is negative

I. Some Q is P;

Part of the Q category is inside the P category.
Q is not distributed; P is not distributed.
Q and P are both particular.
The statement is affirmative.

O. Some Q is not P;

Part of the Q category is outside of the P category.
Q is not distributed; P is distributed.
Q is particular; P is universal.
The statement is negative.

4. Contraries and Contradictories

The relationship between the four types, A, E, I, O, is a square which demonstrates the contrary relationships, the contradictory relationships, and the sub-relationships:



[From Oxford Dictionary of Philosophy]


Here is a table for displaying the same information:
For example, when A is true, its contrary, E is false. When A is true, its contradictory, O is false.





5. Rules and Fallacies

There are some rules for Categorical Syllogisms, and there are associated fallacy names for their false usage:

Rule 1:
Don’t use four terms;
Associated Fallacy: four term fallacy.

Rule 2:
Distribute the middle term for one or more of the premises:
Associated Fallacy: undistributed middle fallacy.

Rule 3:
A distributed term in the conclusion must also be distributed in the premises.
Associated Fallacy: undistributed middle fallacy.

Rule 4:
Do not use two negative premises.
Associated Fallacy: exclusive premise fallacy.

Rule 5:
If one of the premises is negative, then the conclusion is negative as well.
Associated Fallacy: false affirmative from a single negative premise fallacy.

Rule 6.
No particular conclusion is derived from two universal premises.
Associated Fallacy: existential fallacy.

6. Moods and Figures

Here we come to an interesting part.

The structure of syllogism is called the “mood”. For example, the mood of a syllogism might be AOE, meaning that premise 1 is of the form A; premise 2 is of the form O; the conclusion is of the form E.

And the syllogism also has a “figure”, which describes the shape with respect to the middle term, M. (S= subject of the conclusion, aka “minor term”; P=predicate of the conclusion, aka “major term”):

Figure 1:

P1: M – P
P2: S – M
C: S – P
(The middle term is the subject in P1; the predicate in P2).

Figure 2:

P1: P – M
P2: S – M
C: S – P

Figure 3:

P1: M – P
P2: M – S
C: S – P

Figure 4:

P1: P – M
P2: M – S
C: S – P

Now we are finally able to fully define some arguments merely by referring to some codes. For example, AAA-1 means that the argument is formed of three “all Q are P “ types of statements, in the form of Figure 1.

So now we can list the 15 valid arguments and even categorize them with respect to their figures:

7. List of Valid Syllogisms of Standard Form, Categorized by Figure:

Figure 1 Valid Syllogisms:

1. AAA-1 Barbara
2. EAE-1 Celarent
3. AII-1 Darii
4. EIO-1 Ferio

Figure 2 Valid Syllogisms

5. AEE-2 Camestres
6. EAE-2 Cesare
7. AOO-2 Baroko
8. EIO-2 Festino

Figure 3 Valid Syllogisms

9. AII-3 Datisi
10. IAI-3 Disamis
11. EIO-3 Ferison
12. OAO-3 Bokardo

Figure 4 Valid Syllogisms

13. AEE-4 Camenes
14. IAI-4 Dimaris
15. EIO-4 Fresison

Of these, only EIO is valid in all Figures, 1 through 4.

The validity of these arguments has been determined by examining all 256 arguments for violation of the Rules which are listed above. Feel free to repeat that exercise if you wish. (Note 2)

In the next lesson we will investigate the translation of language-based arguments into the coded syllogistic form, so that we can more easily determine the validity of such everyday language arguments.

Notes:
(1) Much of this material was developed with the help of Copi & Cohen, Introduction to Logic, 2005, as well as other sources by Tarski, Frege, Kelley, Tomassi, etc.
(2) I did not do that. Some things are faith-based.