Categorical
Syllogism, Venn Diagrams and Rules for Testing for Validity
As I mentioned previously, a Categorical Syllogism is a two premised deductive argument whose every claim is a categorical claim, and in which three terms appear in the argument exactly twice in exactly two premises and one conclusion.
Example:
Some consumers are not Democrats.
All Americans are consumers.
Therefore
Some Americans are not Democrats
Notice that each of the three terms appear exactly twice in exactly two claims. The terms are to be labeled in the following ways:
· The term that appears as the predicate in the conclusion of the argument is call the major term.
· The term that appears as the subject in the conclusion of the argument is called the minor term.
· The term that appears in both premises of the argument but not in the conclusion is call the middle term.
· The premise which contains the major term is the major premise.
· The premise which contains the minor term is called the minor premise.
Major Term: Predicate of the Conclusion
Minor Term: Subject of the Conclusion
Middle Term: Term appearing in the premises, but not appearing in the conclusion.
Major Premise: Premise containing the Major Term
Minor Premise: Premise containing the Minor Term
So for the argument:
Some consumers are not Democrats.
All Americans are consumers.
Therefore
Some Americans are not Democrats.
|
Quantifier |
Subject |
|
Predicate |
|
|
Some |
Consumers |
are not |
Democrats. |
Major Premise |
|
|
Middle Term (i.e., not in the
conclusion) |
|
|
|
|
All |
Americans |
are |
consumers. |
Minor Premise |
|
Therefore |
|
|
Middle Term (i.e., not in the
conclusion) |
|
|
Some |
Americans |
are not |
Democrats. |
|
|
|
Minor Term (i.e.
Subject of the conclusion) |
|
Major Term (i.e.
Predicate of the conclusion) |
|
The most frequently use symbols used to abbreviate these terms are;
P for the major term
S for the minor term
And
M for the middle term.
Notice:
1. The argument consists of two premises and one conclusion (i.e., It is a syllogism.)
2. Each of the claims is a categorical claim. (Specifically, an O claim, an A claim, and another O claim.)
3. Each of the three terms appear exactly twice in exactly two claims.
Categorical
Syllogisms and “Standard Form.”
For a categorical syllogism to be in “Standard Form” the Major Premise must be on top. And the Minor Premises must be under the Major Premise.
So notice:
All Americans are consumers.
Some consumers are not Democrats.
Therefore
Some Americans are not Democrats.
While this argument is logically identical to the one above, this rendering of the argument is NOT is standard form. The minor premise is one top. The “Mood” of this argument is OAO, NOT AOO. (See below.) Likewise, this argument is in 1st figure, not 4th figure. (See below.) Failing to put the syllogism into standard form can, therefore, be visually misleading.
The Mood of a Categorical
Syllogism
Once a categorical syllogism is in standard form, we can then determine its mood and figure. One aspect of the form of the syllogism is named by listing its “mood.” The Mood of the syllogism we are considering here is OAO. That is, the major premise is an O claim, the minor premises is an A claim and the conclusions is an O claim. Hence, OAO.
Now suppose the argument was presented this way:
All Americans are consumers.
Some consumers are not Democrats.
therefore
Some Americans are not Democrats
You might think the mood of the syllogism is AOO. But you would be wrong. Why? You would have been fooled into thinking this because the syllogism is NOT in standard form here. This is because the Minor Premise is on top and the Major premise is underneath. So to know what the Mood of the syllogism is, one must be certain that the syllogism is in Standard From
The Figure of a
Categorical Syllogism
The figure of a categorical syllogism refers to the arrangement of the middle terms in the premises. The middle terms can be arranged in four possible ways. They are:
|
M P |
R M |
M P |
P M |
|
S M |
S M |
M S |
M S |
|
S P |
S P |
S P |
S P |
The “shirt collar” mneumonic device can be used to remember the four possible figures. The Ms (middle terms) line up as if on the edges of a men’s shirt collar.
|
M P |
R M |
M P |
P M |
|
S M |
S M |
M S |
M S |
|
S P |
S P |
S P |
S P |
· Figures are used in conjunction with the mood to classify categorical syllogisms. Note there are four different types of categorical claims, and each syllogism contains a total of three. So there are only 64 different possible Moods. (e.g. AAA, AAE, AAI, AAO, AEA, AEE, AEI, AEO, etc.)
· Since each mood can be configured in four different figures, that means there are only 256 possible standard form categorical syllogisms.
· Of the 256, only 24 are valid forms.
· Of the 24 valid forms, 15 are unconditionally valid, and 9 are conditionally valid. (More on this below.)
Testing for
Validity Using the Rules:
Five rules apply to determine whether a syllogism is unconditionally valid:[1]
(If a syllogism does not violate rules 1-5, but does violate rule #6, it is said to be conditionally valid.)
Rule 1: In a valid categorical syllogism, the middle term must be distributed in at least one premise.
Rule 2: In a valid categorical syllogism, any term that is distributed in the conclusion must be distributed in the premises.
Rule 3: In a valid categorical syllogism, if the argument has a negative premise, it must have a negative conclusion.
Rule 4: In a valid categorical syllogism, if the argument has a negative conclusion, it must have a negative premise.
Rule 5: In a valid categorical syllogism, there cannot be two negative premises.
Rule 6: (Conditional Requirement) In a valid categorical syllogism, a particular conclusion cannot be drawn from two universal premises. (Unless one assumes existential import. Hence, conditionally valid.)
All and only those arguments that pass each of these tests are valid. Failure to satisfy one or more of the rules renders the argument invalid.
So notice, in many cases, once I know the mood and figure of the argument in question, I can determine whether it is valid quite easily. For instance:
· No argument with the Mood of EEE is valid regardless of figure. (Why? Fails rule 5.)
· No argument with a mood of III is valid. (Why? Fails Rule 1)
· No argument of the mood AAE is valid. (Why? Fails rule 4)
and so on.
So, to test for validity, I suggest you
1. Plot out your argument by mood and figure.
2. Underline any terms that are DISTRIBUTED
3. Apply the rules:
Example:
AAA 1st Figure
All M are P
All S are M
All S are P
Rule 1 Check
Rule 2 Check
Rule 3 Check
Rule 4 Check
Rule 5 Check
Rule 6 Check
AAA 1st Figure is Valid. Indeed, it is unconditionally valid.
How about AAA 2nd Figure?
All P are M
All S are M
All S are P
Rule 1 Failed (M is not distributed.)
Rule 2 Check
Rule 3 Check
Rule 4 Check
Rule 5 Check
Rule 6 Check
No, it is invalid.
How about AAI 1st Figure?
All M are P
All S are M
Some S are P
Rule 1 Check
Rule 2 Check
Rule 3 Check
Rule 4 Check
Rule 5 Check
Rule 6 Failed
Thus AAI 1st Figure is conditionally valid. That it, if we know or can assume that there is at least one S, in other words, make the existential assumption, then we can regard the argument as valid. The premises would give us good reason to accept the conclusion.
Of the 256, only 24 are valid forms. Of the 24 valid forms[2], 15 are unconditionally valid, and 9 are conditionally valid.
Unconditionally valid
|
Figure 1 |
Figure 2 |
Figure 3 |
Figure 4 |
|
|
|
AAA |
AEE |
AII |
AEE |
|
Conditionally valid
|
Figure 1 |
Figure 2 |
Figure 3 |
Figure 4 |
Required condition |
|
AAI |
AEO |
AEO |
S exists |
|
|
AAI |
EAO |
M exists |
||
|
AAI |
P exists |
In this example Socialists is the minor term, Republicans the major and collectivists the middle.
DIAGRAM
We fill in the this diagram just as we have already done for categorical claims. Let's first diagram the major premise.
Next we will diagram the minor premise.
Since diagraming the premises resulted in shading in the entire area where the S and P circles overlap, and since that is exactly how we would diagram the conclusion if we were to do so, we can conclude that this syllogism is valid. In general, an argument is valid only when diagraming the premises automatically result in a diagram of the conclusion.
When one of the premises is an I or an O-claim, there can be a problem about where to put the required X. The following example represents such a problem.
Some S are not M
All P are M
therefore,
Some S are not P
DIAGRAM
An X in either area 1 or area 2 would diagram the minor claim, but which is the better choice? In some cases the decision is made for us. When one premise is a universal and the other is a particular, diagram the universal first.
DIAGRAM
Once the A-claim has been diagramed, there is no longer a choice about where to put the X. Hence the completed diagram for the argument looks like this:
DIAGRAM
This argument is valid.
In some syllogism, the rule that I just explained doesn't help. For example:
All P are M.
Some S are M.
therefore,
Some S are P.
A syllogism like this leaves us in doubt as to where to put the X even after we have diagramed the universal claim. When such a situation arises, here is the rule that I want you to follow. An X which can go in either of two areas, should be put directly on the line that separates the two areas. In essence, an X on the line say that the X belongs in one or the other of the two area but that the information given is not sufficient for determining which. When the time comes to see if the diagram yields the conclusion, we pay attention only to those X's that are entirely within one or another area. X's on the line don't count.
A last point about the diagrams, when both the premises are universals but the conclusion is a particular claim. Where does the X come from? Well this depends on what type of Square of Opposition you are using. Put another way, are you going on the assumption that you are dealing with non-empty sets, or are you accepting the possibility that one or more of the set about which you are talking are empty.
In the case of the former, when you make an existential presupposition, if any circle has all but one area shaded in, an X should be placed in the area. You have already indicated that if the set is not empty then the only place for members to be is that last area, and by you presupposition, you have already indicated that the set must have some members. For these reasons, you are justified in placing the X in the last free area.
On the hypothetical view however, such a move is not justified. It is for this reason that no argument is valid which goes from universal premises to particular conclusion on the hypothetical view. In this case it would not be enough to say simply where an argument is valid or invalid. You would first have to tell me whether you were making the existential presupposition or not and the reason for you choice.
Diagram and Test for Validity
There are legal limits on all gamefish, but carp are not gamefish. So there are no legal limits on carp.
All Gamefish (D) are Fish with legal limits.
No Carp (D) are Gamefish (D).
No Carp (D) are Fish with legal limits (D). (of No Fish with legal limits are Carp.)
Major Term: Fish with Legal Limits
Minor Term: Carp
Middle Term: Game Fish
AEE 1
Rule 1 Check
Rule 2 Check (Fail)
Rule 3 Check
Rule 4 Check
Rule 5 Check
Rule 6 Check
There are legal limits only on gamefish. There are no legal limits on carp, since carp are not gamefish.
There are legal limits on carp or carp are not gamefish. There are no legal limits on carp, so carp are not gamefish.