# Determining validity of Categorical Syllogisms

There are two ways to determine whether a categorical syllogism is valid or invalid. One is to draw a picture of the premises using Venn diagrams (three overlapping circles: one for each category). If the conclusion shows up as a result of drawing the premises, then we know the argument is valid, because that means that the conclusion results necessarily from the premises. The other method is to check the form against a set of rules.

Venn Diagrams

When we analyze a categorical syllogism with Venn diagrams, we need three overlapping circles. Each circle represents one of the three terms (the Major, the Minor, and the Middle).  The diagrams here are all labeled in terms of  S, P, and M:  “S” is the minor term (the Subject of the conclusion); “P” is the major term (the Predicate of the conclusion); and “M” is the middle term (the term occurring only in the premises).  For the sake of being able to talk about the diagrams, I always put them in the same order: the left hand circle is the Minor term, the right hand circle is the Major term, and the circle below them is the Middle term. Then I number the segments starting from the top left.  Make sure you follow this convention if you want to be able to communicate with me about where the shading and x’s should go in your diagrams.

Each premise is diagrammed. If, as a consequence, the conclusion has also been diagrammed, then the premises entailed the conclusion, so the diagram shows the form to be valid.

To diagram the major premise, we look only at the two circles P and M (sections 2-7, excluding 1). To diagram the minor premise, we look only at the two circles S and M (sections 1,2 and 4-7, excluding 3).  To “read” the conclusion, we look only at the circles for S and P (sections1-6, excluding 7).  The drawings, done step by step, are simply a mapping of the two circle drawings of A, E, I and O onto the relevant two circles here.Let’s begin with the AAA-1 as an example.

All M is P

All S is M

All S is P

For the major premise (All M is P), shade away 4 and 7. The area called M has been reduced to sections 5 and 6, both of which are inside P.

For the minor premise (All S is M), shade away sections 1 and 2. The area called S is now reduced to sections 4 and 5, which are both inside the M circle.

The only area that can be called S at this point is area 5.  And area 5 is entirely inside the P circle: All S is P.

Let’s see another example.  EOI 3

E M P      (No M is P)

O M S     (Some M is not S)

I S P      (Some S is P)

The first premise is diagrammed by shading 5 and 6.

The second premise is diagrammed by placing an X inside the remaining M area but outside S –section 7.

Now we look for the conclusion. What are we looking for?  Some S is P:  an X located in section 2.  We find it is not there, so the argument is invalid: this conclusion does not follow necessarily from these premises.  DO NOT INSERT THIS X!  You stop drawing after the two premises, to see what they entail.

Another example will show you how an ambiguity can arise, and how to deal with it.

I A I 2

I P M       (Some P is M)

A S M      (All S is M)

I S P         (Some S is P)

Now we see that there are two areas (5 and 6) which are candidates for the X that the major premise calls for.  In such a case, there is no justification for using more than one X, so the only place it can go is on the line dividing 5 from 6.

Shading away 1 and 2 will diagram the minor premise.

We are looking for  I S P.  ISP shows up when there is an X in section 4 or in section 5.  It is not the case that we have a X in either area, so the conclusion does not show up, and the form is invalid.

This final example shows why, when you have both a particular premise and a universal premise, it can pay to diagram the universal first.

EMP                No M is P

ISM                 Some S is M

OSP                 Some S is not P

If you diagram the universal first, you’ll eliminate 5 and 6, so when you move to diagram ISM, there will only be one possible place for the X: section 4.  If you don’t diagram the universal first, you’ll have to place the X on the line between 4 and 5.

Then when you shade away 5 and 6, you see the need to move your X off that line and into area 4.

So in the end it should look like this:

To submit Venn diagrams to the Discussion Forum in Canvas, open this link to see how to generate the template in Paint so you can shade and mark.

Instructions for Venn diagrams

Here is another power point presentation that shows how to diagram syllogisms:

PPoint from the web Venn Diagram Technique for testing syllogisms

The Five Rules of Syllogisms

The other approach is to test your syllogism against five pretty simple rules.

Two of those rules involve the property called “distribution.”

Consider this form: AAA-2

A P M

A S M

A S P

If you remember the matter of DISTRIBUTION from earlier in the discussion of Categorical Propositions, you will recall that “A” statements distribute their Subject terms, but not their Predicate terms. That, by the way, is why you cannot convert them (switch the order of the subject and predicate, and expect to always have an equivalent truth value to that of the first statement).

Since “A” statements do not distribute their predicate terms, this form shows us a case where the middle term –the one that is supposed to make the connection between the other two terms (the Major and the Minor)– is undistributed. Being undistributed means that the term does not make a reference to all members of the class that it names. And this means that the term fails to really connect the Major and Minor terms. Since there is no connection in the premises between the Major term and the Minor term, no legitimate claim can be made about them in any conclusion. So, nothing follows from a set of premises in which the Middle term is undistributed.

The rule is: The middle term must be distributed at least once. When it is not, we say the argument commits the Fallacy ofUndistributed Middle Term.

Because of similar considerations, there is a rule to the effect that if a term is distributed in the conclusion of the syllogism, that same term must have been distributed in the premise in which it occurred in order for the syllogism to be valid. The reason is this: the presence of a distributed term means that something is being said about every member of the class it names. If the conclusion says something about every member of the class, but the premise does not provide information about every member, then the conclusion claims something that is not founded in the premise, hence something that is not necessarily true. Since the two terms in the conclusion are called the “major” and the “minor,” the fallacy has two names: Illicit major term or Illicit minor term. Respectively, these mean that the syllogism distributes the major term in the conclusion but not in the premise, and that the syllogism distributes the minor term in the conclusion, but not in the premise. Here’s an example:

All M is P

No S is M

therefore No S is P

In this example (AEE 1), the major term, P, is the predicate of a universal negative conclusion, which distributes it. But “P” is also occurring as the predicate of a universal affirmative premise, which does not distribute it. So more is said about the class P in the conclusion than is warranted in the premise. This is an Illicit Major.

Two other rules are extremely easy to apply; both concern negation.

No valid syllogism has two negative premises. Negative statements exclude one class from another, and no information about the relation between two categories can be inferred with certainty from two statements each of which excludes one of them from a third one. The fallacy is known as Exclusive Premises.

The other does not lend itself so readily to a catchy name, but pretty much needs to be spelled out as a conditional statement: If a syllogism has a negative premise, it must have a negative conclusion, and vice versa (meaning that if it has a negative conclusion it must have one -and only one- negative premise). One way you might reduce this to something catchy is in terms of double negatives. Most people “know” that double negatives are “wrong,” so here we could meddle a little with expectations and say syllogisms must obey the Double Negative requirement. The Double Negative requirement: if a syllogism has one negative statement, it must have two. As long as we also know to obey the Exclusive Premises rule, this will steer us in the right direction: one negative premise whenever there’s a negative conclusion, and vice versa.

So, you can test your ability to apply these rules by writing out the figures of these forms. If a syllogism breaks any one rule, it is invalid. If it breaks none, it is valid. You can see here how clearly validity is a matter of the form: nothing you could do with varying the content could help a syllogism that breaks a rule.

1. AAA-2

2. IAA-3

3. OAO-4

4. AAA-3

5. OAI -4

6. EAO-1

7. EEE-2

8. IAI-1

9. AII-3

10. IEO-4

11. AAA-4

12. OAA-1

13. EOI-2

14. AAA-1

15. OAI-2

You can check your answers on the next page.   To double check, do Venn diagrams for each of them as well. Remember to put the Minor term as the left hand circle, and the Major term as the right hand circle.

Of course, to check for Exclusive Premises and Double Negation, you don’t even have to spell out the figure; the mood reveals compliance or non-compliance with those rules.

In this powerpoint, I’ll walk through the analysis of two categorical syllogisms for you. In discussing the first one, you might note that I refer to “five rules of syllogisms,” butI only check for four of them. That’s because I haven’t presented the fifth one yet, and it’s one that we won’t actually worry about in this course, although I will explicate it below in just a minute.

But there is a fifth rule as well, and it’s something of an annoyance, because its triviality is out of proportion to the degree of complexity involved in explicating it. On the other hand, in its favor, it has the catchiest name, often being referred to as the Fallacy of Existential Import. Obviously this brings to mind Existentialism, the 20th century orientation in philosophy that concerns itself with matters like the meaning of life, death, and authenticity. Existentialism and the fallacy of Existential Import have about as little to do with one another as any two subjects can have.

It’s easy to tell that this form, AAI-1, when diagrammed on overlapping circles, is not going to come out looking valid. Why? Because in drawing the universal premises, only shading will happen, but the conclusion calls for an “x” to be inserted. You just don’t use “x” when you diagram a universal. On the other hand, it obeys all of the rules we just spelled out: the middle is distributed at least once, nothing is distributed in the conclusion (so neither “S” nor “P” needs to be distributed in the premises), and there are no negative statements to cause any trouble.

To grasp the issue, we have to return to the interpretation of universal statements as conditionals, and of particulars as existential, i.e., asserting the existence of their subject terms. We introduced this interpretation of these two quantities, in the chapter on Finishing the Square, as a result of seeing that all statements about non-existent things can be called “false” if they are taken to be asserting the existence of the non-existent things. This is problematic because it means that “All leprechauns are mischievous” and its contradictory “Some leprechauns are not mischievous” are both false, which destroys the notion of contradiction. We pretty much agree that that (the destruction of the relation and principle of contradiction) is too high a price to pay. And we find that reading the universal statement as a conditional is an easy enough shift to make, and one that allows the relation of contradiction to be preserved.

If anything is a leprechaun, that thing is mischievous.

If anything is a unicorn, it has one horn.

If anything is a man, it’s mortal.

If anything is a Pope, it’s not a Hindu.

Quite clearly, the “if” neutralizes any inclination to think that it’s important to understand these statements as making a claim about the existence or reality of their subjects. The point of the statements “all men are mortal” and “no Popes are Hindus” lies in the predication, not in the question of reality or being.

Moral: It looks like some forms are valid sometimes and not valid other times. The Existential Fallacy is, in this regard, a little like the informal fallacy called “Equivocation.” Equivocation can cause an argument with a valid syllogistic form to be invalid; it can’t make the form itself invalid, but that particular instance or instantiation of the form can be invalid. Strictly, it’s not the form that is valid sometimes and invalid other times; it is that the instance or instantiation of a valid form can be invalid under specific circumstances.

The Rappahanock River runs through Fredericksburg.

All things that run through Fredericksburg have feet.

So the Rappahannock River has feet.

Clearly this conclusion does not follow, but it’s not because there’s anything wrong with BarbaraBarbara is just fine, but the plurality of meanings of “run” causes these premises to be unable to establish this conclusion.

These two instantiations of AAI-1 show the point:

1. All leprechauns are Irish

Whoever’s Irish is mischievous

So Some leprechauns are mischievous

2. All O’Tooles are Irish.

Whoever’s Irish is mischievous .

So Some O’Tooles are mischievous.

The one is invalid because it results in the claim that something non-existent exists as part of building its predication, and that’s objectionable. The other is valid because it results in the claim that something that exists exists as part of building its predication, and it’s hard to object to that.

Maybe you can see what I meant about this rule being annoying: its triviality is out of proportion to the difficulty of explicating it. There is no reason to even consider this rule unless the syllogism you are considering has both premises universal and the conclusion particular.

To summarize the rules, then, let’s recall the names of the five fallacies:

Undistributed Middle Term

Illicit Major or Minor Term

Exclusive Premises

Double Negation Requirement

Existential Import

When you diagram a syllogism, and you conclude from the diagram that it is invalid, you should also be able to confirm that by identifying the rule it breaks.

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