Deductive Logic and Venn Diagrams

 

Deduction and Validity

Before getting to Venn Diagrams we should simply review here some basic points about deductive arguments generally and validity in particular. Deductive arguments are arguments wherein the conclusion is necessarily true (assuming true premises and a valid form). In other words, it is impossible to have a situation where: (1) the premises of the argument are true, and (2) the form of the argument is valid, and (3) the conclusion is false. The reason for this is very simple: the conclusion of a deductive argument does not contain any new information --it is already contained (in some implicit form) in the premises itself.

Further, we can see from the above that the concept of validity is very important for deductive arguments. The conclusion is guaranteed to be true only if the form of the argument is valid and the premises are true. Hence, we must know how to determine whether or not a deductive argument is valid. NOTE: validity and invalidity apply only to deductive arguments. Inductive arguments are neither valid nor invalid.

So, what is validity? Questions of validity are purely formal. In testing for validity we are not in any way concerned with the actual content of an argument. We are only concerned with its form --the way in which the premises are supposed to provide support for the conclusion. So, the first step in testing for validity will always involve some process of abstraction --that is, some process wherein we ignore the content and focus only on the form of the argument. Abstraction will almost always involve, in this context, the replacement of particular contents with “variables” --usually letters (A, B, C, D). Second, we then arrange the variables in the same form as they had in the argument itself --that is, we arrange them according to the specific form of the argument.

 

Example 1:

 

1. All cows are animals

2. Betsy is a cow.

3. Therefore, Betsy is an animal.

 

Note the “content” of this argument. What are we talking about? Betsy, cows and animals. So, we assign variables.

Cows = C

Animals = A

Betsy = B

 

Now, arrange the variables in the same form as they have in the argument and we get:

1. All C are A

2. B is C

3. Therefore, B is A.

 

Once we have the argument translated into variable form we are going to ask a simple question: given that the premises are true, does the conclusion necessarily follow? There are, generally, two different ways of answering this question, one is impossibly hard, the other is very easy.

The impossibly hard method of answering this question is to go through every possible combination of true premises that meet the form we are testing and determine if it always gives us a true conclusion. What we are looking for is simple: can we find an argument where the premises are true, but the conclusion is false. If we can, then the form is invalid. So, for example, we can replace C with Humans, A with Mortal, and B with Socrates, getting: All humans are mortal, Socrates is a human, therefore Socrates is mortal. Then we can replace C with Dog, A with Mammal, B with Daisy, and this can go on forever. So, while this method works, it takes a very very long time.


The easy method is to simply ask yourself: will this form yield true conclusions given true premises no matter what the specific content is? Testing this question is simple. Again, there are two possibilities here. If you have categorical syllogisms then you test this question by using Venn Diagrams. If you have compound statements using logical operators, then you use Truth Tables. We are going to look only at Venn Diagrams here, but the basic principle is the same: assuming that the premises are true, does the conclusion necessarily follow?

 

Venn Diagrams and Validity

 

A Venn Diagram (V) is a clear cut method for determining the validity or invalidity of any form of categorical Syllogism. A categorical syllogism is a deductive argument that relates (through patterns of inclusion and exclusion) categories of things and that contains two (and only two) premises. (Of course, a categorical syllogism may have one explicit and one implicit or hidden premise.) Now, given that categorical syllogisms articulate relations of inclusion and exclusion between categories, we can determine that there are four (and only four) basic categorical claims. These are:

 

 

Universal Affirmative.

 

This is a statement that claims that everything that falls within one category also falls within some other category.

E.G. “All humans are animals.” Note here that we are saying there cannot be something that is human and is not animal, or that everything we would put in the “human” category, we would also put in the “animal” category. Note further, that this is not a reversible relationship. “All A are B” does not equal “All B are A.”

We can call this the relation of Universal Inclusion.

 

Universal Negative.

 

This is a statement that claims that nothing within one category also falls within some other category. In other words, to put something in one category excludes it from some other category.

E.G. “No human is reptilian.” Note here that we are saying that you cannot put something in both the “human” category and at the same time in the “reptile” category. Note that this is reversible: “No A are B” is the same as “No B are A.”

We can call this the relation of Universal Exclusion.

 

Particular Affirmative.

 

This is a statement that claims there are some members of one category that also are members (or fall within) another category.

E.G. “Some humans are rational.” Note here that we are saying that some of the things that belong to the category of “human” also belong to the category of “rational.” Further note: there is no implication that if some humans are rational that some humans are not rational.

We can call this the relation of Partial Inclusion.

 

Particular Negative.

 

This is a statement that claims that there are some members of one category that are not members (or do not fall within) another category.

E.G. “Some humans are not reptilian.” Note here that we are saying that there are some members of the category of “human” that do not fall within the category of “reptile.” Again, the fact that some humans are not reptilian does not imply that some humans are.

We can call this the relation of Partial Exclusion.

 


These four basic claims form a network of necessary relations. The truth/falsity of one type of claim about some content will necessarily imply the truth/falsity of another type of claim about the same object. This set of necessary relations is known as the “square of opposition” (of which there are two versions). See your textbook, page 165 for an explanation of the square of opposition. We will not spend any time on it here. But, as we work through the process of diagramming a categorical syllogism we will see these relations emerge.

 

Diagramming Categorical Syllogisms (DCS).

The first step in DCS is very simple. Since arguments are never written in a nice easy script, it is often necessary to re-write the argument into a form that can be easily analyzed and evaluated. We have already seen this with casting, and we will see this here as well. The key for categorical syllogisms is to identify the categories being used, and the relations being drawn between those categories.

 

 

Step One:

 

First translate the argument into a categorical structure. This involves identifying the categories that are being related to one another, and the manner of relation (Universal Affirmation, Universal Negation, Particular Affirmation, and Particular Negation).

 

 

Generally speaking this is easy enough to do. Look at the following examples:

 

 

Most apples are red.

 

We identify the categories here as “apples” and “things that are red.” Then, we note that “most” means the same as “some.”

 

 

Some apples are things that are red.

(The formal relations here are bolded, the categories are underlined.)

 

Many people are not as stupid as politicians would have us believe.

 

We identify the categories here as “people” and “those things that are as stupid as politicians would have us believe.” NOTE: that we do not include the not here as part of the category. You should always leave negations (not, no) outside of categories. Then we see that “many” here means the same as “some” and since we see a “not” here we obviously have a “some are not” or particular negation.

 

 

Some people are not among those things that are as stupid as politicians would have us believe.

 

Rather clumsy, I know, but it helps in the diagramming.

 

There are no politicians that are honest.

 

The categories here are clearly “politicians” and “things that are honest”, the “no” indicates we have a Universal Negation.

 

 

No politicians are things that are honest.

 

Everyone who works in Washington D.C. is a fool.

 

The categories are “people who work in Washington D.C.” and “fools” and the relation is clearly universal affirmation.

 

 

All people who work in Washington D.C. are fools.

 

On the whole, this process is rather simple and with practice you will become an expert at it. There are, however, a few simple rules to keep in mind.

 

Rules for Translation (See textbook, pages 161-162)

 

 

Rule One

 

If you have a statement that reads: “All people are not stupid” or any form where you have “All” linked with “are not” then this cannot be translated. This is because such claims can be translated as either “Some people are not stupid” or “No people are stupid.”

 

 

Rule Two

 

When you have an “individual” named in the argument (like Socrates) treat the name as a category. Hence, Socrates becomes “people who are Socrates.”

 

 

Rule Three

 

Adjectives become categories. Thus, “Fred is smart”, becomes “All people who are named Fred are people who are smart.”

 

 

Rule Four

 

Turn verbs into categories as well. Thus, “Fish swim” becomes “All fish are things that swim.”

 

 

Rule Five

 

Categories should always be positive --there is no such thing as a negative category. Hence, negations should always be part of the form of the translation, never the category itself. Two examples here:

 

A. “There are many things that are not human made.” This becomes: “Some things are not things made by humans.”

 

B. “There are many things that are unpleasant.” This becomes “Some things are things that are unpleasant.”

 

Notice that in (B) we leave “unpleasant” alone --this is because “unpleasant” is not a negation --it is not a negation of the whole category but stands as a category separate from those things that are pleasant. Again, this takes practice, but you will get the hang of it.

 

 

After completing this step, you then want to replace the categories with variables (letters):

 

 

Step Two:

 

Replace categories with variables.

 


This is a relatively simple step and many people as they become better and better at this process skip step one altogether and proceed to step two. Very simply it involves assigning some variable to each of the categories. Usually it is best to pick a variable (pick a letter) which is related to the category itself. Thus, use “C” to stand for “cat” and “D” to stand for “dog.” Then, you should rewrite the argument in variable form, much like we did in the first example:

 

 

 

 

1. All cows are animals

2. Betsy is a cow.

3. Therefore, Betsy is an animal.

 

Note the “content” of this argument. What are we talking about? Betsy, cows and animals. So, we assign variables.

Cows = C

Animals = A

Betsy = B

 

Now, arrange the variables in the same form as they have in the argument and we get:

1. All C are A

2. B is C

3. Therefore, B is A.

 

Then, after step two, we are ready for the process of diagramming. The process of diagramming is again very simple. We use over-lapping circles to represent the various categories and their interrelation. Since all categorical syllogisms will have three (and only three) categories, there will always be three circles.

 (See Fig. 1)

 

 

 

Fig 1.

 


The “seven domains” found in this figure are important. Each represents a different possibility of inclusion and exclusion. We shall look at each in turn:

 

 

 

 

Domain One

 

This domain is where you put something which is “A,” but is in neither of the other circles, “B” or “C.”

 

Domain Two

 

This domain is where you put something which is in both “A” and “B” but not in “C.”

 

Domain Three

 

This domain is where you put something which is in “B” but not in “A” or “B.”

 

Domain Four

 

This domain is where you put something that is both “A” and “C” but not in “B.”

 

Domain Five

 

This domain is where you put something that is in all three circles.

 

Domain Six

 

This domain is where you put something that is both in “B” and “C” but not in the “A” circle.

 

Domain Seven

 

This domain is where you put something that is only in “C” but is neither in “A” nor “B.”

 

Notice that these domains all deal with relations of inclusion and exclusion. When diagramming specific claims (e.g. All A are B, No A are B, Some A are B, Some A are not B) we use two (and only two) basic methods: Shading and Putting an X.

 

Shading and Putting and X

 

Shading is only used when dealing with “All” and “No” claims (Universal affirmation and negation), putting an X is used only when dealing with “Some are” and “some are not” claims (Particular affirmation and negation). This gives us then, Steps three and four (with a few corresponding rules):

 

 

Step Three

 

Shade in all the “All” or “No” claims found in the argument. Note that you should always start with the “all” or “no” claims. Never begin diagramming with a “some” or “some are not” claim.

 

Rules for Shading

 

If you shade in a section you are literally saying that “nothing can go in there”

 

 

 

Hence, if we have a claim that says: “All A are B” we are going to shade in all of the “A” circle except where it overlaps the “B” circle (this includes the places where it overlaps the “C” circle). By doing this we are saying that you cannot put something into the “A” circle without at the same time putting it into the “B” circle.

 

 

 

If you have a claim that says “No A are B” then we would shade in all of the area where “A” and “B” overlap (including where both overlap with “C”) in order to indicate that you cannot put something in the place where “A” and “B” overlap.

 

 

 

 

 

Step Four

 

If there are any “some” or “some are not” premises, put an “X” where the premise indicates an “X” should be.

 

Rules for “X”

 

For “some .... are...” claims, you should put an “X” in the overlap between the two categories indicated.

 

 

 

For “some ... are not....” claims, you should put an “X” outside of the overlap between those claims.

 

 

 

If there is more than one domain in which the “X” might go, then you put the “X” on the line between the two domains.

 

 

 

Any time there is an “X” on the line, then you necessarily have an invalid argument.

 

 

 

 

 

Step Five

 

Now, you check for validity. Note, that you only diagram the premises. After having diagramed the premises, the conclusion should be evident. If it is not evident, if you have to do more work to make the conclusion evident, then the argument is clearly invalid.

 

This will be easier if we look at some examples.

 

A simple example first: [Note: I cannot shade or put an X in these diagrams do to the limits of word and my knowledge of how to manipulate it –so, try to figure out how to do these by reading the descriptions in the third column.]

 

 

1. All A are B

2. All B are C

3. All A are C

 

 

 

1. Only diagram the premises (1 & 2)

2. Start with “All” or “No” claims (universal claims) --only universal claims, so no problem there.

3. Shade Universal claims.

4. This is a clearly valid claim because the conclusion --the idea that All A are C is clearly indicated here --there is no way to put something into the A circle without also putting it into the C circle, since only Domain 5 is open to the A circle and this includes the C circle.

 

1. All A are B

2. No C are B

3. No A are C

 

 

 

 

 

 

 

 

 

 

 

 

1. Only diagram the premises (1 & 2)

2. Start with “All” or “No” claims (universal claims) --only universal claims, so no problem there.

3. Shade Universal claims.

4. Once again, this is a clearly valid claim because the conclusion --the idea that No A are C is clearly indicated here. There is no place where the A and the C circle overlap that has not been shaded in.

 

1. No A are B

2. All B are C

3. No A are C

 

 

 

 

 

 

 

 

 

 

 

 

1. Only diagram the premises (1 & 2)

2. Start with “All” or “No” claims (universal claims) --only universal claims, so no problem there.

3. Shade Universal claims.

4. Now, notice, that the conclusion claims that No A are C, this would mean that the area between A and C is completely shaded in (domains 4 & 5), but it is not. Hence, the conclusion is not contained in the premises, so this argument is INVALID.

 

1. No A are B

2. Some Bare C

3. Some C are not A

 

 

 

 

 

 

 

 

 

 

 

1. Only diagram the premises (1 & 2)

2. Start with universal claims (premise 1).

3. Shade universal claims: here we shade out the area between A and B to indicate that there is no possibility of something being both A and B (domains 2 & 5)

4. Now put in the X. Note we are saying that some B are C. So the X should go into the overlap between B and C. There is only one open domain in the overlap between B and C (domain 6). So we put an X there.

5. Now, is the conclusion, that some C are not A here? Where is the X? Is it in the C? Yes. Is it in the A? No. Well, that’s it. That is what the conclusion calls for. So, this is indeed a Valid argument.

 

1. Some A are B.

2. All B are C.

3. Some A are C.

 

 

 

1. Only diagram the premises (1 & 2)

2. Start with the universal claims --here that is premise 2.

3. Shade universal claim: shade all area of B that is outside of C (domains 2 & 3). This shows that something cannot be B without at the same time being C.

4. Put in the X (premise 1). Now, the X should go in A and in B. So, it must go in the overlap. However, there is only one place in the overlap that has not been shaded out --domain 5. So, we put the X there.

5. Is this valid? The conclusion says that Some A are C. With an X in domain 5 we are saying the same thing --so this is valid.

 

1. Some A are B

2. All C are B

3. Some A are C.

 

 

 

 

 

 

 

 

 

 

1. This looks like the last one, but is not.

2. Only diagram the premises, 1 & 2

3. Start with universal claims, again 2.

4. Diagram universal claims --shade in area of C outside of B (domains 4 & 7).

5. Now put an X. The claim is that you need to put an X between A and B. There are two possibilities: domain 2 and domain 5. Are either shaded out? No. So the X could go into either area --since it is not clear which area into which it should go, we put the X on the line between the two areas.

6. This is invalid because there is an X on the line.

 

Okay. Now you just have to practice this. Good luck.