PHIL102: Introduction to Critical Thinking and Logic

4a. Create Venn diagrams as a means to represent and reason about relationships among classes

• What is a Venn diagram?
• What is existential import?
• How do Venn diagrams provide a visual explanation of distribution?
• How do Venn diagrams reveal logically equivalent propositions or indeterminate inferences?

Drawing pictures helps a lot of us understand the logic of the categorical proposition. There are some who find pictorial representations of a proposition's logical structure off-putting. Fortunately, there are plenty of linguistic articulations of this material, but it is also the case that picture-making in categorical logic is systematic, and, once you get the hang of it, very handy. Let's start with a reminder of our four types of categorical proposition:

 

Here is a way to begin visualizing the relationships between the S and P classes, where a lowercase x symbolizes the particular affirmative, "there is at least one", or "some":

Start by focusing on each of the S circles. In the A-claim (All S are P), the entirety of the S class is inside the P class. In the E-claim (No S are P), the S class is entirely outside of the P class (and, it is worth noting, vice versa). Now, take a look at the arrangement of the circles for the two particular claims, Some S are P and Some S are not P. Notice that they overlap somewhat, which creates three internal areas where a relationship between the categories can be mapped:

In the I-proposition, Some S are P, the x is in the overlap area. Another way to read the diagram, without changing the logical structure of the claim is: There is at least one thing that is both an S and a P. In the O-proposition, Some S are not P.

Logicians use this arrangement of circles, which provides a template for all four claim types. The Venn diagram, named after 19th century English mathematician John Venn, provides a visual representation of each claim type's logical structure. Notice that the particular affirmative and particular negative Venn diagrams look just as they do above:

The next question is, how do we diagram the universal propositions? We need a way to express the complete inclusion of S-class members in the P class, and a way to express the complete exclusion of S-class members from the P class. In logic, this is achieved by shading or drawing lines through the area of a circle that is supposed to be empty or, more precisely, that cannot contain any members. It's as if the shading expresses the idea that there's no there there:

Here is another way to visualize the universal claims, where the lines drawn through the area of S that is outside of P, and the overlap area between S and P, for universal affirmative and universal negative propositions, respectively, designates the impossibility of anything being there:

Go back to the original diagram above, in which nested circles represent the universal affirmative proposition, All S are P. Now look at the Venn diagram for the same proposition. You can see that the shaded area of the S category means that there couldn't be an S that isn't included within the P category. Now, do the same for the universal negative proposition, No S are P. In the original diagram above, the two circles are separated, which shows that there isn't even one S that is also a P. The Venn diagram visualizes this relation by way of shading in the area of overlap between the S and P categories.

Let's now bring into the discussion existential import. Recall that universal claims express affirmative or negative relations between the subject and predicate classes. The affirmative relation in the A-proposition is that every S is included in the P category. The negative relation in the E-proposition is that every S is excluded from the P category. Here's a question: Are there any existing members of the S class? In other words, when we claim that all the S's are or are not members of the P category, are we also assuming such members exist? When we say, "Werewolves are scary", or "Leprechauns are gold hoarders", for example, we don't assume that the subject class has existing members – at least, most people agree that there aren't any werewolves or leprechauns. In logic, we can express the universal claims in terms of existential import: The subject class has existing members and is assumed in the Aristotelian or Traditional Logic. This assumption is diagrammed as follows, where the circled x denotes an existing member of the class in question:

So far, we have laid out a number of technical concepts that we will use repeatedly when working in both versions (Aristotelian and modern) of categorical logic. There is one more concept to discuss before we begin to use them in making inferences between propositions (immediate inferences) and from a set of two propositions (categorical syllogisms). It is called distribution. We say that a term (the term that denotes the S class or the P class) is distributed or undistributed. A term is distributed when the proposition refers to the entire class denoted by it:

Here is a summary of the distribution of terms:

 

To sum up, this is what we've learned about categorical propositions, so far:

• Categorical logic consists of four types of categorical proposition:
• Universal affirmative (A-proposition): All S are P
• Universal negative (E-proposition): No S are P
• Particular affirmative (I-proposition): Some S are P
• Particular negative (O-proposition): Some S are not P
• Venn diagrams provide visual representations of each categorical proposition's logical structure
• An x represents, e.g., the word or phrase, "Some", and "There is at least one".
• Shading denotes emptiness, or the impossibility of anything being in the shaded area of a circle;
• The Aristotelian, or traditional, system of categorical logic assumes existential import, that is, assumes that there is at least one existing member of the subject class in the categorical proposition
• The modern system of categorical logic does not assume existential import
• A term is distributed when it is entirely included into, or excluded from, a category:
• Universal propositions distribute the subject class, i.e., the A- and E-propositions distribute the subject class
• A term is also distributed when at least one member of the term-class is excluded from the other class:
• Negative propositions distribute the predicate class, i.e., the E- and O-propositions distribute the predicate class

We are going to start working with the concepts we've just learned:

The way we are going to work with these concepts is understood in terms of what logicians call the (traditional or Aristotelian) square of opposition. This is a diagram that expresses what immediate inferences can be made from one categorical proposition to another. An immediate inference is an inference from one proposition to another. You will likely find some of these inferences are pretty intuitive – they seem to "click" or make sense without you having to think much about them. Still other inferences will feel confusing. It is worthwhile for you to pay attention to each, and ask yourself why you find some so easy to understand, and why others aren't apparently comprehensible.

Before presenting various versions of the traditional square of opposition, let's think through each claim type as a premise for an immediate inference. In other words, beginning with the A-proposition, let's think about the inference to each of the remaining three proposition types, starting with the I-Proposition:

Given the proposition, All S are P, it follows that some S are P:

Notice that the assumption of existential import means that the A-proposition contains within it the I-proposition. This inference is known as subalternation, where the superaltern (the universal proposition) yields its corresponding subaltern (the particular proposition).

Now suppose again that the A-proposition is true. Given this, the O-proposition must be false. These two claims are contradictories: propositions that cannot be simultaneously true or simultaneously false.

Let's think about why A- and O-propositions are contradictories. When we assert the A-proposition is true, we are saying there cannot be even one S outside the P class. This is, however, just what the O-proposition asserts is the case. So, whenever the A-proposition is, or is assumed to be, true, the O-proposition must be false – and vice versa.

Lastly, assume the A-proposition is true. It follows that the E-proposition must be false:

 

You're probably able to recognize that, on the assumption of existential import, the E-proposition contains the O-proposition. So, if the A-proposition is true, the E-proposition must be false. What we cannot infer, however, is that A- and E-propositions are contradictories. We will see that they are not. They are, however, contraries: When a universal proposition is true, its corresponding universal is false. In this case, when A is true, its contrary, E, is false.

In the midst of the discussion about immediate inferences from a true A-proposition, you likely already drew at least one other inference around the traditional square of opposition, namely the inference from the E-proposition to its corresponding particular, the O-proposition. Why? Because, when thinking about the contraries, A-to-E, where we assume A is true, we saw that, assuming existential import, the E-proposition contains the O-proposition:

Similarly, you likely noticed that, just as A and O are contradictories, so also are E and I:

 

Notice that the true E-proposition means that there can't be even one entity in the area of overlap between S and P. The I-proposition, however, claims there is an entity in that area of overlap. So, if the E-proposition is true, the I-proposition cannot be true. The two claim types are contradictories.

Lastly, E and A are contraries, which means, assuming E is true, A must be false. That is because the E-proposition both denies that there can be anything in the overlap area between S and P, and because the E-proposition maintains there is at least one thing in the area of S outside of P. The A-proposition in the traditional interpretation of the universal, that is, on the assumption of existential import, maintains both that nothing is in the area of S outside of P, and that there is at least one S:

We have two more claim types to think about, in terms of immediate inferences around the traditional square of opposition: I and O. Let's start with the I-proposition. We already know that I- and E- propositions are contradictories, so let's start with that inference. In other words, if the I-proposition is true, the E-proposition must be false.

The remaining two inferences are a bit more complicated to think about. That's because the inference from a true I to either an A or an O is undetermined. In other words, if the I-proposition is true, there is no necessary inference: The resulting A-proposition, the superaltern, might be true or it might be false. Similarly, the resulting O-proposition, the subcontrary, might be true or it might be false. The structure of the original claim does not "force" the inference. Here is the Venn for the undetermined superalternation of the I-proposition:

Notice that the I-proposition does not include shading in the area of the S class outside of the P class. So, whether or not there could be anything in the area of S outside of P is an open question. Here are two ordinary language examples that show how the inference from the I-proposition to its corresponding superaltern could yield a true proposition or a false one:

Similarly, a true I-proposition does not yield a necessarily true or necessarily false O-proposition. In other words, subcontraries may be true at the same time, and a true I-proposition may yield a false subcontrary:

There may or may not be anything in the area of overlap between S and P in the O-proposition. Hence, its truth-value is undetermined. Here are two ordinary language examples that show how the inference from the I-proposition to its corresponding subcontrary could yield a true proposition or a false one:

The pattern of inferences is likely becoming clear at this point, as we move into considering the last set of immediate inferences from the assumption that the initial claim is true. A true O-proposition mirrors its subcontrary, the I-proposition. In other words, what is the case about the immediate inferences from the I-proposition to the three other claim types is mirrored in the corresponding particular claim, Some S are not P.  Here are the three inferences from the assumption that the O-proposition is true:

and

and

Below is the traditional square of opposition. As you look at it, think about each inference in terms of beginning with the assumption that the premise – the first categorical proposition – is true:

Notice that we haven't yet discussed inferences from a false premise. In other words, we have not discussed what inferences we may make when the initial categorical proposition is false. In fact, you already know two sets of inferences from a false premise: contradictories. If the A-proposition is false, the O-proposition must be true, and vice-versa. If the E-proposition is false, the I-proposition must be true, and vice-versa. Think about these relations in terms of the Venn diagrams:

We can also infer, based on the other inferences we know, the following necessities:

What is left undetermined is the truth-value of a universal whose corresponding contrary is false. Here are a couple of examples to show the problem:

If you think the traditional (or Aristotelian) square of opposition is complicated, you're correct – at least compared with the modern square of opposition. What makes the traditional square so complicated is the fact that, in most cases, you must know the truth-value of the initial proposition in order to determine the value of the inference – and even then, there are instances where the inference is undetermined.

The modern interpretation of the universal claim type – the A- and E-propositions – does not assume existential import. In other words, there is no assumption of an existing member of the subject class in the universal claim. By suspending judgment about the existence of members in the subject class of a universal claim, the number of inferences in the square of opposition is severely restricted. In fact, there are only two sets of inferences that can be made on the modern square of opposition: contradictories.

Recall the two ways of diagramming the universal claim:

Here is the modern square of opposition:

We have seen some inferences we can make on the traditional and modern interpretation of existential import. More specifically, we've seen when we must infer one claim type from another. Now we turn our attention to what happens when we make internal changes to the quantity and quality of a categorical proposition, as well as the order of the subject and predicate terms. Let's start with an overview of the three types of inferences: conversion, contraposition, and obversion.

All S are P

Converse: All P are S
Obverse: No S are non-P
Contrapositive: All non-P are non-S

No S are P

Converse: No P are S
Obverse: All S are non-P
Contrapositive: No non-P are non-S

Some S are P

Converse: Some P are S
Obverse: Some S are not non-P
Contrapositive: Some non-P are non-S

Some S are not P

Converse: Some P are not S
Obverse: Some S are non-P
Contrapositive: Some non-P are not non-S

Let's begin with taking the converse of a proposition. The inference involves simply switching the subject and predicate positions. The quantity and quality of the proposition are left untouched. The inferences are as follows:

Some of the inferences will feel mentally off, while others will feel obviously correct. Here are the evaluations:

*We will see shortly that, on the assumption of existential import, that is, on the traditional or Aristotelian interpretation of the universal, conversion by limitation makes possible the conversion of the A-proposition.

A few examples may help you think through the valid and invalid inferences. Consider whether or not converting a claim results in one that is logically equivalent to it:

Here are the Venn diagrams that provide a visual demonstration, on both interpretations of the universal – that is, assuming and not assuming existential import, of equivalence and non-equivalence:

 

The only way to successfully convert an A-proposition is by limitation. In conversion by limitation, an inference is made to the I-proposition – subalternation. Conversion of the I-proposition is valid, as we will see momentarily. Hence, by first limiting the A-proposition through subalternation, the resulting conversion is successful (valid). It is important to remember that conversion by limitation is possible only on the assumption of existential import, that is, on the traditional or Aristotelian interpretation of the universal. That is what makes possible the inference to the I-proposition, that is, the A-proposition's subaltern. Such an inference is never valid on the modern interpretation of the universal. We know this because subalternation is not a valid inference.

Notice that the conversion of the I-proposition and the conversion of the O-proposition is the same under either interpretation. Remember that the concept of existential import applies only to universal claims. Here are Venns for the converted particular claims, I and O:

 

When the diagrams do not match up, we can see that the inference from a proposition to its converse is invalid – the two propositions do not make the same claim.

Contraposition is the mirror inference for A- and O-propositions. Just as conversion is valid for E- and I-propositions, contraposition is valid for A- and O-propositions. (Moreover, just as conversion by limitation is valid for the A-proposition, and only on the Aristotelian or traditional interpretation of the universal, contraposition by limitation is possible for the E-proposition.) The internal manipulation is, however, more complex. First, let's walk through the steps, which can be taken in any order, but we will begin with what we do when converting a claim:

Let's try to make sense of each inference, by way of an example: The contrapositive of all dogs are animals is that all non-animals are non-dogs. Another way to put this is in conditional form: If it's not an animal, then it's not a dog. We can see that the contraposed A-proposition says the same thing as the original claim. This is not the case with the contraposed E-proposition. Let's take the example, no dogs are cats. The contrapositive is, no non-cats are non-dogs. This means that whatever is a non-cat is also a non-dog. Surely, however, that can't be correct. It can't be correct to say, no dogs are cats is equivalent to saying, if it's not a cat then it's not a dog. The shading in the area outside of both circles, on both the traditional and modern interpretations, provides us with a visualization of diagramming the class complement of a universal negative.

Contraposition by limitation is possible for the E-proposition only on the assumption of existential import, that is, on the Aristotelian or traditional interpretation of the universal. First, the subaltern is inferred, since as the diagram shows, a true E-proposition contains its corresponding particular, the O-proposition. The contraposition of the O-proposition is equivalent to the original, as we will see now.

 

As with the conversion of particular claims, notice that the contrapositive of the I-proposition and the contrapositive of the O-proposition is the same under either interpretation. Remember that the concept of existential import applies only to universal claims. Here are Venn diagrams for the contraposed particular claims, I and O:

The contraposed I proposition asserts that there is a non-P – there is something outside the P-class – that is also a non-S – something that is outside the S-class. Hence, the x in the area outside both categories. The contraposed O-proposition asserts that there is at least one non-P that is also not a non-S – which is to say, there is at least one non-P that is an S. It is the equivalent of the original.

Lastly, obversion is a valid inference on both interpretations of the universal. It is achieved by a two-step process:

 

Here are some examples to illustrate and elucidate the process of obversion:

Here is a table that sums up the last three immediate inferences, and their evaluations:

	All S are P	No S are P	Some S are P	Some S are not P
Converse:	All P are S Not equivalent to the original (invalid) except on the traditional interpretation, where the I-prop. is inferred and then converted.	No P are S Equivalent to the original (valid)	Some P are S Equivalent to the original (valid)	Some P are not S Not equivalent to the original (invalid)
Obverse:	No S are non-P Equivalent to the original (valid)	All S are non-P Equivalent to the original (valid)	Some S are not non-P Equivalent to the original (valid)	Some S are non-P Equivalent to the original (valid)
Contrapositive:	All non-P are non-S Equivalent to the original (valid)	No non-P are non-S Not equivalent to the original (invalid) except on the traditional interpretation, where the O-prop. is inferred and then contraposed.	Some non-P are non-S Not equivalent to the original (invalid)	Some non-P are not non-S Equivalent to the original (valid)

To review, see:

• Categorical Logic and The Venn Test of Validity for Immediate Categorical Inferences
• Basic Notation
• Universal Statements and Existential Commitment

 

4b. Evaluate the validity of arguments using Venn diagrams

• How do Venn diagrams show an argument is valid or invalid?

A Venn diagram is one way to determine whether or not a categorical syllogism is valid. Here are the steps for completing a Venn diagram:

1. Diagrams are completed only for the premises
2. If one premise is a universal, and the other is a particular, diagram the universal premise first. That's because the shaded area may preclude diagramming an x in a given area
3. When the premises do not force diagramming an x entirely inside or outside an area for a particular claim, the x is placed on the line of a relevant circle

You already know how to diagram individual categorical propositions. You will bring that skill to bear in the process of diagramming the premises of a categorical syllogism. First, rather than diagramming the relevant elements of two overlapping circles, a Venn diagram for the categorical syllogism involves three:

The circles need not be arranged in the order above, as you can see here:

 

A valid argument is one whose premises contain the conclusion. So, when diagramming the premises of a valid argument, the conclusion appears. Below are several examples, following the modern interpretation of the universal. Some are valid, some are invalid.

Example 1:

Example 2:

Example 3:

To review, see:

• Venn Validity for Categorical Syllogisms
• Syllogism
• The Logic of Venn Diagrams

 

4c. Describe the limitations of Venn diagrams as assessment tools

• What are some limitations of the Venn diagram as an assessment tool?

Diagramming categorical propositions is a powerful tool for determining both equivalence and validity. As we have seen, however, the relevant elements are specific. For example, a categorical syllogism is a three-term argument. So, anything more complicated becomes cumbersome for the Venn process. Moreover, the Venn diagram is limited to classes of objects, and so cannot represent individual things.

To review, see Limitations of Venn Diagrams.

 

Unit 4 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• categorical proposition
  
• complement
• contradictories
• contraposition
• contraposition by limitation
• contrapositive
• contraries
• converse
• conversion by limitation
• distribution
• existential import
• immediate reference
• modern system
• obversion
• square of opposition
• subaltern
• subalternation
• subcontraries
• superaltern
• traditional (Aristotelian) system
• undetermined
• universal propositions
• Venn diagram