Friday, March 16, 2007

Venn Diagrams - Part IV

For the last essay in this series, I want to take a couple of real life examples and analyze them using Venn diagrams. If you stumble across this in the backwaters of the internet and have a question about logic or Venn diagrams, please feel free to add a comment.

My friend Terry West complains about an argument for believer's baptism. He writes,

Premise 1. Believers are to be baptized.
Premise 2. Infants cannot believe.
Conclusion 3. Therefore Infants cannot be baptized.

Now, as has been stated several times and in several ways in the last two articles, the most glaring fallacy is that the subject of premise one and two are not the same, therefore what is true of the subject of premise one cannot be said to be untrue of the subject of premise two.

We can analyze the argument using Venn diagrams and not worry about a verbal justification for any objections. The terms are Believers, Baptized, and Infants. Obviously we need three circles to represent these terms.

And if we take the first premise as All Believers are to be baptized, then we must shade out the part of the Believers circle that falls outside the Baptized circle. The shading represents emptiness, so the diagram below means that there are not any believers who are not baptized.

The second premise says that infants are not believers, so we shade all of the Infants circle that falls within the Believers circle. This will show that all the infants are not believers.

Looking at the results in the above diagram, we see that part of the Infants circle that falls within the Baptism circle remains unshaded. Thus the conclusion, Infants cannot be baptized is unsupported.

But we want to rethink this argument a bit. Obviously the first premise, "all believers are baptized" is untrue. The thief on the cross is an example, or Abraham. The dispute is about who is properly to be baptized, so we should say "all believers are properly to be baptized." But really, this doesn't get at the force of the argument, so we should really put the major premise this way: only Believers are Properly Baptized.

Or, more correctly: All Properly Baptized Persons are Believers So now the argument will look like this:

All Properly Baptized Persons are Believers;
no Infants are Believers;
therefore no Infants are Properly Baptized Persons.

The major premise of this argument would look like this when diagramed:

This first diagram shows that all properly baptized persons are believers. Now we add shading to represent the minor premise, which is, "no infants are belivers."

With the argument revised like this, the conclusion "no infants are properly baptized persons" is valid. If the premises are true, then the conclusion is certainly established.

Terry West objects that this is circular reasoning. The syllogism itself is not circular. But the argument does assume the truth of the major premise, "all properly baptized persons are believers." If that premise is put in dispute, (or the minor premise, for that matter) then it is, of course, incumbent on the arguer for believers' baptism to support that premise. Terry would object that the major premise is perhaps untrue and unsupported. But considering this question is taking us beyond the use of Venn diagrams into the question of Christian baptism. (I remain a credo-baptist, by the way.) I just wanted to look at the diagrams to show how they would be drawn with a slightly different argument. The exercise is also useful to show that one must be very careful in defining the terms for the syllogism and constructing the premises accurately. The diagrams themselves are extremely useful in analyzing and justifying critiques of categorical syllogisms.

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Wednesday, March 14, 2007

Venn Diagrams - Part III

The standard categorical syllogism has three categorical statements or propositions: two premises — major and minor — and a conclusion. These three statements contain three terms. In the examples in the previous essay in this series, oranges, citrus fruit, apples, basketball players, people over 7 feet tall, F, and G are all terms. In a standard categorical syllogism, each term occurs in exactly two of the statements.

As an exercise, identify the terms in this syllogistic argument:

All citrus fruit is acidic,

All oranges are citrus fruit;

Therefore all oranges are acidic.

The terms are citrus fruit, acidic, and oranges. Notice that each of the terms occur in two of the statements: citrus fruit occurs in the major premise and the minor premise; acidic occurs in the major premise and the conclusion, and oranges occurs in the minor premise and the conclusion.

In a Venn diagram, the circles represent the terms (i.e., in the above argument, citrus fruit, acidic, and oranges) of the syllogistic argument. So, to diagram the citrus fruit argument, we need three circles, as follows:

The first statement says citrus fruit is acidic. As we saw in part II, in order to represent this in the diagram, we shade — remember, shading represents emptiness — the part of the citrus fruit circle that falls outside the acidic circle, thereby showing that there is no citrus fruit that is not acidic.

The second statement, oranges are citrus fruit, is shown in a similar way; we shade all of the area of the oranges circle that falls outside the citrus fruit circle.

All we need to do now is examine the diagram to see what conclusions are supported. At a glance we can see that all of the oranges circle that is not empty (not shaded) falls within the acidic things circle, and thus the conclusion all oranges are acidic is proved by the premises.

In the next installment, I will use an example from a dispute about Christian baptism to demonstrate how to spot a bad argument using Venn diagrams.

Friday, March 09, 2007

Venn Diagrams - Part II

This brief essay, the second in my series on Venn diagrams, is based on the work of W. V. Quine, Methods of Logic, 4th ed., and Irving M. Copi and Carl Cohen, Introduction to Logic, 10th ed. A categorical syllogism, in its classic form, contains three categorical statements (Quine) or propositions (Copi & Cohen). Two such statements constitute the premises -- a major premise and a minor premise -- and the third constitutes the conclusion.

A premise or conclusion will be in one of four forms: A (all F is G), I (some F is G), E (no F is G), or O (some F is not G). The traditional logical system built up labels for these premises with rules for proper inference, i.e. methods for constructing and evaluating arguments. The labels and rules may impress one as arcane and tedious, while Venn diagrams are simpler, much less tedious, and more enlightening. Venn diagrams do have their limits, but for most simple arguments they are a thing of beauty.

To represent the four types of categorical statements using Venn diagrams, we need two circles. The following statement is an A statement:

All oranges are citrus fruit.

We need a circle for oranges and another intersecting circle for citrus fruit. Like so:

To show that all oranges are citrus fruit, we need to shade the part of the oranges circle that falls outside the citrus fruit circle.

Remember from part I of this series that shading represents emptiness. So the diagram above shows that there are no oranges that are not citrus fruit. That is, the area of the oranges circle that falls outside the citrus fruit circle is empty. (As an interesting aside, the shape that is shaded in the above diagram was called a lune by Quine.)

Let's diagram a different kind of statement:

No apples are citrus fruit.

This is an E statement: No F is G. It is diagramed as follows:

The shape that we shaded here (Quine called the shape a lens) shows that the area of overlap between the apples circle and the citrus fruit circle is empty. That is, there are no apples that are citrus fruit (and, conversely, no citrus fruit are apples). The Venn diagram shows this relationship beautifully and automatically shows us the valid inference no citrus fruit are apples.

The third type of statement is the I statement: some F is G. A typical statement might be some basketball players are over 7 feet tall. For this statement, we would put a cross in the lens between the circle representing basketball players and the circle representing people over 7 feet tall.

The cross shows us that there is at least one basketball player who is over 7 feet tall. And we can automatically see the converse, that at least one person who is over 7 feet tall is a basketball player.

Finally, the O statement, some F is not G is represented by the following diagram:

Notice that we put the cross in the lune of the F circle to show that there is at least one F that is not also a G. A typical statement of this type might be some basketball players are not over 7 feet tall.

The next part in this series will show how a full categorical syllogism can be diagramed.

Thursday, March 08, 2007

Venn Diagrams - Part I

I profess to be examining the logic of the theology of John Calvin. In order to do that, we have to have some ideas about logic itself and we have to have tools for analyzing logic. One common form of argumentation is Aristotelian or categorical syllogisms. Categorical syllogisms are those syllogisms that have a major premise, a minor premise, and a conclusion. The famous one runs like this:

All men are mortal;

Socrates is a man;

Therefore Socrates is mortal.

In my opinion, one of the best and simplest ways to analyze categorical syllogisms is Venn diagrams. They make the analysis simple and are also wonderful for making a visual presentation of one's analysis. Other techniques are required for more complicated arguments. But even complicated arguments can often be broken down into a series of simple syllogisms that are susceptible to analysis by Venn diagrams.

I thought I would put together a couple of short essays on the use of Venn diagrams. The material here is based on W. V. O. Quine, Methods of Logic, and Copi Cohen, Introduction to Logic, 10th edition.

In Venn diagrams, shading represents emptiness. So to represent "there are no unicorns," we would make a diagram like this:

A cross in a circle means "there is one or more." So to represent "there is at least one unicorn," we would make a diagram like this:

What does an empty circle mean? In modern logic, an empty circle means a lack of information. So the following diagram means, "there are zero or more unicorns."

More to follow in the next essay.