The Magic 8 Ball

The Magic 8 Ball novelty item has been around since the 1950s. I had one when I was a kid in the 1970s, and as a business executive, I always had one on my desk. Sometimes colleagues would come to see me just to consult the Magic 8 Ball. 

The Magic 8 Ball is a plastic ball styled as an oversized 8 ball as used in the game of pool. The ball is hollow, and filled with an opaque blue liquid. You ask the Magic 8 Ball a yes-or-no question, shake it vigorously, and then turn it over to peer through a small round window on its underside. Through the opaque blue liquid, a triangle appears with an answer to your question. The triangle is one side of a buoyant 20-sided die that floats to the surface of the blue liquid when you turn over the Magic 8 Ball. 

Online Virtual Devices

Try out the Interactive Magic 8 Ball, and the Google Dice Roller. Familiarize yourself with the way these work. Note that the Google Dice Roller includes a 20-sided die. So, a comparison of the two virtual devices goes some ways to explaining the mechanism of a physical Magic 8 Ball. Also check out this 3D model of a regular icosahedron. A 20-sided die is a regular icosahedron. If available, experiment with a physical Magic 8 Ball, and a physical 20-sided die. 

Randomness

Note that the Magic 8 Ball uses randomization. It has 20 answers that it can display. One of those answers is chosen at random through the appropriate mechanical usage of the device. A big part of the entertainment value of the Magic 8 Ball comes from its randomness. In the exercises below, I want you to lean into the kind of randomness that the Magic 8 Ball introduces. 

Specific Regarding Form

The exercises work because they rely on certain formal characteristics of the activities. There are formal characteristics of the Magic 8 Ball, and the rolling of dice. There are formal characteristics of English grammar. And there are formal characteristics of logical inference. The following exercises will be very specific regarding form.  

Randomization of Content

Within critical thinking, the obverse of "form" is "content." The "content" of a short argument is what what the argument is about. The main content is the factual claim that constitutes the conclusion. The conclusion (or thesis) is some claim about the world. In order to understand that claim, you have to rely on your experience of the world. With content there a nuances of meaning that come into play, because each of us experiences the world in a unique way. 

Deductive logical form is not nuanced. It is precise. And it is precisely the same for everyone. Furthermore, it has nothing to do with content. But ordinary deductive arguments naturally arise with both form and content infused into one another. Thus, it is quite challenging to see the content and form as distinct from one another. The following exercises will help to overcome that challenge. 

The exercises help to dissociate form from content by randomizing the content of the arguments that will be created. Lean into that randomization of content. If you follow the directions correctly, the arguments concocted should by and large be paradoxically absurd. On the one hand, they won't make sense in terms of content. But, on the other hand, they will be perfectly understandable in terms of their logical form. 

Group Exercises

The exercises below are designed as group exercises, because it helps to randomize the content of the arguments created. However, it is possible to complete the exercises solo (see instructions within the first exercise).

Magic 8 Ball Questions

Example

You can ask the Magic 8 Ball any yes-or-no question, and through its formal mechanical features, it will give you an answer that is grammatically appropriate. For example, you could ask: 

Will Leslie win the 27th annual ping pong tournament? 

And then the Magic 8 Ball might answer: 

It is decidedly so. 

Exercise

Complete the Magic 8 Ball Questions Exercise in order to create a random list of yes-or-no questions.

Magic 8 Ball Propositions

A "proposition" is a claim of fact about the world. A proposition must be logically either true or false (but not both). In formal deductive logic, the only type of sentence allowed is a proposition. 

Example

We can take any yes-or-no question appropriate to the Magic 8 Ball, and covert it into a proposition. For example, the question: 

Will Leslie win tomorrow's ping pong tournament?

can be transformed into the factual claim: 

Leslie will win tomorrow's ping pong tournament. 

Note how this conversion is largely just a matter of rearranging the words. This is typical. 

Exercise

Complete the Magic 8 Ball Propositions Exercise in order to create a list of propositions appropriate to formal deductive logic. 

Magic 8 Ball Arguments

In deductive logic, an "argument" is a set of propositions. One of the propositions is the "conclusion" of the argument. And the remaining propositions are collectively the "premises." An argument asserts that the truth of the premises guarantees the truth of the conclusion. 

There are several classic argument forms that are studied in deductive logic. We will examine just a few of those. 

Valid Forms

An argument form is said to be "valid" if the truth of the premises (each and every one being true all together) in fact does guarantee the truth of the conclusion. 

Modus Ponens

One valid form of argument is "Modus Ponens." The pattern (form) looks like this: 

If P, then Q.
P.
Therefore, Q. 

The variables 

P

and 

Q

can be replaced by any propositions. The resulting argument will be valid. In terms of form, that is as good as any argument can get. 

Since we can substitute in any propositions, we can even use our Magic 8 Ball propositions. 

Example

For example, let's consider the following list of propositions. 

1. Leslie will win tomorrow's ping pong tournament.
2. Leslie will add another trophy to their trophy cabinet the day after tomorrow.

Now, let's associate these propositions with the variables of the form. Let: 

P = Leslie will win tomorrow's ping pong tournament.
Q = Leslie will add another trophy to their trophy cabinet the day after tomorrow. 

The Modus Ponens argument then is: 

If Leslie will win tomorrow's ping pong tournament, then Leslie will add another trophy to their trophy cabinet the day after tomorrow. Leslie will win tomorrow's ping pong tournament. Therefore, Leslie will add another trophy to their trophy cabinet the day after tomorrow. 

More schematically, we can give the argument as: 

Premise 1: If Leslie will win tomorrow's ping pong tournament, then Leslie will add another trophy to their trophy cabinet the day after tomorrow.

Premise 2: Leslie will win tomorrow's ping pong tournament. 

Conclusion: Therefore, Leslie will add another trophy to their trophy cabinet the day after tomorrow. 

Although Premise 1 is composed of two more basic propositions, it itself is a proposition. The argument consists of three propositions: Premise 1, Premise 2 and the Conclusion. 

Intuitively we can see that this argument reasonable. But it is more than reasonable. The logical inference from the truth of the premises to the truth of the conclusion is valid. And a valid inference is as strong as any infererance can be. It is absolutely strong. This absolute logical strength is purely a matter of the form of the argument. —But what does that mean? That is what we are trying to get at in the exercises here.  

Example

The list of two propositions above have an obvious logical connection. But note that in our preceding exercises we went to pains to ensure that our list of propositions would have obvious logical connections purely by chance. And many pairs of propositions would have no clear connection. So, let's consider an example like that. Let's add to our list to get: 

1. Leslie will win tomorrow's ping pong tournament.
2. Leslie will add another trophy to their trophy cabinet the day after tomorrow.
3. Christopher Nolan will win for "Best Picture" at the Academy Awards next year.

As I said above, the Modus Ponens form is valid for any propositions that we plug in for the variables. So, let:

P = Leslie will win tomorrow's ping pong tournament.
Q = Christopher Nolan will win for "Best Picture" at the Academy Awards next year.

Putting the propositions into the Modus Ponens form, we get: 

If Leslie will win tomorrow's ping pong tournament, then Christopher Nolan will win for "Best Picture" at the Academy Awards next year. Leslie will win tomorrow's ping pong tournament. Therefore, Christopher Nolan will win for "Best Picture" at the Academy Awards next year.

Like the previous Modus Ponens argument, this Modus Ponens argument is valid — because they have the same form. Only the form is considered when evaluating validity. 

Now, proving that Modus Ponens is universally valid takes a bit of work. See chapters 1 through 19 of my Logic 0. Let's just take it as stipulated that any argument fitting the Modus Ponens form is valid. 

Nonetheless, there is something wrong with this second example of Modus Ponens above. More on that later (see "Soundness" below). 

Modus Tollens

Another valid argument form is "Modus Tollens." It looks like this: 

If P, then Q.
It is not the case that Q.
Therefore, it is not the case that P.

Example

Let:

P = Leslie will win tomorrow's ping pong tournament.
Q = Leslie will add another trophy to their trophy cabinet the day after tomorrow.

Then plugging these into the Modus Tollens form, we get: 

If Leslie will win tomorrow's ping pong tournament, then Leslie will add another trophy to their trophy cabinet the day after tomorrow. It is not the case that Leslie will add another trophy to their trophy cabinet the day after tomorrow. Therefore, it is not the case that Leslie will win tomorrow's ping pong tournament.

This argument is valid. With this particular example it might be a little easier to see why it is valid. Recall that an argument is valid only if the truth of the premises guarantees the truth of the the conclusion. In other words, with a valid argument form, if we assume all of the premises are each true, then the conclusion must be true. 

Follow along with the thought process here. First, assume: 

Premise 1: If Leslie will win tomorrow's ping pong tournament, then Leslie will add another trophy to their trophy cabinet. 

Take this as something that you know to be factually true. You know how Leslie is. 

Now, let's say that — in a science fiction twist — we travel five days into the future, and take a look at Leslie's trophy cabinet. The trophy is not there. So, we know by experience the fact that:

Premise 2: It is not the case that Leslie will add another trophy to their trophy cabinet the day after tomorrow.

Once we see the trophy is missing from the cabinet we know for a fact that:

Conclusion: Therefore, it is not the case that Leslie will win tomorrow's ping pong tournament.

There is no logically possible way that both premises are true, but the conclusion can be false. That's validity. 

Example

For an example that maybe has some features of Nolan's film Tenet, follow the Modus Tollens form, letting: 

P = Christopher Nolan will win for "Best Picture" at the Academy Awards next year.
Q = Leslie will win tomorrow's ping pong tournament.

You can contemplate the logic on your own. Just keep in mind that it is a valid argument. Any propositions will work. Counterfactual scenarios can be made logical sense of, if we apply interpretive approaches appropriate to science fiction and fantasy literature. 

Invalid Forms 

There are two invalid argument forms that are a regular part of the study of formal logic. 

Denying the Antecedent 

If P, then Q.
It is not the case that P.
Therefore, it is not the case that Q.

This form looks very simlar to Modus Ponens and Modus Tollens. But it is not valid. It is possible for the premises to both be true while the conclusion is false. 

Example

Let:

P = Leslie will win tomorrow's ping pong tournament.
Q = Leslie will add another trophy to their trophy cabinet the day after tomorrow.

Taking the Denying the Antecedent form, we get: 

Premise 1: If Leslie will win tomorrow's ping pong tournament, then Leslie will add another trophy to their trophy cabinet the day after tomorrow. 

Premise 2: It is not the case that Leslie will win tomorrow's ping pong tournament. 

Conclusion: It is not the case that Leslie will add another trophy to their trophy cabinet the day after tomorrow.

This form is a regular part of the study of formal logic, because it is possible that both premises are true, and that the conclusion is true. But that possibility does not satisfy the definition of validity. Validity says that it is impossible that both premises are true, and that the conclusion is not true. 

Counterexample 

This argument form can be proven to be invalid by a "counterexample" — a single example that demonstrates the invalidity of a logical inference. It might be that Leslie will also win tomorrow's pinball tournament. In this case both Premise 1 and Premise 2 can be taken as true, but the Conclusion will be false. 

Affirming the Consequent

If P, then Q.
Q.
Therefore, P.

Again this looks very similar to Modus Ponens and Modus Tollens. And again it is invalid. 

Example

Let:

P = Leslie will win tomorrow's ping pong tournament.
Q = Leslie will add another trophy to their trophy cabinet the day after tomorrow.

The Affirming the Consequent form gives: 

Premise 1: If Leslie will win tomorrow's ping pong tournament, then Leslie will add another trophy to their trophy cabinet the day after tomorrow. 

Premise 2: Leslie will add another trophy to their trophy cabinet the day after tomorrow.

Conclusion: Leslie will win tomorrow's ping pong tournament. 

Again it is possible that both premises are true, and that the conclusion is also true. However ... 

Counterexample 

Again it might be that Leslie will win tomorrow's pinball tournament. 

Counterfactual Examples

Counterfactual examples play an important role in revealing the formal character of deductive reasoning. 

But there are a couple of problems that counterfactual examples present. First, when I concoct fanciful counterfactuals, students are tempted to think that I have expertly mainuplated the content in order to produce the desired effect. And that is not the impression that I want. Clearly, the solution to this difficulty is to have the students create counterfactual examples for themselves. But then there is a second problem that arises: the students then have a natural resistance against concocting pairs of  propositions that have no obvious logical connection with one another. They just feel that there is something wrong about it, and are tempted to fix the examples — ending up with non-counterfactuals. 

This is why we have gone to great lengths to create a list of random propositions. We can now use the random list of propositions to create randomized arguments. 

Exercise

Continue to lean into the randomness as you complete the Magic 8 Ball Arguments Exercise. The randomization of the process should produce some silly counterfactual examples. And you can be sure that I did not engineer the content. I have stipulated the form — but not the content. 

Soundness

As we have seen above, a deductive argument can be formally valid, but still not be really all that great. That is because there is more to a deductive argument than just its form. 

Example

For example, we encountered earlier the following argument: 

Premise 1: If Leslie will win tomorrow's ping pong tournament, then Christopher Nolan will win for "Best Picture" at the Academy Awards next year. 

Premise 2: Leslie will win tomorrow's ping pong tournament. 

Conclusion: Therefore, Christopher Nolan will win for "Best Picture" at the Academy Awards next year.

This follows the Modus Ponens form, and therefore is perfectly valid. However, there is something amiss. The problem is with Premise 1. This premise is not true. 

Note that the definition of validity is couched within the hypothetical condition that the premises are all true. An argument is valid if when assuming the truth of the premises, the conclusion must be true. And in this example, assuming both Premise 1 and Premise 2 are true, the conclusion must be true. —But Premise 1 is not true. That doesn't change the fact that the argument is valid. But it reveals the issue of "soundness." 

An argument is "sound" if it is both valid, and also has all true premises. 

So, although the above example is valid, it is not sound. That is what is wrong with it. 

If an argument is sound, then it is unassailable — and we must accept the conclusion. This is the ideal of argumentation. 

Critical Analysis of Soundness

Given the ideal of soundness, there are two ways — and only two wayts — that a deductive argument can be criticized: (1) on the basis of validity, and (2) on the basis of the truth of premises. If an argument is invalid, then it is a bad argument (and it doesn't matter whether or not its premises are true). But if an argument is formally valid, then only the truth of its premises can be challenged. If it is found that at least one of the premises is false, then the argument is unsound. And an unsound argument is no good. We don't have to accept its conclusion.  

Exercise

Let's practice the critical analysis of soundness by completing the following Google form for each of the four Magic 8 Ball arguments you have created (make four submissions). 

 Criticizing the Soundness of a Deductive Argument 

Review of the Exercise

If you followed all the above instructions (throughout this webpage), then your Modus Ponens argument, and your Modus Tollens argument should both be valid. Whether each of these two is sound depends on the truth of the premises. Given the randomization of the content of the arguments, it is relatively likely that at least one of the two is an unsound argument. The other two arguments — your Denying the Antecedent and Affirming the Consequent arguments — should both be invalid. And for these two, the truth of the premises do not matter — because an invalid argument is unsound. 

Seeing Content as Distinct from Form

Deductive arguments in the English lanaguage come with both form and content. To realize that the logical inference of a deductive argument is purely formal, we need to be able to abstract the formal structure of the argument away from the content of the propositions. This takes experience of seeing the content as distinct from the form. One way to gain this experience is through analyzing arguments for which we know that the content is randomized. The Magic 8 Ball and dice have provided a means of effectively producing randomized content for arguments where the form is very precisely prescribed. And you are fully assured that the content is truly randomized, because you have part of the randomization process at each stage of the process.