A syllogism

How to discuss an argument’s content, if you have a predicate argument (a syllogism).

 

When we evaluate arguments, we are interested in both the argument’s form and its content. We evaluate its form in terms of validity. We evaluate its content in terms of truth. We need to examine each statement in an argument individually to determine whether it is true or false. So, how do we do this? This depends, in part, on the type of statement we have.

 

General information

1.\When you discuss arguments in your paper, treat them the way they are treated in the course packet. Put each premise on its own line. Put the premises on top of the conclusion. And if you can, use some sort of notation to separate the premises from the conclusion.

 

This is good:   All dogs are cats

Some cats are not poodles

No poodles are dogs.

 

This is not good: All dogs are cats. Some cats are not poodles. No poodles are dogs.

 

We don’t like to treat statements in an argument horizontally. It obscures the form of the argument. It also makes it harder to distinguish the argument from your discussion of the argument. Do not put arguments in a paragraph with the rest of your discussion.

 

2. Remind the reader what the statements of the argument say. When you discuss a premise, remind the reader what this premise said. Don’t make them turn back the pages to find out what it said. This is annoying. And in real life, no one will do you the courtesy of seeing going back to an earlier page to see what you said.

 

3. If you have a sentence that goes on and on and on and on, cut it up into shorter sentences. If you have a sentence that goes on for a paragraph, you generally are committing two mistakes at one time—your sentence says too much and it doesn’t say enough. Your sentence says too much because you are giving the reader too much information at one time. And your sentence says too little because generally you leave out something that would make your point clearer.

 

4. Transitional sentences are good. Let the reader know what you are planning to discuss. If you are focusing on the form of the argument, let the reader know this. If you are explaining why the premises are true or false, let the reader know this. Don’t make the reader guess what you are doing. This generally results in a lower grade.

 

5. Examples are important. We are training you to give examples. And quite often, an example is more important than your explanation. This is because the example provides the evidence or proof that justifies your explanation. It is enough to say something true, you want to back up your true statements with evidence.

So, how do you know when to give examples? Generally, the following cases hold, but not in every case.

You can prove that a particular statement is true by giving an example. Suppose we have this: Some valid arguments do not have a false conclusion. To prove this is true, you just need to give one example of a valid argument that does not have a false conclusion. This example would work:

 

All cats are poodles

Some cats are dogs

Some dogs are poodles.

 

You can prove that a universal statement is false by giving an example. Suppose we have this: All valid arguments are arguments with a false conclusion. To prove this is false, you just need to give one example of a valid argument that does not have a false conclusion. This example would work:

 

All cats are poodles

Some cats are dogs

Some dogs are poodles

 

 

When you need to prove a particular statement is false, you need to give an explanation. Examples generally cannot prove that a particular statement is false. Suppose we have this: Some valid arguments are arguments with true premises and a false conclusion. You cannot prove that this is false by giving an example of one valid argument that does not have true premises and a false conclusion. You would have to provide every valid argument and show that not a single one has true premises and a false conclusion. This would expand the pages of your paper quite significantly (and you probably include every valid argument that exists.) In this case, you would want to focus on the definition of validity.

 

When you need to prove a universal statement is true, you need to give an explanation. Examples cannot prove that a universal statement is true unless you give every single example of the relevant sort. Suppose we have this: No valid arguments are arguments with true premises and a false conclusion. One example is not sufficient to prove that no such argument exists. Again, you would need to focus on the definition of validity.

 

 

For categorical syllogisms, we have six types of statements. We have the four categorical statements;

• All S are P
• No S are P
• Some S are P
• Some S are not P

And we have two singular statements:

• s is P
• s is not P

‘S’ stands for the subject of the statement while ‘P’ stands for the predicate. Basically, in each statement, we are making a claim about S (or s). We are claiming that S possesses or does not possess a certain relationship with P. When we discuss the truth of a statement, we must make S hold. If S=valid arguments, then we must talk about valid arguments. If S=dogs, then we must discuss dogs.

 

I am going to show you various ways to handle truth in terms of various examples. These are all methods that students have used successfully in the past.

 

Example 1

The first premise of the argument says this: All valid arguments are arguments with true premises. This premise is false. Consider the following example:

All M are P                     All dogs are cats

All S are M                     All mice are dogs

All S are P                      All mice are cats

The argument instance has a valid form. But it is not a sound argument. A sound argument has a valid form and true premises; but this argument has false premises.

 

Discussion of this example: Note that the subject of the statement was ‘valid arguments’. This means that we must discuss valid arguments. If we discuss anything else, such as invalid arguments, then we have changed the subject and are discussing some other statement.

 

Example 2

The second premise of the second syllogism says No argument with mood and figure III-4 has true premise and a false conclusion. This premise is false. An argument with mood and figure III-4 has this form:

Some P are M

Some M are S

Some S are P

This form has instances with true premises and a false conclusion.

Some dogs are black animals

Some black animals are cats

Some cats are black animals

General comment: To prove that a universal statement is false, we generally provide examples

 

Discussion of this example: Our subject was ‘arguments with mood and figure III-4’. This means that we had to discuss this type of argument. Generally, to prove a universal statement is false, we need to provide an example. We also explain why this example proves the statement is false. In other words, we provide both an explanation and an example.

 

Example 3

The first premise says this:

No sound argument has a false conclusion.

This premise is true. By definition, a sound argument has both a valid form and true premises. By the definition of validity, we know that when a valid argument has true premises, it must have a true conclusion.

 

Discussion of this example: Generally, but not always, we cannot prove a universal statement is true by providing an example. If we provided an example of a sound argument and pointed out that its conclusion is true, all we would have accomplished is to show that one sound argument lacks a false conclusion. This is not the same as showing that no sound argument has a false conclusion. And let’s face it, there is no way you are going to be able to examine every sound argument in your paper.

 

Example 4

The last premise says this: No argument with mood and figure III-4 is an argument with mood and figure OOO-1. Compare the two forms:

Some P are M                                 Some M are not P

Some M are S                                 Some S are not M

Some S are P                                  Some S are not P

III-4                                                   OOO-1

As we can see, these forms are not identical. The middle terms occurs in different locations and the statements are not the same type.

 

Discussion: While you generally cannot prove a universal statement is true by giving an examples, there is an exception to every rule; and this counts as an exception. Putting the two forms side by side makes it obvious that they are not the same.

 

Example 5

The second premise of the first syllogism is this: Some arguments with true premises and a true conclusion are not valid. This premise is true.

Some cats are not dogs.

No dogs are elephants         .

Some elephants are not cats

This instance has true premises and a true conclusion but it also has an invalid form.

 

Example 6

The first premise of our sorites is this: Some valid arguments are not sound. This is true. Consider the following argument:

All cats are dogs

All mice are cats

All mice are dogs

This argument has a valid form (it is AAA-1); but it also has false premises. A sound argument is not only valid but it also has true premises. So, any valid argument with false premises will be unsound.

 

Discussion: You can generally prove particular statements are true by providing examples. You want to explain why these examples are relevant.

 

Example 7 (a)

The second premise of our enthymeme is this: Some sound arguments are valid.

This is true. Consider this example.

All dogs are animals

All poodles are dogs

All poodles are animals

This argument is sound and it also has a valid form.

 

Example 7(b)

The second premise of our enthymeme says this: Some sound arguments are valid. This is true. By definition, a sound argument must have a valid form (and true premises). So, obviously, some sound arguments are valid. In fact, all sound arguments are valid.

 

Discussion: Both 7(a) and 7(b) are fine. I think 7(b) is a better answer because it provides the strongest case. Remember, a particular affirmative statement such as ‘some sound arguments are valid’ does not imply that there are some sound arguments that are not valid. A particular statement that talks about some members of a class could still be true even if the statement were true about every member of the class.

 

Example 8

Our conclusion says this:

Some arguments with mood and figure AAA-1 are not valid.

This premise is false. AAA-1 is a valid form.

All M are P                  M-1 OK

All S are M                  S-2 OK

All S are P                   P-0 OK

0=0 (or no negative statements)

As we have just shown, AAA-1 has a valid form. Because it is valid, by definition, it cannot have even one instance with true premises and a false conclusion.

 

Discussion: if a particular statement is false, we need to give an explanation. We cannot rely solely on examples. Telling me that you were unable to find an instance of AAA-1 with true premises and a false conclusion is not sufficient. You need to explain why it is impossible to come up with such an instance.

 

Example 9

The second premise of our first syllogism says this:

This argument is valid.

If we examine the argument in which this premise occurs, we see that it has this form:

All M are P

All S are M

All S are P

Since this form is valid, this premise is true.

 

Discussion: singular statements are tough. It is generally easier to explain why a statement such ‘this argument is valid’ is true or false rather than a statement such as ‘all members of the class of this argument are valid’. Make your life easier. You can treat singular statements one way when you put them in standard logical form; but you can discuss an easier version when you are explaining why they are true or false.