LESSON 14

The  Square of Opposition

Reading assignment: 4.3 (p. 211) 4.5 (pp. 225-230)

Click here to skip the following discussion and go straight to the assignments.

Modern Square of Opposition:

In this lesson, as with the Venn diagrams, I think it is simpler to introduce both the traditional and modern squares of opposition at the same time. A square of opposition helps us infer the truth value of a proposition based upon the truth values of other propositions with the same terms. By now you should be familiar with the difference between the Boolean and Aristotelian interpretation of categorical propositions. Suffice to say that because of this difference, there are more inferences available to you if you can certify that at least one individual represented by the S class of universal statements exists.

If you can't certify this, the only one of the following relations you can rely on to preserve truth value is the contradictory relation, which makes the Modern Square of Opposition really an X of Opposition. Go figure...

Traditional Square of Opposition

You will need to memorize the traditional square of opposition:

Note that you only use it when you can certify that the S terms in the propositions actually exist, to avoid committing the existential fallacy. If you can't certify this, you must use the Boolean square. (In other words just the contradictory relation)

 

Notice the following points:

Universals on top v particulars on bottom

Affirmatives on left v negatives on right

Contradictories (diagonals) always have the opposite truth values--you will always be able to determine the truth value of contradictories.

Contraries cannot both be true at the same time. It cannot be the case that both all cats are animals and that no cats are animals. So if you know that one is true, then you also know that the other is false. (They can, however, both be false simultaneously.)

Sub-contraries are exactly the opposite of contraries. It is always the case that at least one is true. If you know that one is false, then you automatically know that the other is true. (They can however both be true simultaneously.)

With sub-alternation, you should remember that you can go down with true (i.e. if A is true, then I is true as well) and up with false (i.e., if I is false, then A is false as well.) But not vice versa.

Example:

                If it is true that all cats are animals, then it is also true that some cats are animals. This is true by          
                "default"--some means "at least one" it does not necessarily imply that "some are not."

                Conversely, if it is false that some cats are dogs, then it is certainly false that all cats are dogs.

 

You should recognize that the two strongest categorical propositions are a true universal and a false particular. With either of these you can derive the truth value of all the other standard form types. From a false universal and a true particular, the weakest type of categorical propositions, you can only determine the truth of the contradictory.

If you are presented with an argument in which an inference is drawn along a line of the table that didn't preserve truth value, you call the fallacy committed "illicit X" where X is the line that joins the two statements.

 

 

Proofs

Pay close attention to the last four paragraphs of this section. Hurley shows you how to prove that arguments are true, using the tools you learned in the last couple of sections. This is your first introduction to the concept of a logical proof. Since this will become very important in chapter seven, it is worth investing some time on it now--your payback will be making chapter seven easier to understand. 

Remember that your goal is to get from the premise you are given to the conclusion, using only the changes you can legally make with the operations you have just learned. 

1) First, write out the problem, substituting Letters for your S and P terms. Don't forget to indicate any false statements and term compliments. Use a double slash to separate your conclusion from your premise. 

2) As you work the problem, make a valid inference, and write the new statement on a line below the one you changed. Then you write the name of the operation you just did to justify that line of the proof. 

3) Keep changing each new statement by making valid inferences. Once you manage to make the change that lets you write out a statement that exactly matches the conclusion, you are done. You have shown that using only operations that preserve truth value and are known to be valid inferences, the premise follows from the conclusion.

Helpful Tip:

As my first step in working one of these problems  look at all the differences between the premise and the conclusion and list them off to the side of the problem, and check them off after they are done. This helps me plan my approach to the problem.

For instance, if I need TWO term compliments, I know I'll be using contraposition at some point. If I need to change the quantity of the premise, I know I'll be using one of the vertical operations on the Square, (subalternation or contradictory) Similarly, If I need to change the truth value I know I'll need to use an operation from the square. If I need to switch S & P terms, I'll need to use either conversion or contraposition.

There can be more than one way to do these, especially the longer ones. 

EXAMPLE:

It is false that some ficus benjaminas are untemperamental  house plants.
Therefore, all ficus benjaminas are temperamental house plants.

List of changes:
Truth value
quantity
1 term compliment, P term

Discussion: (This is a stream-of-consciousness expose of my reasoning as I approach the problem. Ignore it if it confuses you) I figure that since I need to change the truth value I'll be using the square of opposition. Since I need to change the quantity too, contradictory suggests itself. (neither contrary nor subcontrary, the other operations that change truth value, change quantity at the same time.) But wait!... If I use contradictory I'll change the quality of the statement too, that isn't on my list... Hmmm. OH. If I change the quality, that will be a good thing because then I'll be able to do obversion to get the term compliment of P. That will change the quality back to affirmative.

So now I'll work the problem:

(F) Some F are non-T    //     All F are T

     No F are non-T     contradictory

    All F are T          obversion

 

 

 

Logic Coach Assignment: 4.5 I 1-3, II all, IV all

Assignment 1: (10 points each)

Use the traditional square of opposition to determine whether the following arguments are valid or invalid, i.e., is the conclusion necessarily justified by the premise? If not, state the fallacy committed.

1. Some pirates are daring men. Therefore, some pirates are not daring men.

2. Some wasps are unfriendly insects. Therefore, it is false that no wasps are unfriendly insects.

3. It is false that all professors are ignorant fools. Therefore, no professors are ignorant fools.

4. It is false that some muffins are not wholesome foods. Therefore, some muffins are wholesome foods.

5. All dictionaries are useful books. Therefore, it is false that no dictionaries are useful books.

 

Use the traditional square of opposition, together with conversion, obversion and contraposition to prove that the following arguments are valid. Show each intermediate step in the deduction.

                            1) No feminists are male chauvinist pigs.
                                Therefore some advocates of equality are not anti-feminists. 

                                (note: "advocates of equality" and "male chauvinist pigs" are term compliments)

 

                             2) It is false that no unhealthy things to ingest are food additives.
                                 Therefore, some food additives are not healthy things to ingest.   

 

              

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