7.2.1 Finishing the Square and Immediate Inferences

Finishing the Square, and Immediate Inferences


The Square shows four sets of relations:

Contradictory propositions: A and O; E and I

Contrary propositions: A and E

Subcontrary propositions: I and O

Super/ sub-altern propositions: A and I; E and O.

A certain amount of information about the truth values of other propositions can be deduced if one knows (or assumes) the truth value of any one statement, as a result of what these four relations mean.

“Contradictory” means that statements have opposite truth values.

“Contrary” means that they cannot both be true.

“Subcontrary” means that they cannot both be false.

“Subalternation” means that if the universal is true, the particular of the same quality is true.

“Superalternation” means that if the particular is false, the universal of the same quality is false.

 

So if we assume that our A statement is true, what follows?

that E is false, O is false, and I is true.

Do you see why? (The first is the contrary –they can’t both be true; the second is the contradictory –they have opposite values; the third is the subaltern.)

And if we assume our A is false?

Then O is true, but we can’t infer anything about E or I.

Do you see why?  (Contraries can both be false; subalterns of false universals can be true or false.)

Assume I is false:

O is ___

A is ___

E is ___.

 

Here are the answers and reasons why:

 

Assume E is true:

A is ___

I is ___

O is ___.

And here are the answers for these:

 

This is information about the relations entailed amongst categorical propositions that Aristotle first articulated, though he did not draw it out in a picture of a square.

This is known as the traditional square of opposition: “traditional” because during the 20th century, it has become clear that the relations mapped here are all valid only if we are talking about categories that are not empty. If we talk about categories that are empty, and say things like

All talking eggs are rude

we encounter a special situation: the statement and its contrary (No talking eggs are rude) are both false, but so are their contradictions (Some talking eggs are rude; Some talking eggs are not rude)!

This is impossible, right?

It seem like it explodes the square!

 

When you see why this happens, you can begin to address it. It happens because any statement which claims or implies the existence of something that does not exist is going to be a candidate for being called “false.” Regardless of the predication it makes, any statement with a subject term that names an empty set is going to be a false one, if by the mere fact of naming the class we are making a claim that the class is non-empty. So, if we could assume that none of the terms we use names empty or memberless categories, then we can apply all the information contained in the four relations that make up the square.

But what do you think about this:

The cure for AIDS is worth spending billions of research funds on.

If anyone thinks that’s a true statement, it is not because the assumption that the cure for AIDS exists is a correct one. There is, today, no cure for AIDS. Nonetheless, this seems like a statement that is not condemned to being false, despite the fact that it is inappropriate to assume that its subject term names a class that exists.

Perhaps you’ll be a little sympathetic to the modern interpretation of A statements, which (unlike the traditional one) takes them to be captured quite adequately as conditionals. Rewritten as a standard form categorical propositon, the statement about the cure for AIDS amounts to this:

All cures for AIDS are cures worth spending billions on.

Rewritten as a conditional statement, this says,

If something is a cure for AIDS, then it is worth spending billions on.

The power of “if” is compelling, isn’t it? With that little word, the notion that we are committed to asserting the existence of what we are talking about is set aside. Back then to our worry about the validity of the Square of oppositions: can we reason according to it or not? The answer is that we can, unless it is a mistake to assume the existence of the subject term. Why? Because the A-I relation makes it seem that if A is true, I will be too. Consider:

A: All unicorns have one horn

(If anything is a unicorn, it has one horn)

therefore

I: Some unicorns have one horn.

(There is at least one thing that is a unicorn, and it has one horn)

This means “There is at least one unicorn, and it has one horn,” which is clearly false.

So in modern or contemporary logic, the Aristotelian or traditional interpretation of the existential assumptions of all statements has been recognized and dealt with: it is not an appropriate assumption. Therefore, the only relations between the categorical propositions that REALLY holds is the one called “Contradiction.” The others all break down over the controversies generated by treating all statements as having existential import.

 

 

Immediate Inferences:

“Immediate Inference” is a phrase that denotes something that follows from something else without anything additional. “To mediate” means to be in-between in some fashion. “Im-” is a negative prefix, so “immediate” means “without mediation,” or “directly.”

Are there inferences that can be made immediately? Well, once we have language and we’ve agreed to represent all predication in one of the four forms of A, E, I or O statements, then, yes, certainly. From knowing that  No Senators are Presidents, I can immediately, without any further information, infer that No Presidents are Senators.

 

Here’s an illustration of this. Let me draw you a picture of it.

 

This is the generic picture for any E statement:  two overlapping circles, one of which represents all the Senators, the other of which represents all the Presidents. The area that overlaps between them has been filled in. In categorical logic, we use the filling in to mean “erasing” or “scratching out.” That is, by filling in that area where the two overlap, we have eliminated it, so that no overlap can be found or seen.

As soon as you “see” that No Senators are Presidents, you see that No Presidents are Senators. The same picture shows both statements. They are not really distinct claims; the second is just a different way of saying what the first says, and vice versa.

 

If that Venn diagram is convincing, then this one, diagramming an I statement will be as well, for it shows that Some S is P is equivalent to Some P is S.

This is Some S is P

It’s very easy to see the symmetry of these two types of statements, to see that they are reversible without any change of truth-value. Saying that Some sharks are predators and that Some predators are sharks is saying two different things –these statements don’t actually mean the same thing; but the truth of the one is certain if the other is true. The truth of the one can be immediately inferred from the other.

In categorical logic, this inference is called Conversion. If you think of Conversion as the operation by which you switch the positions of the subject and predicate terms, then we can say that Conversion is a Valid Operation for E and I statements. The pictures we just looked at showed us that every E in which the order of S and P is reversed is going to look just like the E before they were reversed. The same for every I.

 

What about every A, or every O?

Here’s a picture of an A.

All S is P  (All Sharks are Predators)

 

And here what it looks like if we reverse the order of the terms:

All P is S   (All Predators are Sharks)

These two pictures are clearly not identical to one another, unlike what happened when we drew the pictures of E and I converted (reversed).

And there are plenty of examples that should come to mind easily enough: All Popes are Catholics is true, but All Catholics are Popes is not.

Does that example show us that whenever we convert an A statement that was true, the result is one that is false? Consider this one:

All bachelors are unmarried men.

All unmarried men are bachelors.

There’s a case where an A statement is true, and its converse is true as well. So if there are cases where an A is true and its converse if false, and other cases where an A is true and its converse is true, we can’t infer anything about the truth value of an A via the operation of conversion alone. We’ll have to say that Conversion yields no conclusion in the case of A statements. In other words, it is not a valid operation for A statements, whereas it is valid for E and I.

When you see the pictures of an O statement and its converse, you’ll see that conversion is also invalid for O:

Some S is not P

Some P is not S

 

These pictures are not identical to one another, and that means that these statements are not equivalent to one another.

Venn diagrams are a convenient way to represent what is going on in categorical logic, and we’ll return to them shortly to see a visual way to diagnose the validity or invalidity of categorical syllogisms.

 

For now, we have more to say and appreciate about some other ways we can manipulate the content of categorical propositions.

Two other operations upon them are valuable to know: Obversion and Contraposition.

 

Obversion is a valid immediate inference for all four propositional forms. It consists of two steps, which in effect, cancel each other out, but yield a new way of saying what the original statement said.

No philosophers are golfers.

All philosophers are non-golfers.

 

All rabbis are Jews.

No rabbis are non-Jews.

 

Some women are mothers.

Some women are not non-mothers.

 

Some men are not fathers.

Some men are non-fathers.

 

With the exception of the first pair, each statement here is true, and it is pretty obvious that each statement in each pair is equivalent to the other one in that pair. What does the operation called “Obversion” consist in? Two changes: changing the quality of the original statement (from affirmative to negative or vice-versa) and forming the complement of the predicate class (the complement being the class of everything that is not included in the predicate class).

 

The other operation we can impose on categorical propositions is called “Contraposition,” and it also involves two changes: forming the complement of each class, and reversing their order, so that the term that was predicate becomes subject complemented, and the one that was subject becomes predicate complemented:

All birds are animals.

All non-animals are non-birds.

Pretty obvious that these are equivalent.

 

No Popes are Buddhists.

No non-Buddhists are non-Popes.

Wait a minute! That would mean that I’m the Pope, since I’m a non-Buddhist. If a false conclusion can follow from a true premise, the argument is invalid.

 

Some men are non-fathers.

Some fathers are non-men.

That doesn’t follow either.

 

How about this one:

Some animals are not mammals.

Some non-mammals are not non-animals.

Or try:

Some flowers are not geraniums.

Some non-geraniums are not non-flowers.

 

At this point it begins to get exhausting. Our minds get overloaded with arbitrary examples that might, in principle, be able to help us see the relationships, but the triviality of these inferences intuitively occurs to us and we object “It is not really useful to know that there is at least one thing that’s not a geranium and that is also a flower (i.e., not a non-flower).”

 

OK, and I won’t pretend otherwise. But we can just sum this all up in a few universal statements:

Conversion, which consists of switching the order of the subject and predicate terms, is valid only for E and I.

Obversion, which consists of forming the complement of the predicate and changing the quality of the statement, is valid for all four.

Contraposition, which consists of forming the complements of both terms and switching their order, is valid only for A and O.

With a set of rules like that, we could play a little game: we could start with a true A statement, contrapose it, obvert that, convert that, and say something about its contradictory and its contrary on the square.

Or we could start with a false I, convert it, obvert that, and draw a conclusion about its contradictory.

And what could we expect about the contrapositive of the obverse of the converse of an E statement?

Or: what is the obverse of the converse of the obverse of any statement called?

 

Homework on Immediate Inferences and the Square:

Build yourself a chart of 10 inferences beginning from a statement that’s true, like All cats are animals, or No angels are demons, and convert it, obvert that, state the contrary (or subcontrary) of that, contrapose it, then state its contradictory. Say what you think you can about the truth value at each step, and why (in terms of the rules).  Do this again with a statement that is false; do 10 more inferences.  Provide comments to justify your calculations, as you see below.

Here’s an example:

1. All candidates are citizens  T

2. All citizens are candidates  U (undetermined: we can’t draw this inference because conversion is not valid for A statements)    There’s no point in working out more transformations from #2, since its truth-value is unknown, so will be all equivalent statements and all relative statements. So #3 builds off of #1 again.

3. No candidates are non-citizens  T (obverse of 1: valid)

4. All candidates are non-citizens  F (contrary of 3: can’t both be true)

5. Some candidates are not citizens F (contradictory of 1)

6. All non-citizens are non-candidates T (contrapositive of 1: valid for A)

7. No non-citizens are non-candidates  F (contrary of 6; can’t both be T)

8. Some candidates are citizens T (subaltern of 1)

9. Some citizens are candidates T (converse of 8.)

10. No candidates are citizens  F (contradictory of 8, also contrary of 1)

11. All candidates are non-citizens F (obverse of 10; valid, hence same truth-value).

12. Some candidates are not non-citizens T (contradictory of 11)

13. Some citizens are not non-candidates T (contrapositive of 12)

 

More exercises for practice:

HW on square Feb 12

HW on square Feb 12 solutions

Using the Square of Opposition and the Immediate inferences, determine the truth values of the following, or write “U” for “undeterminable.”

 

Assume  No S is P to be true:

1. subaltern

2. obverse of 1

3. contradiction of 2

4. contrary of 3.

5. contrapositive of 4

6. subaltern of 5

7. subcontrary of 6

8. contrapositive of  7

9. contradiction of 8

10. converse of 9

 

Assume  All S is P to be true:

1. subaltern

2. obverse of 1

3. contradiction of 2

4. contrary of 3.

5. contrapositive of 4

6. subaltern of 5

7. subcontrary of 6

8. contrapositive of  7

9. contradiction of 8

10. converse of 9

 

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