1. Categorical propositions

2. categories By categories, it is simply meant groups of things.
We can put into categories, or categorize, pretty much anything.
Tables, pizzas, tools, televisions, animals, blond people…
Categorical logic concerns how we categorize things and their relationships.

3. By using Venn Diagrams, we can represent categories.
Blond People
Electronic Devices
Mammals
The circles represent categories.
Each category contains whichever objects, people, or whatever else we want.

4. Unless otherwise specified, a category contains all objects designated.
For example, the category of blond people contains all the blond people that we like: all the blond in the world of in North America, etc.
Blond People

5. By using an X or by shading the area of a circle, I can indicate various things:
For example, if I place an X inside a category, I indicate that the category is not empty.
I indicate that there is something there.
X = at least 1.
The X inside the circle indicates that something is a blond person.
There could be more than 1.
X means there is at least 1 thing in the category, and it is a blond person. X
Blond People

6. X
By placing an X outside the circle, I indicate that something is not a blond person.
There is at least one thing besides blond people, and it is not a blond person.
Blond people

7. Now if I shade completely the area of the circle, I indicate that there aren’t any blond people.

8. Blond people
And if I shade the area outside the circle, I indicate that everything is blond people.

9. The objects, people, animals, etc
The objects, people, animals, etc. designated by a category may have logical relations with other categories.
for example I can say that all blond people are humans.
As you recall, sentences have a Subject and a Predicate.
We use 2 circles, one for the Subject, one for the Predicate.
There are only 4 possibilities to indicate how the Subject relate to the Predicate.

10. 1: All members contained in the category of the Subjects are members of the category of the Predicate: A-Form.
2: No members contained in the category of the Subjects are members of the category of the Predicate: E-Form.
3: Some members contained in the category of the Subjects are members of the category of the Predicate: I-Form.
4: Some members contained in the category of the Subjects are not members of the category of the Predicate: O-Form.

11. 4 logical areas S P
Area 1: This area contains only members designated by the subject.
Area 2: This area contains only members designated by the subject that are also members of the predicate category.
Area 3: This area contains only members designated by the predicate.
Area 4: This area contains other objects—but not the objects in areas 1, 2, and 3.

12. Now by using shading and an X, I can represent all 4 categorical forms.
For the subject, let’s use “mammals” and for the predicate let’s use “cute”
Let’s start with,
A-Form:
All S are P
By shading completely area 1, I indicate that all mammals are cute.

13. E-Form:
No S are P
By shading area 2, I indicate that no mammals are cute.

14. I-Form:
Some S are P
By placing an X in area 2, I indicate that some mammals are cute—and also that some cute things are mammals.

15. O-Form:
Some S are not P
By placing an X in area 1, I indicate that some mammals are not cute.
By placing an X in area 3, I would indicate that some cute things are not mammals.

16. Quantity & quality Quality Quality = = Negative Affirmative
Quantity = Universal A: All S are P E: No S are P I: Some S are P O: Some S are not P Quantity = Particular
Quality =
Negative
Quality =
Affirmative

17. Distribution

18. Categorical propositions are statements that describe classes (groups) of objects designated by the subject and the predicate terms.
A class is a group of things that have something in common (birds, light bulbs, desks, etc.)
Categorical statements describe the ways in which things are related.
For example, the categorical statement “All screwdrivers are tools,” says that if we look into the class of tools, we will see that all screwdrivers in the world are inside it.

19. A proposition may refer to classes in different ways: to all members or some members.
The proposition “All senators are citizens” refers to all senators, but not to all citizens: All senators are citizens, but not all citizens are senators!
Notice that this proposition does not affirm that all citizens are senators, but it does not deny it either.

20. To characterize the way terms occur in categorical propositions, we use the term “Distribution.”
Distribution of a term:
A distributed term is a term of a categorical proposition that is used with reference to every member of a class.
An undistributed term is a term of a categorical proposition that is not being used to refer to each and every member of a class.

21. A: All S are P:All pizzas are things that have tomato sauce on top.
Spaghetti
with sauce
Pizza with cheese
Tomato
Pizzas
Lasagna
Pizza Bianca
Eggplant
Parm
Subject – S: Pizzas
Predicate – P: Things that have tomato sauce on top
Since all pizzas are things that have tomato sauce on top, the subject (Pizzas) is Distributed.
In other words: How many pizzas are things that have tomato sauce on top? All of them!
Although all pizzas are things that have tomato sauce on top, not all things that have tomato sauce on top are pizzas. Thus the predicate (Things that have tomato sauce on top) is Undistributed.
In other words: How many things that have tomato sauce on top are pizzas? Not all of them!

22. E: No S are P:No pizzas are things that have tomato sauce on top.
Spaghetti
with sauce
Pizza with cheese
Lasagna
Pizza Bianca
Eggplant
Parm
Subject – S: Pizzas
Predicate – P: Things that have tomato sauce on top
Since no pizzas are things that have tomato sauce on top, the subject (Pizzas) is Distributed.
In other words: How many pizzas are pizzas? All of them!
And since no things that have tomato sauce on top are pizzas, the predicate (Things that have tomato sauce on top) is Distributed as well.
In other words: How many things that have tomato sauce on top and are not pizzas are things with tomato sauce on top and are not pizzas? All of them!

23. I: Some S are P:Some pizzas are things that have tomato sauce on top.
Spaghetti
with sauce
Pizza with cheese
Lasagna
Tomato
Pizzas
Pizza Bianca
Eggplant
Parm
Subject – S: Pizzas
Predicate – P: Things that have tomato sauce on top
Since some pizzas are things that have tomato sauce on top, the subject (Pizzas) is Undistributed.
In other words: How many pizzas are things that have tomato sauce on top? Some of them!
And since some things that have tomato sauce on top are pizzas, the predicate (Things that have tomato sauce on top) is Undistributed as well.
In other words: How many things that have tomato sauce on top are pizzas? Some of them!

24. O: Some S are not P:Some pizzas are not things that have tomato sauce on top.
Spaghetti
with sauce
Pizza with cheese
Tomato
Pizzas
Lasagna
Pizza Bianca
Eggplant
Parm
Subject – S: Pizzas
Predicate – P: Things that have tomato sauce on top
Since some pizzas are not things that have tomato sauce on top, the subject (Pizzas) is Undistributed.
In other words: How many pizzas are not things that have tomato sauce on top? Some of them!
And since all things that have tomato sauce on top are things that have tomato sauce on top, the predicate (Things that have tomato sauce on top) is Distributed.
In other words: How many things that have tomato sauce on top are things that have tomato sauce on top? All of them!

26. opposition

27. OPPOSITIONS
Let’s apply our knowledge of Venn Diagrams to describe the relations among propositions.
The way categorical propositions relate is called OPPOSITION.
OPPOSITION is the logical relation between any two categorical propositions.
There are 5 ways in which they relate (They are opposed):

28. CONTRADICTORIES
Two propositions are said to be contradictories if one is the denial of the other—they cannot both be true or both false.
The A proposition “All judges are lawyers” and O “Some judges are not lawyers” are contradictories.
They cannot both be true: It is impossible that all judges are lawyers but some are not! These statements cannot both be true.
Also, if it is false that all judges are lawyers, then it is true that some judges are not lawyers— cannot both be false.

31. More examples:
A: All books are boring—true! O: Some books are not boring—false!
O: Some books are not boring—True! A: All books are boring—false!
E: No cats are brown—true! I: Some cats are brown—false!
I: Some cats are brown—true! E: No cats are brown—false!

32. 2. CONTRARIES
Two propositions are said to be contraries if they cannot both be true, but both may be false:
An A proposition “All judges are lawyers” and E, “No judges are lawyers,” are contraries.
It’s not possible that all judges are lawyers but no judges are lawyers! If one is true the other is false.
However, it is possible that both statements are false: Think about it! Some judges are lawyers and some judges are not lawyers. So, if some are and some aren’t, it is false that all are and it is false that none are.

33. More examples:
A: All cats are grey—true! E: No cats are grey—false!
E: No cats are grey—true! A: All cats are grey—false!
But as we know, in the world some cats are grey and some cats are not grey. So,
A: All cats are grey—false! E: No cats are grey—false!

34. 3. SUBCONTRARIES
Two propositions are said to be subcontraries if they may both be true but cannot both be false:
An I proposition, “Some judges are lawyers” and O, “Some judges are not lawyers” are subcontraries.
This is evident: Since some judges are lawyers and some are not, I and O may both be true.
However, if it is false that some judges are lawyers (False I), then it follows that some judges are not lawyers—so O is true!
In other words, I and O may both be true but cannot both be false.
I: Some judges are lawyers—true! O: Some judges are not lawyers—true!
I: Some judges are lawyers—false! O: Some judges are not lawyers—true!

36. 4. SUPERALTERNATES
When two propositions have the same subject and predicate and agree in quality (Both affirms or both deny) but differ in quantity (One universal the other particular) they are said to be CORRESPONDING propositions.
An A, “All spiders are eight-legged animals” has a corresponding proposition, I “Some spiders are eight-legged animals.” Both affirm = same quality; One is universal the other particular = differ in quantity.
Propositions A and I are said to be superalternates.
Superalternation is the relationship between the universal statements A and E and their corresponding particular statements I and O. In this relationship, the truth of the universal statements implies the truth of the particular statements, but not the other way around—the truth goes down.

37. So, “All spiders are eight-legged animals” (A) implies that some spiders are eight legged animals (I).
If it is true that all spiders in the world have 8 legs, obviously it must be true that some spiders have 8 legs.
Remember that “some” means “at least one.”
However, the other way around does not work: “Some spiders are eight-legged animals” does not imply that all spiders are eight-legged animals.

38. This is obvious:
If you take some spiders, say 10, and see that they have 8 legs, can you declare that all spiders in the world have 8 legs? No!
So, superalternation says that any true universal statement implies the truth of its corresponding particular statement.
True A implies a true I & True E implies a true O.
However be careful:
True I does not imply true A & True O does not imply a true E.

39. More examples:
If all shoes are comfortable (True A) then it is true that some shoes are comfortable (True I).
But if you take some shoes, say, 5 pairs, and they all are comfortable (True I), it does not follow that all shoes in the world are comfortable (A = ?)
If all teachers are good, it follows that some teachers are good.
But if some teachers are good, it does not mean all are.

41. More Examples:
If no socks are made of cotton, it follows that some socks are not made of cotton.
But if some socks are not made of cotton, I may not assume that no socks are made of cotton.
If no music is good, some music is not good.
But if some music is not good, it does not mean that none is good.

42. 5. SUBALTERNATES
If superalternation is the relationship between the universal statements A and E and their corresponding particular statements I and O, SUBALTERNATION is the relationship between the particular statements I and O and their corresponding universal statements A and E.
In the relationship of subalternation, the falsity of the particular statements I and O implies the falsity of the corresponding universal statements A and E, but not the other way around.
So, a false I implies a false A: If it is false that some people are blond, it must be false that all people are blond. However, a false A does not imply a false I: if it’s false that all people are blond, it does not imply that it’s false that some are.

43. More Examples:
If it’s false that some days are holidays, then it is false that all days are holidays. (Some days are not holidays. Then not all days are holidays!)
But if it’s false that all days are holidays, this does not imply the falsity that some days may be holidays.
Think about it, if you deny that all are holidays, you don’t assert that none are holidays. Obviously it is possible that some days may be holidays.

45. The modern square of opposition

46. The traditional square of opposition enables us to determine a number of relations among the four categorical forms.
But these relations depend on whether we make what is called an existential assumption or existential import.
An existential assumption (or import) means assuming that the entities indicated by the subject are in existence.
A-form asserts that all members in the category of the subject are members of the category of the predicate. In other words, if the subject is “Martians” and the predicate “blond,” then A- form says, “All Martians are blond.”
This statement is interpreted as, “There exists Martians, and they’re all blond.”

47. E-form, “No Martians are blond,” means there exist Martians, but none are blond.
For modern logicians, the traditional square is limited.
We need to make assertions of things that don’t exist.
“All dinosaurs are extinct” does not assume that there are dinosaurs!
The modern interpretation “All Martians are blond” is a hypothesis: “If anything is a Martian, then they are all blond.”
We don’t need to assume Martians exist.
So one might propose to rescue the traditional interpretation by considering the square contingently in the sense that we should use it only for entities in existence.

48. And what about this statement: “All shoplifters are prosecuted.”
Curiously, if there are no shoplifters, the proposition is true! In other words, statements like this require that the class of the subject be empty.
A-form and O-form are contradictories: they have opposite truth value. But if the entities of the subject do not exist, then A is false—and so is O!
That is, if it is not true that “All Martians are blond,” then it is not true that “Some Martians are not blond.”
But this is impossible because contradictories must have opposite truth values.
Consequently, we must not make that assumption. The modern interpretation preserves the relationships of the contradictories.

49. A-form and E-form are no longer contraries
A-form and E-form are no longer contraries. Contraries cannot both be true.
However, if we use the modern interpretation, and there are no shoplifters, both A and E are true.
Similarly, I and O are no longer subcontraries.
Subcontraries cannot both be false. Because I and O have an existential assumption by definition, they both assert that the entities described by the subject exist.
But if the entities in question don’t exist, then I and O are both false!

50. A no longer implies I and E no longer implies O
A no longer implies I and E no longer implies O. These pairs no longer exhibit superalternation or subalternation.
Given that the modern interpretation, A-form means “If anything is a shoplifter they’re all prosecuted.” But then A cannot imply that “There is in existence at least one shoplifter and it is prosecuted” which is exactly what I-form asserts.
According to the modern interpretation A and E are interpreted hypothetically so are always true because they cannot be falsified.
I and O by nature have an existential assumption, and if the subjects don’t exist, A and E are true and I and O false. Consequently the pairs no longer exhibit superalternation.
And the falsity of I and O no longer implies the falsity of A and E. if there are no shoplifters, then I and O are false. However, it does not follow that A and E are false. As we have seen, A and E are always true.