1. Deductive Arguments: Categorical Logic
Chapter 7 Lecture Notes
Deductive Arguments:
Categorical Logic

2. Chapter 7
This chapter focuses on the G (good grounds) condition of ARG and deals with simple deductive arguments. Deductive Relationships One statement deductively entails another if and only if it is impossible for the second one to be false, given that the first one is true. (page 178) The nature of the impossibility is logical impossibility.

3. Chapter 7
Another way to think of this is to say that an argument is deductively when it is impossible for all the premises to be true and the conclusion false. This means that deductively valid arguments satisfy the R and G conditions of ARG. Many argument are valid based on their logical form. (178) We are going to focus on the logic of categories.

4. Chapter 7
When evaluating deductively valid argument we need to realize that the argument will satisfy the R and G conditions, but may fail to satisfy the A condition of acceptable premises. Sometimes arguments are valid not because of their form, but because of the mean of the word involved (179) Robert is a brother logically entails that Robert is male. This is because of the meaning of brother.

5. Chapter 7
We are going to be dealing with Categorical Logic in this chapter. To start, we need to learn the four categorical forms.
In categorical logic, the terms all, no, some, and not are the basic logical terms.
The four categorical forms are:
A: All S is P
E: No S is P
I: Some S is P
O: Some S is not P

6. Chapter 7
The A form categorical statement is: A: All S is P This is a universal affirmation statement. It says that all the members of the S category are members of the P category. For example: All sisters are female persons. All my sandwiches are peanut butter.

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The E form categorical statement is: E: No S is P This is a universal negation statement. It says that none of the members of the S category are members of the P category (Ss are excluded from the P category). For example: No sisters are male persons. None of my sandwiches are pastrami.

8. Chapter 7 The I form categorical statement is: I: Some S is P
This is a particular affirmation statement. It says that at least one member of the S category is a member of the P category.
For example:
Some sisters are pilots.
Some of my sandwiches are pre-made.

9. Chapter 7
The O form categorical statement is: O: Some S is not P This is a particular negation statement. It says that at least one member of the S category is not a member of the P category (some Ss are excluded from the P category). For example: Some sisters are not pilots. Some of my sandwiches are not pre-made.

10. Chapter 7
The Square of Opposition shows the logical relationship between the different categorical forms. A: All S are P E: No S are P I: Some S are P O: Some S are not P A and O are contradictories: they have to have opposite truth values. The same goes for I and E.

11. Chapter 7
Other than being contradictory, there are other logical relationships between the statements. One of contritely. A and E statements are contraries because they cannot both be true, but they can both be false. I and O statements are referred to as subcontraries statements because they can both be false, but they cannot both be true. See page 181 for examples.

12. Natural Language and Categorical Forms
Chapter 7
Natural Language and Categorical Forms
One has to be very careful when converting English sentences into categorical forms. See pages for more examples of the Universal Affirmative, A form.
All S are P can be expressed in English by the following:
Any S is P (Any friend of yours is a friend of mine)
An S is P (A turkey is a bird)
The Ss are all Ps (The kittens are all spayed)
Each S is a P (Each day is blessing)
There are many other examples and stylistic variants.

13. Chapter 7 The Universal Negative, E form. No S are P Examples,
Not a single whale can fly.
No whale can fly.
Whales cannot fly.
No whales are creatures that can fly.
Whales are not able to fly.
Context and other issues will matter in translating these English sentences into their categorical forms.

14. Chapter 7 The Particular Affirmative, I form. Some S are P Examples:
Some S is P just means that there is at least one thing S that is part of P. We also have to take into account context with things starting with ‘a’ or ‘an’. For example,
A pianist gave a concert look a lot like
A turkey is a bird.
Notice that (1) is an I form statement, but (2) is an A form statement. Context and meaning can confuse issues.

15. Chapter 7 The Particular Negative, O form. Some S is not P
The word ‘not’ has to function in such a way to preclude some members of the S (subject) category from the P (predicate) category. So,
No all famous plays were written by Arthur Miller is really
Some famous plays are not plays written by Arthur Miller.
See page for more O from examples.

16. Chapter 7
Venn Diagrams One can represent the categorical statements using circles to represent the categories. These are called Venn Diagrams, named after the 19th Century English philosopher and logician John Venn. We are going to show the relationship between the categories in the forms with these circles. Since there are four basic form, A, E, I, and O, there will be four corresponding Venn Diagrams

17. Chapter 7
We shall represent the A form statement with a Venn Diagram. Below is figure 7.3, and it represents visually the claim All S are P. This Venn Diagram indicates that all the S things are part of the P category. So, all the Ss are P in this diagram. So, if all my sandwiches are peanut butter. There is nothing in the S circle so we shade it out.

18. Chapter 7
We shall represent the E form statement with a Venn Diagram. Below is figure 7.4, and it represents visually the claim No S are P. We shade the middle part of the circles because there are no things that are both S and P. The visual representation indicates that nothing is both S and P, and this is what is conveyed in the E form categorical statement.

19. Chapter 7
We shall represent the E form statement with a Venn Diagram. Below is figure 7.5, and it represents visually the claim Some S are P. This diagram indicates that there is at least one thing that is both S and P. For this reason we put an x in the area where the S circle and the P circle overlap. This indicates that there is something that is both S and P.

20. Chapter 7
We shall represent the E form statement with a Venn Diagram. Below is figure 7.6, and it represents visually the claim Some S are not P. In this case we need to indicate that there is at least one thing that is S, but is not P. We do this with an x in the S circle that is not part of the P circles. Some of my friends are not democrats indicates that Some S is not P.

21. Chapter 7
Immediate Inferences An immediate inference is a direct move from one categorical statement to another validly. We will look at several immediate inferences, the first is conversion. To generate the converse of a categorical proposition all you need to do is switch the subject and predicate terms. Sometimes when you do this, you get valid inferences, and sometimes you do not. An immediate inference from a categorical statement to its converse is valid for E and I forms, but not A and O. We shall see why.

22. Chapter 7
Here are the conversion diagrams for E and I form statements. The immediate inference is valid because the diagrams are the same or contain the same information. Review images 7.7 and 7.8 below. E: No S are P has the converse No P is S I: Some S are P has the converse Some P is S

23. Chapter 7

24. Chapter 7
Not all conversion, immediate inferences are valid. The move from A and O form categorical statements to their converse are not valid. A: All S are P converts to All P is S O: Some S are not P converts to Some P are not S. The Venn Diagrams are not the same and thus, the inferences are not valid. See figure 7.9 below.

25. Chapter 7

26. Chapter 7
There are three other simple immediate inference types that can occur: contraposition, obversion, and contradiction. Each has its own set of rules and have logical equivalences. Here is a chart of the logical equivalencies and those that are not logically equivalent.
Form
Conversion
Contrapostion
Obversion
Contradiction A
NLE LE E I O

27. Chapter 7 Summary of Rules of Immediate Inference
Conversion. (To create the converse of a statement, transpose its subject andpredicate.) All E and I statements are logically equivalent to their converse. No A or O statements are logically equivalent to their converse.
Contraposition. (To create the contrapositive of a statement, transpose its subject and predicate and place non in front of both.) All A and O statements are logically equivalent to their contrapositive. No E or I statements are logically equivalent to their contrapositive.
Obversion. (To create the obverse of a statement, change its quality from positive to negative or from negative to positive and place non in front of the predicate.) All statements in categorical form are logically equivalent to their obverse.
Contradiction. If A is true, then O is false, and vice versa. If E is true, then I is false, and vice versa (see Table 7.1).

28. Chapter 7
We also need to look at the classic Categorical Syllogism. There are several instances of how a categorical syllogism can be created, but it involves exactly two premises and a conclusion. The S and P terms are indicated in the conclusion, and the middle term is the term in the premises that are not in the conclusion. The predicate term in the conclusion is the major term and the premise with that term is the major premise. The subject term is the minor term and the premise with the subject term from the conclusion is the minor premise.

29. Chapter 7
A Venn Diagram that allows for diagramming the premises of a categorical syllogism requires three distinct circles that overlap in the following way. See figure 7.11 below.

30. Chapter 7
To test to see if a categorical syllogism is valid, all you need to do is check to see if the information represented in the diagram for the preemies is the same as the information represented in the diagram for the premises. Refer to pages 201 to 203 for examples of valid categorical syllogisms and their diagrams. For more detailed rules that govern the categorical syllogism, see page 206.

31. Chapter 7
An important term to know is enthymeme. Enthymemes are argument that have either an unstated premise or an unstated conclusion. These are often found in categorical syllogisms. Enthymemes can give rise to other kinds of paradoxical inferences called heaps or the Greek term is sorites. Review pages for a complete explanation of these issues.

32. Chapter 7
Terms to review: Binary thinking Categorical logic Categorical syllogism Complementary predicate Contradictory Contraposition Contrary Contrary predicates Conversion Distribution Enthymeme Existential Interpretation Fallacy of the undistributed middle False dichotomy

33. Chapter 7
Hypothetical interpretation Immediate Inference Logical Equivalence Major term Minor term Middle term Obversion Particular affirmative (I) Particular Negative (O) Sorites Square of opposition Stereotyping Subcontrary statements Universal affirmative (A) Universal negative (E) Venn Diagram