Categorical Arguments, Claims, and Venn Diagrams

1. Categorical Arguments, Claims, and Venn Diagrams • Sign In! • Review • Group Abstractions! • Categorical Arguments • Types of Categorical Claims • Diagramming the claims • For next time: Read Chapter 8 pages 257-263

2. Review • Last time we had begun to make the transition from informal logic to formal logic • We did that by practicing and reviewing things we already knew: identifying claims and abstracting them • We also were reacquainted with categorical claims and categorical arguments • Let's take another quick look at categorical arguments

3. Review • Categorical claims were claims that relate two different groups (categories) with one another • When we translate sentences into categorical claims we must make sure to clarify the groups involved • Ex: “All students pay fees” • How would we translate this claim to make the groups clearer • We also began to diagram some of these arguments using Venn Diagrams

4. A quick note about Venn Diagram • We have been using Venn Diagrams all quarter to visualize the relationships between claims • It's important to keep in mind what each overlap in a Venn Diagram is telling us when we attempt to use diagrams to prove validity • The Image on the right demonstrates those relationships

5. Warm Up • For each sentence, translate it into a categorical claim to make the groups clearer along with the relationship between the groups • “Everybody who is ineligible for Physics 1A must take Physical Science 1” • “No students who are required to take Physical Science 1 are eligible for Physics 1A” • The two claims look similar but are in fact quite different

6. Warm Up • “Everybody who is ineligible for Physics 1A must take Physical Science 1” • All students who are ineligible for Physics 1A are students who must take Physical Science 1 • All As are Bs • “No students who are required to take Physical Science 1 are eligible for Physics 1A” • No students who must take Physical Science 1 are students who can take Physics 1A • No Bs are Cs • To see why these claims are different we can diagram them

7. Warming up even more • It's important that we practice these skills • Translating claims into categorical claims (for now) and later into claims in sentential logic will need to become second nature • Take a minute to practice the following claims. Make the groups as clear as possible: • “There are frogs wherever there are snakes” • “Not every lizard is a salamander” • “Some aardvarks are not mammals”

8. Subject and Predicate Terms • Every Categorical Claim has the following basic form: • Indicator [group] indicator [group] • All As are/are not Bs • The first group in a categorical claim is typically called the subject term • The second group in a categorical claim is typically called the predicate term • Why do they have these names? A predicate is a property attributed to subjects in a sentence • Ex- The child (subject) is tall (predicate)

9. The Four Categorical Claim Types • Last time, we saw a lot of different indicators that we can use to express categorical claims • All of those different kinds of categorical claims are reducible (i.e. can be translated into without changing the meaning of a claim) to one of four types of categorical claims: • 1. A-Claim: All As are Bs [A = All] • 2. E-Claim: No As are Bs [E = Excludes] • 3. I-Claim: Some As are Bs [at least one A is a B] • 4. O-Claim: Some As are not Bs [ at least one A is not a B]

10. Affirmative and Negative Claims • A-Claims and I-Claims are sometimes called affirmative claims because they state a positive relationship between two groups • Conversely, O-Claims and E-Claims are called negative claims because they state a negative relationship between two groups (the claim is negative because two groups are being excluded) • We'll need these distinctions later but for now it's good to keep them in mind as we practice translating

11. Examples and Diagrams • Let's diagram each of these different claim types: • A-Claim: All As are Bs • ex- All zombies are undead • E-Claim: No As are Bs • ex- No living humans are zombies • I-Claim: Some As are Bs • ex- Some undead beings are zombies. [At least one undead being is a zombie] • O-Claim: Some undead beings are not zombies. [At least one undead being is not a zombie]

12. Practice makes perfect • For each of the following: 1) figure out the subject and predicate terms 2) decide what the sentence is saying about the groups and then 3) translate the argument into standard categorical form; finally 4) diagram the claim • “Minors are not eligible” • “Some veeblefetzers are carbostats” • “Idiots would support the measure, but no one else would” • “Coffee is a stimulant, since coffee contains caffeine”

13. Working Backwards • What categorical claims are implied by the Venn Diagram on the right? • What conclusion(s) can you draw based on this diagram? • Can you create a valid argument using the information here?

14. Group Exercises • In groups of 3-4, what Categorical Claim is implied by this diagram? • Are there any other claims implied by the diagram?

15. Group Exercises! • Identify all of the categorical claims that are implied by the diagram on the right • There may be many categorical claims, make sure to note whether they are A, E, I, or O claims

16. Group Exercises (last slide) • Translate the claims below into standard categorical claim form THEN represent those claims with a single Venn Diagram • “No members of the club are people who took the exam” • “Some people who did not take the exam are members of the club” • “None of the people who gave blood were tested, so everybody who gave blood must have been untested”

17. For next time • Read Chapter 8 pages 257-263 • Quiz!