“Greatest Hits” of Formal Logic

1. “Greatest Hits” of Formal Logic Overview of deductive reasoning

2. Formal Logic is the science of deductive reasoning • Definition: “reasoning from known premises, or premises presumed to be true, to a certain conclusion.” • In contrast, most everyday arguments involve inductive reasoning. • reasoning from uncertain premises to probabalistic conclusions • “inference-making”

3. Formal logic cannot establish the truth of the premises. The truth of the premises must be presumed, or taken as a given. Some premises may be proven or authenticated by scientific testing, reference to external sources, etc. Some premises may be granted or stipulated by all the parties to an argument Some premises may have been established as the conclusion of a previous argument DNA testing and paternity If a DNA sample is collected and analyzed properly and, If the DNA is an exact match with the alleged father, Then that person is the father. Structural validity versusmaterial truth

4. There is no middle ground. A deductive argument can’t be “sort of” valid. By contrast, everyday arguments enjoy degrees of probability--plausible, possible, reasonable, believable, etc. In deduction, proofs are always valid or invalid.

5. The form or structure of a deductive argument determines its validity • The fundamental property of a valid, deductive argument is that IF the premises are true, THEN the conclusion necessarily follows. • The conclusion is said to be “entailed” in, or contained in, the premises. • If all pigs have curly tails • And Nadine is a pig • Then Nadine has a curly tail

6. The terms used in a syllogism must be defined precisely • If the meanings of key terms are vague or ambiguous, or change during the course of a deductive argument, then no valid conclusion may be reached. • Major premise: All pitchers hold water • Minor premise: Tom Glavin is a pitcher • Conclusion: Therefore, Tom Glavin holds water (the term “pitcher” has two different meanings in this argument, so no valid conclusion can be reached)

7. Example of a valid deductive argument major premise: All cats have 9 lives minor premise: “Whiskers” is a cat conclusion: Therefore, Whiskers has 9 lives (Note: it doesn’t matter whether cats really have 9 lives; the argument is premised on the assumption that they do.)

8. “Validity” versus “Soundness” • An argument is valid if its structure conforms to the rules of formal logic. • An argument is sound if it is valid, and its premises are true. • Thus validity is a prerequisite for soundness, but an argument needn’t be sound to be valid. • If sound, then valid too • If valid, not necessarily sound

9. Example of a valid, but unsound argument major premise: All cats are pink minor premise: Felix is a cat conclusion: Therefore, Felix is pink (Cats aren’t pink, which makes the first premise untrue. Validity, however, presumes the truth of the premises.) Example of a valid and sound argument major premise: Anthrax is not a communicable disease minor premise: Communicable diseases pose the greatest threat to public health conclusion: Therefore, anthrax does not pose the greatest threat to public health (The premises are true and the conclusion is valid, that is, it necessarily follows from the premises) Validity versus soundness

10. Syllogistic reasoning The syllogism is a common form of deductive reasoning. There are different types of syllogisms categorical (universal premises) hypothetical (if-then premises) disjunctive (either-or premises) All follow the basic form: major premise minor premise conclusion

11. Categorical syllogisms rely on universal premises Example of a valid categorical syllogism: major premise: All Christians believe Jesus is the son of God. minor premise: Biff is a Christian. conclusion: Biff believes Jesus is the son of God. (Note: validity isn’t affected by whether the premises are true or not. Obviously, other religions don’t accept Jesus as the son of God.)

12. Hypothetical syllogisms use “if-then” premises Example of a valid hypothetical syllogism: Major premise: If Biff likes Babbs, then he’ll ask her to the prom. Minor premise: Biff likes Babbs, Conclusion: Therefore, he’ll ask her to the prom.

13. Disjunctive syllogisms use “either-or” premises Example of a valid disjunctive syllogism: Major premise: Either Babbs will get her navel pierced, or she’ll get a tongue stud. Minor premise: Babbs didn’t get her navel pierced. Conclusion: Therefore, Babbs got a tongue stud.

14. Practice syllogism Major premise: Any creature with six legs is an insect. Minor premise: . Dr. Gass has six legs. Conclusion: Therefore, Dr. Gass is an insect. • What kind of syllogism is this? (categorical, hypothetical, or disjunctive) • Are the premises true? • Is the conclusion valid? • Is the argument sound (true premises and a valid conclusion) Answer: Valid, but unsound

15. Well-known forms of deductive invalidity • Affirming the consequent • Invalid Example: • If A, then B • B • Therefore, A • Invalid Example: • Students who plagiarize are expelled from school • Rex was expelled from school • Rex must have plagiarized

16. More deductive invalidity • Denying the antecedent • Invalid example: • If A, then B • Not A • Therefore, not B • Invalid example: • If you exceed the speed limit, you’ll get a ticket. • I’m not exceeding the speed limit. • Therefore, I won’t get a ticket.

17. More deductive invalidity • Undistributed middle term: • Valid example: • All A are B • All B are C • Therefore, all A are C • Invalid example • All A are B • All C are B • Therefore, all A are C The middle term, B, must serve as the subject of one premise, and the predicate of another premise, but cannot occur in the conclusion

18. Undistributed middle term: • Invalid example: • All humans need air to breathe • All dogs need air to breathe • Therefore, all humans need dogs

19. All rock stars want to become movie stars Morton wants to become a movie star Therefore, Morton must be a rock star affirming the consequent denying the antecedent undistributed middle term valid syllogism What, if anything, is wrong with this syllogism? Answer: Undistributed Middle Term

20. Anyone who has lived in California for more than a few years has experienced an earthquake Nadine has lived in California for more than a few years Nadine has experienced an earthquake affirming the consequent denying the antecedent undistributed middle term valid syllogism What, if anything, is wrong with this syllogism? Answer: Valid Syllogism

21. Anyone who has tried heroin has tried marijuana Naomi hasn’t tried heroin Therefore, Naomi hasn’t tried marijuana affirming the consequent denying the antecedent undistributed middle term valid syllogism What, if anything, is wrong with this syllogism? Answer: Denying the Antecedent If A, then B Not A Therefore, not B

22. All Christian fundamentalists are opposed to abortion Nadine is opposed to abortion Nadine is a Christian fundamentalist affirming the consequent denying the antecedent undistributed middle term valid syllogism What, if anything, is wrong with this syllogism? Answer: Affirming the Consequent If A, then B B Therefore, A