PHIL 120: Third meeting

1. PHIL 120: Third meeting • What to know for Test 1 (in general terms). • Symbolizing compound sentences (cont’d) • Paying attention to English punctuation when symbolizing into SL and when and how to use SL punctuation: () and [] • Special phrases and terms • Basic notions: their inter-relationships and implications continued • Open: review questions

2. Part 1 Preparing for Test 1

3. Test 1 • You need to know: • The definitions of basic notions of logic introduced in lecture on Chapter 1 (see glossary p. 27; you may ignore ‘inductive strength’) and what they imply • The recursive definition of SL and what it implies about what is (and is not) a sentence of SL, that sentences with connectives have a main connective, etc. • The characteristic truth table for each of the 5 connectives of SL • How to symbolize simple and compound sentences into SL

4. I Vitamin C cures all colds. Vitamin C never cures colds. ------------------------------- The moon is made of green cheese. II There are 385 days in a year February has 31 days ------------------------------- A rose is a rose III I’m here and nobody is here -------------------------- 2 + 2 = 5 For example, you need to know that and why the following arguments are deductively valid

5. Part 2 (a) Symbolizing compound sentences continued

6. Terminology • Sentences of the form P  Q are called material conditionals. The sentence that follows the logical operator ‘if’ and is symbolized to the left of the horseshoe is called the antecedent. The sentence that follows the logical operator ‘then’ and is symbolized to the right of the horseshoe is called the consequent. • The sentences connected by the & in a conjunction are called conjuncts. • The sentences connected by the v in a disjunction are called disjuncts.

7. Paying attention to English punctuation • ‘If Sarah skis regularly and Otis does too,then it is not the case that Sarah jogs regularly’. • Choose atomic sentences to symbolize the simple declarative sentences, e.g.,: S: Sarah skis regularly O: Otis skis regularly J: Sarah jogs regularly • The coma after ‘does too’ suggests that all that comes before it is a compound sentence (S & O) that is the antecedent of a conditional, the consequent of which is ~J • So we can use: (S & O)  ~J

8. ‘If Sarah skis regularly and Otis does too, then either it is not the case that Sarah jogs regularly or it is not the case that Otis does’ OR • If Sarah skis regularly and Otis does too, then it is not the case that both Sarah jogs regularly and Otis does’ • Use the coma again to identify the main connective as ‘if, then’ () and to identify the antecedent and consequent of this conditional • Using T for ‘Otis jogs regularly’ • We can use: (S & O)  (~J v ~T) OR (S & O)  ~(J & T)

9. Truth tables of the two forms of the consequent demonstrate that the sentences are logically equivalent

10. Paying attention to English punctuation • ‘Alison works hard although Mark doesn’t; but if Mark is a success, then Alison is too’ • The semi-colon indicates that there is a sentence before and after it. • ‘But’ indicates that we use an & to connect the sentences before and after the semi-colon. A: Alison works hard M: Mark works hard S: Mark is a success D: Alison is a success • So we can use: (A & ~M) & (S  D)

11. Paying attention to English punctuation • ‘Either Michael, or Roxanne, or Shirley works hard; but if Michael works hard, then either Roxanne doesn’t or Shirley doesn’t’ • Use the semi-colon to identify the sentence as a conjunction whose left conjunct is a disjunction and right conjunct is a material conditional M: Michael works hard R: Roxanne works hard S: Shirley works hard The left conjunct can be symbolized as (M v R) v S OR M v (R v S)

12. Using the punctuation of SL ‘Either Michael, or Roxanne, or Shirley works hard; but if Michael works hard, then either Roxanne doesn’t work hard or Shirley doesn’t’ Left conjunct can be symbolized as: M v (R v S) OR (M v R) v S The right conjunct can be symbolized as: M  (~R v ~S) OR M  ~(R & S) As each conjunct includes a binary connective, we need to use brackets so that it is clear which sentences are combined using &

13. Using the punctuation of SL If we don’t use brackets we have the following: M v (R v S) & M  (~R v ~S) (or one of the other versions) But this is not a sentence of SL because it has no main connective. We have no idea what conditions would make it true or false. We want one of the versions that includes brackets: [M v (R v S)] & [M  (~R v ~S)] Here we have a conjunction, with a disjunction as the left conjunct and a material conditional as the right conjunct.

14. Using the punctuation of SL So we can use any of the following (logically equivalent) symbolizations: [M v (R v S)] & [M  ~(R v S)] [M v (R v S)] & [M  (~R & ~S)] [(M v R) v S] & [M  ~(R v S)] [(M v R) v S] & [M  (~R & ~S)]

15. What is and what is not a sentence of SL These are sentences of SL: ~~A A22 ~(A  B) [A  (B v A)]  M These are not sentences of SL (why not?) A ~ B ~(A  B A  (B v A)]  M

16. Part 2 (b) “Special” phrases and terms

17. ‘Only if’ Compare: (a) ‘If the operation is a success, then the patient survives’ (or ‘Provided that the operation is a success, then the patient survives’) with: (b) ‘Only if the operation is a success, the patient survives’ The only time (a) will be false is when the operation is a success but the patient does not survive. If the operation is cancelled or a failure, the conditional is true.

18. ‘Only if’ (a) ‘If the operation is a success, the patient survives’ (b) ‘Only if the operation is a success, the patient survives’ The only time (b) will be false is if the patient survives and the operation was not a success. Let’s use: O: the operation is a success P: the patient survives • can be symbolized as ‘O  S’ • can be symbolized as ‘S  O’

19. Truth table for (a) and truth table for (b):Note differences in rows 2 and 3

20. Why ‘if and only if’ works as it does We said that a sentence of the form P  Q is logically equivalent to the conjunction of 2 material conditionals Take the sentence A  B It is logically equivalent to the sentence (A  B) & (B  A) ‘If A then B (A B), and if B then A’ OR ‘If A then B, and A only if B’ ‘B  A’ symbolizes both paraphrases of the right conjunct.

21. ‘If and only if’

22. ‘Unless' ‘Mary jogs unless she is sick’ M: Mary jogs. S: Mary is sick. Can be paraphrased and symbolized as EITHER: ‘Either Mary jogs or Mary is sick’ M v S OR ‘If Mary is not sick, then Mary jogs’ ~S  M OR ‘If Mary does not jog, then Mary is sick’ ~M  S

23. Truth table for ‘unless’

24. ‘Either/or but not both' The v and ‘either/or’ reflect the inclusive sense of ‘or’. So consider the sentence: ‘Either Sarah plays poker well or Jack does, but not both’ S: Sarah plays poker well J: Jack plays poker well. Left conjunct can be symbolized as: S v J Right conjunct can be symbolized as: ~S v ~J OR ~(S & J) The whole sentence can be symbolized as: (S v J) & (~S v ~J) OR (S v J) & ~(S & J)

25. ‘Either/or and not both’(or at most one)

26. ‘Neither/nor' Consider the sentence: ‘Neither Alice nor Bruce plays poker’ This is logically equivalent to: ‘Alice doesn’t play poker and Bruce doesn’t play poker’ which we paraphrase as: ‘It is not the case that Alice plays poker and it is not the case that Bruce plays poker’. It is also logically equivalent to: ‘It is not the case that either Alice or Bruce plays poker’.

27. ‘Neither/nor' ‘Neither Alice nor Bruce plays poker’ A: Alice plays poker. B: Bruce plays poker. ‘It is not the case that Alice plays poker and it is not the case that Bruce plays poker’ ~A & ~B ‘It is not the case that either Alice plays poker or Bruce plays poker’ ~(A v B)

28. ‘Neither/nor’

29. Connectives that are not truth-functional A connective is truth functional if and only if it determines the truth value of a sentence given the truth values of the sentence’s immediate components. ‘Because’ is not a truth functional connective. It can connect 2 sentences that are each true to form a true sentence: ‘Jan. 19 celebrates MLK because he was a great American’ Or connect 2 true sentences that are each true to form a false sentence: ‘Jan. 19 celebrates MLK because 2 + 2 = 4’

30. Implications of logical notions • Main connectives matter because they determine the truth value of a given sentence (T or F) based on their characteristic truth tables and the truth values (T or F) of a sentence’s immediate components on each possible truth value assignment • Compare: (A v B)  ~A • with: A v (B  ~A)

31. The sentences’ truth tables demonstrate that they are not logically equivalent (rows 1 and 2)

32. Part 4 Open review