INTRO LOGIC

1. INTRO LOGIC DAY 05

2. Schedule for Unit 1 40% of Exam 10 arguments 4 pts each 60% of Exam 12 translations5 pts each

3. CHAPTER 4 TRANSLATIONS IN SENTENTIAL LOGIC

4. Basic Idea • In order to uncover logical forms, • we paraphrase English sentences • so that they contain only • standard connectives. We also use special abbreviations and special punctuation.

5. Abbreviation Scheme • atomic sentences are abbreviated by upper case letters (of the Roman alphabet). standard connectives are abbreviated by special symbols (logograms). compound sentences are abbreviated by algebraic combinations of 1 and 2.

6. Standard Connectives connective symbol and & or  not  if–then 

7. … but … … although … … or … but not both neither … nor … … if … … only if … … if and only if … … unless … … is necessary for … … is sufficient for … … if … otherwise … …unless…in which case Non-Standard Connectivesa few examples

8. Simple Transformations • Some simple sentences are straightforwardly equivalent to compound sentences. Jay and Kay are students Jay is a student and Kay is a student J & K Kay is a rich student Kay is rich and Kay is a student R & S

9. Pitfall #1 • Kay is a former student Kay is former and Kay is a student ? F & S

10. Pitfall #2 • Jay and Kay are married ambiguous [‘ambi’ (also ‘amphi’) means “both” and “around”] other words containing ‘ambi’ and ‘amphi’ ambidextrous ambivalent ambience (ambiance) ambitious amphibian amphitheater

11. First Reading • Jay and Kay are married • Jay is married (to someone) • and • Kay is married (to someone) J & K

12. Second Reading Jay and Kay are married • Jay and Kay are married-to-each-other M • later, in predicate logic, • further parts are revealed Mjk

13. Pitfall #3 • Odd use of ‘and’ • Sometimes ‘and’ is used as follows. • keep trying, and you will succeed • keep it up buster, and I will clobber you • give him an inch, and he will take a mile • give me the money, and I’ll give you the product • you do that, and I’ll kill ya

14. Other Toughies

15. Pitfall #3, continued • What happens if we symbolize • keep trying and you will succeed • as: • K & S ??? K you will keep trying S you will succeed K & S you will keep trying and you will succeed

16. appropriate paraphrase • (1) if you keep trying, then you will succeed K  S (2) if you keep it up buster, then I will clobber you K  C (3) if you give him an inch, then he will take a mile G  T (4) if you give me the money, then I’ll give you the product M  P (5) if you do that, then I’ll kill ya D  K

17. Exclusive ‘Or’ • Ram xor Sam will win (the election) • Ram will win, or Sam will win, butnot both at least one of them win ( R  S ) also & both of them will win…not! ( R & S ) alternatively, Ram will win, and Sam will not win, ( R &S ) or  Sam will win, and Ram will not win. ( S &R )

18. ‘Neither...Nor’ • it is neither raining nor sleeting • it is not raining, and it is not sleeting • R &S neither Jay nor Kay is sleeping Jay is not sleeping, and Kay is not sleeping J &K

19. Alternative Formulation • it is neither raining nor sleeting • it is not raining or sleeting • it is not true that ( it is raining or it is sleeting) • ( R  S ) •  • R &S

20. Conditionals • standard expression: • if A then C • which is symbolized: A  C A is the antecedent C is the consequent

21. Simple Variants of ‘if…then…’ • if A, then C • if A, C • C if A all of these are symbolized the same: A  C ‘if’ always introduces the antecedent ‘then’ always introduces the consequent

22. Other Variants of ‘if’ • C provided A • C in case A • C supposing A provided A, C in case A, C supposing A, C all symbolized the same way A  C

23. ‘Only If’ • ‘ only if’ is not equivalent to ‘ if ’ I will get an A onlyif I take all the exams vs. I will get an A if I take all the exams T F I will get an A if I get a hundred vs. I will get an A onlyif I get a hundred T F

24. How does ‘only’ work? employees only authorized personnel only cars only right turn only • ‘only’ operates as a dual-negative modifier. • for example, (1) means to • exclude anyone who is • not an employee.

25. a more lyrical example of ‘only’ “I only have eyes for you” The Flamingos 1959 lyrics by Al Dubin music by Harry Warren 1934

26. ‘only’ is focus-sensitive Ionlyhave eyes for you

28. alternative focus for ‘only’ Ionlyhave eyes for you

29. Sam, world’s ugliest dog

30. alternative focus for ‘only’ I onlyhave eyes for you

32. How does ‘only’ modify ‘if’? • ‘only’ modifies ‘if’ • by introducing two negations A onlyif B not A ifnot B ifnotB then notA B A

33. Example I will get an A onlyifI take all the exams A onlyif E I will not get an A ifI do not take all the exams not A ifnot E ifI do not take all the examsthen I will not get an A ifnot E thennot A E A

34. ‘If And Only If’ • Two approaches to ‘if and only if’ • treat it as a simple connective () • treat it as a complex connective • We will concentrate on (2), according to which • if and only if • consists of 3 components: • if • and • onlyif • each of which we already know how to paraphrase.

35. Example of ‘if and only if’ I will pass if and only if I average 50 P if and onlyif A P ifA and P onlyifA if A, then P not P ifnot A ifnot A thennot P ( A  P ) & ( A P )

36. ‘Unless’ • The following are all equivalent. • I will pass only if I study P only if S • I will not pass unless I study not P unless S • I will not pass if I do not study not P ifnot S onlyif has twobuilt-in negations unless has one built-in negation if has nobuilt-in negation unless = ifnot

37. Example of ‘unless’ I will not pass unless I study notP unless S notP if not S notP if not S if not S then notP S  P

38. THE END