Propositional Logic

1. Propositional Logic Propositional Language Translations Truth Tables Propositional Proofs Appendix: Model Theory

2. Introduction • Propositional logic studies arguments whose validity depends on “if-then,” “and,” “or,” “not,” and similar notions. • In the following, firstly we will introduce a formal language that sets up the framework of propositional logic, and then explain the relationship between this language and natural languages. • Secondly, we will introduce a simple model of it, i.e. the one made by truth-tables. • Finally, we will introduce inference rules and show how to construct a proof.

3. 1. Propositional Language • vocabulary • formation rules PL

4. Vocabulary • Logical constants: logical connectives • “,” “,” “,” “,” and “.” • Logical variables: propositional variables • P, Q, R…, and so on. (if necessary with subscripts appended—like ”P1”) • Auxiliary Signs: brackets • “ (,”and “).”

5. The Function of Brackets • Consider the case in mathematics: • 2+35=? • Either (2+3)5=25 or 2+(35)=17. • Similarly consider the following case: • What does “PR” mean? • It means either that (P)R or that (PR). • The function of brackets is to disambiguate the meaning of wffs (well-formed formula).

6. Formation Rules • Any capital letter is a well-formed formula. • The result of prefixing any wff with “” is a wff. • The result of joining any two wffs by “,” “,” “,” or “”and enclosing the result in parentheses is a wff. • Only that which can be generated by the rules (i)-(iii) in a finite number of steps is a wff in PL.

7. Some examples • PP • P(RP) • R • ((PR))Q • ((PR)(Q))P • PP • P(RP) • R • ((PR))Q • ((PR)(Q))P

8. Construction Tree (Top-Down) • For any wff in PL, we can construct a tree for it. • Ex. PP PP (iii, ) P (i)P (ii, ) P (i)

9. Construction Tree (Bottom-Up) • Ex. P(RP) P (i) R (i) P (i) RP (iii, ) P(RP) (iii, )

10. Main Connectives • The main connective of a formula is at the top of the tree (Top-down)—at the bottom if bottom-up. • In another words, if the whole sentence was constituted at last by one connective, then we call this one as main connective of this sentence. • Note: the leaf of tree must be atomic.

11. Some examples • (PR) • (P(RP)) • (P)(R) • ((PR))Q • ((PR))P • PP • P(RP) • R • ((PR))Q • ((PR)(Q))P

12. 2. Translations • As mentioned earlier, propositional logic studies arguments whose validity depends on “if-then,” “and,” “or,” “not,” and similar notions. • It follows that PL must contain these notions in order to represent natural languages to certain degree. • In a way, we can translate arguments in natural languages into wffs in PL.

13. Connectives • “” stands for “not.” • “” stands for “and.” (or “but,” “though”) • “” stands for “or.” (inclusive-or) • “” stands for “if-then.” • “” stands for “if and only if.”

14. Some examples • It will rain tomorrow. • It will not rain tomorrow. • It will rain tomorrow and I will bring my umbrella. • If it rains tomorrow, I will bring my umbrella. • Either it rains tomorrow, or it will not rain. • It will rain heavily if and only if the sky will be covered with dark clouds.

15. More… • Not both Alan and Bill like to play baseball. • If Alan took this course and Bill dropped this course, I would take this course. • Only if Cindy took this course, I would take this course. • Not either Alan drops this course or Bill drops this course.

16. 3. Truth Tables • Logical connectives are represented by some truth functions—by which we can calculate the truth table of each compound wff. • Before we introduce logical connectives, let’s see what a function is and what kind of function is called “truth functional.”

17. Set • A set is a collection of entities (or objects). • The principle of extensionality: • A set is defined by the members it contains. • Ex. Suppose our domain is {Alan, Bill, Cindy} • M: {Alan, Bill} (M: {x|x is a man}) • W: {Cindy} (W: {x|x is a woman})

18. Membership • Suppose we use “a” for Alan, “b” for Bill, and “c” for Cindy: • a  M and b  M, but c M. • c  W, but a  W and b  W. • Suppose that B: {x|x is Cindy’s brother}. Alan is Cindy’s brother, and Bill is not. • a  B and b  B.

19. Subset • If A is a subset of B, then if x  A, then x  B. • A  B • If A is a proper subset of B, then if x  A then x  B and, for some y, such that y  B and y  A. • A  B • Ex. • Given that M: {a, b} and B: {a}, we call B is a proper subset of M—symbolized as B  M.

20. Intersection and Union • AB : {xxA and xB} • AB : {xxA or xB} • Ex. • {1, 2, 3}{2, 4, 6}={1, 2, 3, 4, 6} • {1, 2, 3}{2, 4, 6}={2}

21. Function: Many-one Relation between Domain and Range brotherhood Alan Cindy Bill

22. Suppose both Alan and Bill are Cindy’s brothers. Alan Cindy Bill

23. This is not a function Alan Cindy Bill

24. Function 1 1 0 0

25. Compound Function 1 1 1 0 0 0

26. 1 1 1 0 0 0

27. 1 1 1 0 0 0

28. 1 1 1 0 0 0

29. Logical Connectives—

30. 1 1 0 0

31. Logical Connectives—

32. 1 1 1 0 0 0

33. Logical Connectives—

34. 1 1 1 0 0 0

35. Logical Connectives—

36. 1 1 1 0 0 0

37. Logical Connectives—

38. 1 1 1 0 0 0

39. Truth Function • Connectives which give rise to sentences whose truth value depends only on the truth values of the connected sentences are said to be truth-functional. • Therefore, “not” (), “or” (), “and” (), “if-then” (), and “if and only if” () are truth-functional.

40. Some examples • U  F • U  T • U  F • F  U • F  U • (T)  U

41. U  F

42. U  T

43. Complex Truth Tables • (P  (Q  R)) • ((P  Q)  R) • ((P  Q)  Q) • ((P  Q)  R) • (P  (Q  R))

44. (P  (Q  R))

45. Functional Completeness • A system of connectives can express all truth functions is said to be functionally complete. • Ex. Define “” by “” and “.” • Actually, we can use “” and “” to represent others, and we call these two functions functionally complete.

46. Can you find another system of connectives which is also functionally complete?

47. The Truth-table Test • Recall how valid and invalid are defined for arguments: • VALID = no possible case has premises all true and conclusion false. • INVALID = some possible case has premises all true and conclusion false. • Now, we can use the truth-table test on a propositional argument.

48. (D  A), D/A

49. (D  A), D/A

50. More example • It is in your left hand or your right hand. • It is not in your left hand. •  It is in your right hand. (L: it is in your left hand; R: it is in your right hand) • L  R • L •  R