Review of Propositional Calculus

Transcription

1 CS 1813 Discrete Mathematics Review of Propositional Calculus 1

2 Truth Tables for Logical Operators P Q P Q P Q P Q P Q P Q P 2

3 Semantic Reasoning with Truth Tables Proposition (WFF): ((P Q) (( P) Q)) P Q (P Q) ( P) (( P) Q) ((P Q) (( P) Q)) If prop is when all variables are: P, Q ((P Q) (( P) Q)) Formal statement about meaning Proved via truth table double turnstile Some : prop is Satisfiable If they were all : Tautology All : Contradiction (not satisfiable) 3

4 Fig 2.1, Hall/O Donnell Discrete Mathematics with a Computer Springer, 2000 Rules of Inference 4

5 Some Intrinsic Data Structures in Haskell Sequences (aka lists in primitive PLs, done as linked lists) [x 1, x 2, ] all x s must have same type Examples: [t] means a sequence with elements of type t [1, 9, 3, 27] [And A B, Or P Q, B] type: [Integer] type: [Prop] Tuples (like structs or records in other programming languages) (c 1, c 2 ) pair components may have different types (c 1, c 2, c 3 ) 3-tuple (longer tuples OK must be at least 2 components) Examples: (t 1, t 2 ) means a pair where component k has type t k (7, And A B) type: (Integer, Prop) (AndEL (Assume(A `And` B)) A, Assume B) type: (Proof, Proof) Proof Proof 5

6 Some Theorems in Rule Form a b { Comm} b a And Commutes a b b c { Chain} a c Implication Chain Rule a b b {modtol} a {nomiddle} a ( a) Modus Tollens Law of Excluded Middle a b { Comm} b a Or Commutes a b { F } ( a) b Implication Fwd a a { +&- } NeverBoth (a b) { ( )Comm} (b a) Not Or Commutes a b {conpos F } ( b) ( a) Contrapositive Fwd ( a) b { F } a b Implication Bkw 6

7 More Theorems in Rule Form (a b) {DeM F } ( a) ( b) DeMorgan Or Fwd (a b) {DeM F } ( a) ( b) DeMorgan And Fwd ( a) ( b) {DeM B } (a b) DeMorgan Or Bkw ( a) ( b) {DeM B } (a b) DeMorgan And Bkw a b a {disjsyll} b Disjunctive Syllogism ( a) { F } a Double Negation Fwd a { B } ( a) Double Negation Bkw 7

8 From Fig 2.1, Hall & O Donnell, Discrete Math with a Computer, Springer, 2000 Laws of Boolean Algebra page 1 8

9 From Fig 2.1, Hall & O Donnell, Discrete Math with a Computer, Springer, 2000 Laws of Boolean Algebra page 2 9

10 Some Theorems in Boolean Algebra (a b) b = b { absorption} (a b) b = b { absorption} (a b) c = (a c) (b c) { imp} 10

11 Boolean Equations and Tautologies Tautology WFF that is true for all combinations of values of its atomic consituents Let p, q stand for arbitrary WFFs Suppose Boolean laws prove p = q Then the following WFFs are tautologies p q q p p q 11

12 Theorems and Tautologies Let p, q, stand for arbitrary WFFs p q If this is a theorem, then p q is a tautology p, q, r s p If this is a theorem, then (p q r) s is a tautology If this is a theorem, then p is a tautology 12

13 What It All Means Notions of Consistency in Formal Systems If a b, then a = b (a, b arbitrary WFFs) Completeness in Formal Systems If a = b, then a b (a, b arbitrary WFFs) If b is true whenever a is, there is a proof of a b Propositional Logic Consistency propositional logic is consistent Inference preserves tautologies Inconsistency would make all WFFs tautologies Some WFFs aren t tautologies QED (consistency) Completeness propositional logic is complete That s no surprise, is it? 13

14 Kurt Gödel the first computer scientist Arithmetic is consistent comforting, but not surprising Arithmetic is not complete The recognition of Kurt Gödel as the first computer scientist is an insight of John Allen, a present-day logician and computer scientist, author of Anatomy of Lisp. Some statements about numbers are consistent with all the axioms, but cannot be proved A humongous surprise for mathematicians in the 1930s Consistency and Completeness Impossible goal No formal system can be both Except stripped-down systems (less powerful than arithmetic) In essence, this has nothing to do with numbers It has to do with the limits of formal systems (computers) Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I Gödel, definitions + many lemmas precede main point Translation by Meltzer $6.25, Amazon.com 14

15 Practice Problem ( a) b a b Implication Backward a { B } ( a) b ( a) {disjsyll} b { I} a b 15

16 Practice Problem ( a) ( b) b a contrapositive backwards ( a) ( b) ( ( b)) ( a) {conpos F } { F } { F } ( b) ( a) b a { F } { I L } { I R } ( ( b)) ( a) ( b) a ( b) a { E} ( b) a { B } b a That s ugly! There must be a better way. 16

17 Practice Problem ( a) ( b) b a contrapositive backwards Better way b { B } ( a) ( b) ( b) {modtol} ( a) { F } a { I} b a 17

18 Practice Problem a (( a) b) = a b a (( a) b) = (a ( a)) (a b) { dist/ } = (a b) { comp} = (a b) { commutes} = a b { id} 18

19 The End 19