1 Propositional Logic - Axioms and Inference Rules

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1 1 Propositional Logic - Axioms and Inference Rules Axioms Axiom 1.1 [Commutativity] (p q) (q p) (p q) (q p) (p q) (q p) Axiom 1.2 [Associativity] p (q r) (p q) r p (q r) (p q) r Axiom 1.3 [Distributivity] p (q r) (p q) (p r) p (q r) (p q) (p r) Axiom 1.4 [De Morgan] (p q) p q (p q) p q Axiom 1.5 [Negation] p p Axiom 1.6 [Excluded Middle] p p T 1

2 Axiom 1.7 [Contradiction] p p F Axiom 1.8 [Implication] p q p q Axiom 1.9 [Equality] (p q) (p q) (q p) Axiom 1.10 [or-simplification] p p p p T T p F p p (p q) p Axiom 1.11 [and-simplification] p p p p T p p F F p (p q) p Axiom 1.12 [Identity] p p 2

3 Inference Rules p 1 p 2, p 2 p 3 p 1 p 3 Transitivity p 1 p 2 E(p 1 ) E(p 2 ), E(p 2 ) E(p 1 ) Substitution q 1, q 2,..., q n, q 1 q 2... q n (p 1 p 2 ) E(p 1 ) E(p 2 ), E(p 2 ) E(p 1 ) Conditional Substitution 3

4 2 Propositional Logic - Derived Theorems Equivalence and Truth Theorem 2.1 [Associativity of ] ((p q) r) (p (q r)) Theorem 2.2 [Identity of ] (T p) p Theorem 2.3 [Truth] T Negation, Inequivalence, and False Theorem 2.4 [Definition of F] F T Theorem 2.5 [Distributivity of over ] (p q) ( p q) ( p q) (p q) Theorem 2.6 [Negation of F] F T 4

5 Theorem 2.7 [Definition of ] ( p p) F p (p F) Disjunction Theorem 2.8 [Distributivity of over ] (p (q r)) ((p q) (p r)) ((p (q r)) (p q)) (p r) Theorem 2.9 [Distributivity of over ] p (q r) (p q) (p r) Conjunction Theorem 2.10 [Mutual definition of and ] (p q) (p (q (p q))) (p q) ((p q) (p q)) ((p q) p) (q (p q)) ((p q) (p q)) (p q) (((p q) p) q) (p q) Theorem 2.11 [Distributivity of over ] p (q r) (p q) (p r) 5

6 Theorem 2.12 [Absorption] p ( p q) p q p ( p q) p q Theorem 2.13 [Distributivity of over ] (p q) ((p q) p) ((p q) (p q)) p p (q p) (p q) Theorem 2.14 [Replacement] (p q) (r p) (p q) (r q) Theorem 2.15 [Definition of ] (p q) (p q) ( p q) Theorem 2.16 [Exclusive or] (p q) ( p q) (p q) Implication Theorem 2.17 [Definition of Implication] (p q) ((p q) q) ((p q) (p q)) q (p q) ((p q) p) ((p q) (p q)) p 6

7 Theorem 2.18 [Contrapositive] (p q) ( q p) Theorem 2.19 [Distributivity of over ] p (q r) ((p q) (p r)) Theorem 2.20 [Shunting] p q r p (q r) Theorem 2.21 [Elimination/Introduction of ] p (p q) p q p (q p) p p (p q) T p (q p) q p (p q) (p q) (p q) p F p F p T Theorem 2.22 [Right Zero of ] (p T) T Theorem 2.23 [Left Identity of ] (T p) p 7

8 Theorem 2.24 [Weakening/Strengthening] p p q p q p p q p q p (q r) p q p q p (q r) Theorem 2.25 [Modus Ponens] p (p q) q Theorem 2.26 [Proof by Cases] (p r) (q r) (p q r) (p r) ( p r) r Theorem 2.27 [Mutual Implication] (p q) (q p) (p q) Theorem 2.28 [Antisymmetry] (p q) (q p) (p q) Theorem 2.29 [Transitivity] (p q) (q r) (p r) (p q) (q r) (p r) (p q) (q r) (p r) Theorem 2.30 [Monotonicity of ] (p q) (p r q r) 8

9 Theorem 2.31 [Monotonicity of ] (p q) (p r q r) Substitution Theorem 2.32 [Leibniz] (e f) (E(e) E(f)) Theorem 2.33 [Substitution] (e f) E(e) (e f) E(f) (e f) E(e) (e f) E(f) q (e f) E(e) q (e f) E(f) Theorem 2.34 [Replace by T] p E(p) p E(T) p E(p) p E(T) q p E(p) q p E(T) Theorem 2.35 [Replace by F] p E(p) p E(F) E(p) p E(F) p E(p) p q E(F) p q Theorem 2.36 [Shannon] E(p) (p E(T)) ( p E(F)) 9

10 3 Propositional Logic - Examples and Exercises 10

11 4 Predicate Logic - Axioms Axiom 4.1 [Definition of ] i : m i < n : p i (m n) F (m < n) i : m i < n : p i i : m i < n 1 : p i p n 1 Axiom 4.2 [Definition of ] i : m i < n : p i (m n) T (m < n) i : m i < n : p i i : m i < n 1 : p i p n 1 11

12 Axiom 4.3 [Range Split] (m 1 m 2 m 3 ) i : m 1 i < m 2 : p i i : m 2 i < m 3 : p i i : m 1 i < m 3 : p i (m 1 m 2 ) (n 1 n 2 ) i : m 1 i < n 1 : p i i : m 2 i < n 2 : p i i : m 1 i < n 1 : p i (m 1 m 2 m 3 ) i : m 1 i < m 2 : p i i : m 2 i < m 3 : p i i : m 1 i < m 3 : p i (m 1 m 2 ) (n 1 n 2 ) i : m 1 i < n 1 : p i i : m 2 i < n 2 : p i i : m 1 i < n 1 : p i Axiom 4.4 [Interchange of Dummies] i : m 1 i < n 1 : ( j : m 2 j < n 2 : p i,j ) j : m 2 j < n 2 : ( i : m 1 i < n 1 : p i,j ) i : m 1 i < n 1 : ( j : m 2 j < n 2 : p i,j ) j : m 2 j < n 2 : ( i : m 1 i < n 1 : p i,j ) 12

13 Axiom 4.5 [Dummy Renaming] i : m i < n : p i j : m j < n : p j Axiom 4.6 [Distributivity of over ] (p ( i : m i < n : q i )) i : m i < n : (p q i ) Axiom 4.7 [Distributivity of over ] p ( i : m i < n : q i ) (m < n) i : m i < n : p q i i : m i < n : p i i : m i < n : q i i : m i < n : (p i q i ) Axiom 4.8 [Distributivity of over ] (p ( i : m i < n : q i )) i : m i < n : (p q i ) Axiom 4.9 [Distributivity of over ] p ( i : m i < n : q i ) (m < n) i : m i < n : (p q i ) i : m i < n : p i i : m i < n : q i i : m i < n : (p i q i ) Axiom 4.10 [Universality of T] i : m i < n : T T 13

14 Axiom 4.11 [Existence of F] i : m i < n : F F Axiom 4.12 [Generalized De Morgan] ( i : m i < n : p i ) i : m i < n : p i ( i : m i < n : p i ) i : m i < n : p i Axiom 4.13 [Trading] (m i < n) p i i : m i < n : p i (m i < n) p i i : m i < n : p i Axiom 4.14 [Definition of Numerical Quantification] Ni : m i < n : p i (m n) 0 (m < n) p n 1 (m < n) p n 1 Ni : m i < n : p i Ni : m i < n 1 : p i Ni : m i < n : p i Ni : m i < n 1 : p i

15 Axiom 4.15 [Definition of Σ] Σi : m i < n : e i (m n) 0 (m < n) Σi : m i < n : e i Σi : m i < n 1 : e i + e n 1 Axiom 4.16 [Definition of Π] Πi : m i < n : e i (m n) 1 (m < n) Πi : m i < n : e i Πi : m i < n 1 : e i e n 1 15

16 5 Predicate Logic - Derived Theorems Theorem 5.1 [Definition of ] i : m < i n : p i (m n) F (m < n) i : m < i n : p i i : m+1 < i n : p i p m+1 Theorem 5.2 [Definition of ] (m n) i : m < i n : p i T (m < n) i : m < i n : p i i : m+1 < i n : p i p m+1 16

17 Theorem 5.3 [Definition of Σ] Σi : m < i n : e i (m n) 0 (m < n) Σi : m < i n : e i Σi : m+1 < i n : e i + e m+1 Theorem 5.4 [Definition of Π] (m n) Πi : m < i n : e i 1 (m < n) Πi : m < i n : e i Πi : m+1 < i n : e i e m+1 17

18 6 Some Simple Laws of Arithmetic Throughout this compendium, we assume the validity of all simple arithmetic rules. Examples of such rules are all simplification rules, e.g x+x 2 x x+y y x (x/3) 3 x 0 x 0 1 x x x x x 2 0 x 0 1 x 1 (2 x+10 20) (x 5) (x+y < 2 y) (x < y) (x+y x+z) (y z) x (y +1) (x+z) (x y z) Following is a collection of theorems that might be used. The list is not exhaustive but intend to show the level of complexity that you can specify theorems on. 18

19 Theorems on < and (x < y) (y > x) (x < y) (y x) (y < x) (x < y) (x y) (x < y) (x y) (x y) (x < y) (y z) (x < z) (x y) (x > y) (x x) T (x y) (y z) (x z) (x y) (y < z) (x < z) (x y) (x < y) (x < y) (x y 1) (x < y) (x y) (y x) T (x y) (y < x) T (x y) (x < y) (x y) Theorems on properties about + and (x < y) (x < y +1) (x < y +1) (x y) (x < y) (z y < z x) (0 < x) ( x < 0) (x 1 < x) T (x y 1) (x < y) (x y) (x 1 < y) (x 1 < y 1 ) (x 2 < y 2 ) (x 1 +x 2 < y 1 +y 2 ) (x 1 y 1 ) (x 2 y 2 ) (x 1 +x 2 y 1 +y 2 ) 19

20 Theorems on properties about and / (0 < x) (0 < 2 x) (0 < x) (x < 2 x) (0 < x) (x 2 < x) (0 x/2) (0 x) (x 0) (x y 0) 2 (x/2) x Theorems on equivalence relation (x x) T (x y) (x y) (y x) Theorems about odd(n) and even(n) odd(x) ((x 1) 2 (x 1)/2) even(x) (x 2 x/2) odd(x+2 y) odd(x) even(x+2 y) even(x) odd(x) even(x) odd(x) ((x 1) (x 0)) odd(x) (x 0) F 20

21 7 Predicate Logic - Examples and Exercises 21

22 8 Arrays - Axioms 8.1 Axioms Axiom 8.1 [Assignment to Array Element] (i j) (e f) ((b; i : e)[j] f) (i j) (b[j] f) Axiom 8.2 [Definition of Arithmetic Relations] (b[i : j] x) ( k : i k < j +1 : b[k] x) (b[i : j] < x) ( k : i k < j +1 : b[k] < x) (b[i : j] > x) ( k : i k < j +1 : b[k] > x) (b[i : j] x) ( k : i k < j +1 : b[k] x) (b[i : j] x) ( k : i k < j +1 : b[k] x) (b[i : j] x) ( k : i k < j +1 : b[k] x) x b[i : j] ( k : i k < j +1 : x b[k]) 22

def: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.

def: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system. Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true