Logic As Algebra COMP1600 / COMP6260. Dirk Pattinson Australian National University. Semester 2, 2017


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1 Logic As Algebra COMP1600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2017
2 Recap: And, Or, and Not x AND y x y x y x OR y x y x y NOT x x x / 41
3 Visualisation: Venn Diagrams the boxes are the space of all situations where x and y are true or false labelled circles describe those situations where x and y are true red area describes those situations where the formula is true. 2 / 41
4 Memories from High School... Analogy: Algebraic Terms. Given a set V of variables, algebraic terms are constructed as follows: 0, 1,... and all variables x V are algebraic terms if s and t are algebraic terms, then so is s + t and s t. Example. 5 + (3 x) x (x (3 (7 + 5))) + z Usual precendence. binds more strongly than +: x 3 + y reads as (x 3) + y Crucial Aspect. Terms can be evaluated given values for all variables. 3 / 41
5 Back to Boolean Functions Definition. Given a set of V of variables, boolean formulae are constructed as follows: T (true) and F (false) and all variables x V are boolean formulae if φ and ψ are boolean formulae, then so are φ ψ and φ ψ. if φ is a boolean formula, then so is φ. Examples. T (x ( y)) x x (T (F x) Precedence. binds more strongly than binds more strongly than : x y z reads as (( x) y) z Crucial Aspect. Boolean formulae can be evaluated given (boolean) values for all variables. 4 / 41
6 Equations Examples from Algebra. x (3 + y) = x 3 + x y 25 + (18 y) = 18 y + 25 Boolean Equations. have boolean formulae on the left and right: x (y x) = x T (y x) = x y Valid Equations. For all values of variables, LHS and RHS evaluate to same number. Applies to both algebraic terms and boolean formulae! 5 / 41
7 Valid Boolean Equations. Associativity a (b c) = (a b) c Commutativity a b = b a Absorption. a (a b) = a Identity. a F = a a (b c) = (a b) c a b = b a a (a b) = a a T = a Distributivity. a (b c) = (a b) (a c) a (b c) = (a b) (a c) Complements. a a = T a a = F 6 / 41
8 Equational Reasoning in Ordinary Algebra (x + 12) 3(x + 17) = (x + 12) (3 x ) (distributivity) = (x + 12) (3 x + 51) = x (3 x + 51) + 12 (3 x + 51) (distributivity) = x 3 x + x x ) (distributivity) = 3 x x + 36 x (commutativity) = 3 x 2 + ( ) x (distributivity) = 3 x x each step (other than addition / multiplication of numbers) justified by a law of arithmetic pattern matching against algebraic laws 7 / 41
9 Proving Boolean Equations Example. We prove the law of idempotence: x x = (x x) T (identity) = (x x) (x x) (complements) = x (x x) (distributivity) = x F (complements) x (identity) Rules of Reasoning. All boolean equations may be assumed (with variables substituted by formulae) may replace formulae with formulae that are proven equal equality is transitive! 8 / 41
10 Two faces of boolean Equations Truth of boolean equations: A boolean equation φ = ψ (where φ, ψ are boolean formulae) is true if φ and ψ evaluate to the same truth values, for all possible truth values of the variables that occur in φ and ψ. Equational Provability of boolean equations: A boolean equation is provable if it can be derived from associativity, commutativity, absorption, identity, distributivity and complements using the laws of equational reasoning. Q. How do these two notions hang together? 9 / 41
11 Soundness and Completeness Slightly Philosophical. Truth of an equation relates to the meaning (think: truth tables) of the connectives, and. Equational provability relates to a method that allows us to establish truth of an equation. They are orthogonal and independent ways to think about equations. Soundness. If a boolean equation φ = ψ is provable using equations, then it is true. all basic equations (associativity, distributivity,... ) are true the rules of equational reasoning preserve truth. Completeness. If a boolean equation is true, then it is provable using equations. more complex proof (not given here), using the socalled Lindenbaum Construction. 10 / 41
12 Challenge Problem: The De Morgan Laws De Morgan s Laws (x y) = x y (x y) = x y In English if it is false that either x or y is true, they must both be false if it is false that both x and y are true, then one of them must be false. Truth of De Morgan s Laws: Easy to establish via truth tables. Provability of De Morgan s Laws if the completeness theorem (that we didn t prove!) is true, then an equational proof must exist however, it is quite difficult to actually find it! 11 / 41
13 Criticism of Equational Proofs The good. Completeness tells us that if an equation is true, we can prove it. The bad. Sometimes need lots of ingenuity to find a proof! E.g. x x = (x x) T = (x x) (x x) = x (x x) = x F = x The ugly. Equational reasonign is not natural, i.e. it doesn t mirror the meaning of, and. 12 / 41
14 Towards Propositional Formulae and Natural Deduction New Connective. Implication, written In English. x y means if x is true, then so is y. Truth Table. Informally, think of x y as a promise. the promise is that y is true if x is true x y evaluates to F if the promise is broken x y x y F F 13 / 41
15 Towards Propositional Formulae and Natural Deduction New Connective. Implication, written In English. x y means if x is true, then so is y. Truth Table. Informally, think of x y as a promise. the promise is that y is true if x is true x y evaluates to F if the promise is broken x y x y F F T F T 13 / 41
16 Towards Propositional Formulae and Natural Deduction New Connective. Implication, written In English. x y means if x is true, then so is y. Truth Table. Informally, think of x y as a promise. the promise is that y is true if x is true x y evaluates to F if the promise is broken x y x y F F T F T T T F 13 / 41
17 Towards Propositional Formulae and Natural Deduction New Connective. Implication, written In English. x y means if x is true, then so is y. Truth Table. Informally, think of x y as a promise. the promise is that y is true if x is true x y evaluates to F if the promise is broken x y x y F F T F T T T F F T T 13 / 41
18 Towards Propositional Formulae and Natural Deduction New Connective. Implication, written In English. x y means if x is true, then so is y. Truth Table. Informally, think of x y as a promise. the promise is that y is true if x is true x y evaluates to F if the promise is broken x y x y F F T F T T T F F T T T 13 / 41
19 Interlude: Logic to English Exercise. Use the predicates I I m going surfing, Y you re going surfing, and W there ll be a big wave that kills us all, to translate the following statements to English: 1. I Y W 2. (I W ) (Y W ) Possible Answer. 14 / 41
20 Interlude: Logic to English Exercise. Use the predicates I I m going surfing, Y you re going surfing, and W there ll be a big wave that kills us all, to translate the following statements to English: 1. I Y W 2. (I W ) (Y W ) Possible Answer. 1. If both of us are goign surfing, then there ll be a big wave that kills us all. 14 / 41
21 Interlude: Logic to English Exercise. Use the predicates I I m going surfing, Y you re going surfing, and W there ll be a big wave that kills us all, to translate the following statements to English: 1. I Y W 2. (I W ) (Y W ) Possible Answer. 1. If both of us are goign surfing, then there ll be a big wave that kills us all. 2. If both of us are goign surfing, then there ll be a big wave that kills us all. (Both formulae have the same truth table!) 14 / 41
22 Propositional Formulae Definition. Given a set of V of variables, propositional formulae are constructed as follows: T (true) and F (false) and all variables x V are boolean formulae if φ and ψ are boolean formulae, then so are φ ψ and φ ψ and φ ψ if φ is a boolean formula, then so is φ. Precedence. binds more strongly than binds more strongly than binds more strongly than : x y z reads as ( x) (y z) Boolean Formulae vs Propositional Formulae propositional formulae are boolean formulae with addition of is expressible using boolean formulae: x y = x y but is included as implication is used very frequently 15 / 41
23 Contradictions and Contingencies Types of Propositional Formulae. A propositional formula is valid, if it evaluates to true under all truth value assignments. a contradiction if it evaluates to F under all truth value assignments, and a contingency if there are (necessarily different) truth value assigments for whic it evaluates to T and to F. Example. John had toast for breakfast is a contingency. John had toast for breakfast John had toast for breakfast is a contradiction. p ( q p) (p q) r can be complicated 16 / 41
24 Example proof using truth tables Statement to be proved: φ (p (q r)) ((p q) r)) For all 8 (= 2 3 ) possibilities of p, q, r, calculate truth value of the statement p q r q r p (q r) p q (p q) r φ T T T T T T T T T T F T T T T T T F T T T F T T T F F F F F F T F T T T F F T T F T F T F F F T F F T T F F T T F F F F F F F T Always exponential in size! 17 / 41
25 Natural Deduction Truth Tables. Can be exponential Equational Proofs. Can be very unintuitive Natural Deduction formal system that imitates human reasoning explains one connective at a time: intro and elim rules used to prove validity of formulae. also used in all formal theorem provers 18 / 41
26 Informal Proof Goal. Show that φ (p (q r)) (q s) p is valid. Informal Proof. 1. First,assume that p (q r) (is true) and show that (q s) p. 2. under this assumption, we have that p (is true). 3. still under this assumption, (q s) p (is true). 4. That is, p (q r) (q s) p without assumptions. Formal Natural Deduction Proof. 1 p (q r) Assumption 2 p E, 1 3 (q s) p I, 2 4 (p (q r)) ((q s) p) I, / 41
27 Conjunction rules And Introduction ( I) p q p q as p is true, and q is true, we have that p q is true. And Elimination ( E) p q p p q q as p q is true, we have that p is true. as p q is true, we have that q is true. 20 / 41
28 Example Example. Commutativity of conjunction (derived rule) p q q p assuming that p q (is true), we (also) have that q p (is true). Informal Proof. 1. Assume that p q. 2. because of p q, we have p. 3. because of p q, we have q. 4. therefore, we also have q p. Natural Deduction Proof. 1 p q 2 p E, 1 3 q E, 1 4 q p I, 2, 3 21 / 41
29 Implication rules Implication Introduction ( I) [ p ]. q p q if q is true under the assumption p, then p q is true without the assumption p. [... ] means that the assumption p is discared (no longer made). Implication Elimination ( E) p p q q if both p and p q hold (are true), then so does q. 22 / 41
30 Example  transitivity of implication (derived rule) p q q r We prove p r Informal Proof. 1. fix the assumption p q. 2. fix the assumption q r. 3. additionally assume p (and show r) 4. because p and p q, we have q. 5. because q and q r, we have r. 6. hence p r holds without assuming p Natural Deduction Proof. 1 p q 2 q r 3 p 4 q E, 1, 3 5 r E, 2, 4 6 p r I, 3 5 lines 1 and 2 are assumptions, can be used anywhere line 3 is an assumption we make, can be used only in scope (l 3 5). 23 / 41
31 Aside: Justification of Proof Steps Silly Proof. (we prove what we already know!) 1 p q 2 p 3 q E, 1, 2 4 p q I, 2 3 Line Number Notation. E,1,2 means that rule E proves line 3 from lines 1 and 2 I,23 means rule I proves line 4 from the fact that we could assume line 2 and (using that assumption) prove line 3. In I, 2 3 is the entire scope of the assumption p. 24 / 41
32 Rules involving assumptions 1 p q 2 q r 3 p 4 q E, 1, 3 5 r E, 2, 4 6 q r WRONG I, 4, 5 statements inside the scope of an assumption depends on that assumption. we only know that they are true if the assumption is true! we have assumed p and proved q r, but q r depends on p. Indentation and vertical lines indicate scoping Similar to programming: p is a local variable. 25 / 41
33 Useless assumptions You can assume anything, but it might not be useful. 1 p q 2 q E, 1 3 (p q) q I, p You are a giraffe 2 You are a giraffe E, 1 3 p You are a giraffe You are a giraffe I, / 41
34 Disjunction rules Or Introduction ( I) p p p q q p if p (holds), then so do p q and q p Or Elimination ( E) [p] [q].. p q r r r assuming that we have a proof of p q and for the case that p holds, we have a proof of r for the case that q holds we have a proof of r then we have a proof of r just from p q. 27 / 41
35 E template 1. know that p q 2. in case p is true... a. we know that r. b. and in case that q is true... c. we also know that r d. so we know r as long as p q! 1 p q 2 p.. a r b q.. c r r E, 1, 2 a, b c 28 / 41
36 Example: commutativity of disjunction (derived rule) p q q p Informal Proof. 1. fix the assumption p q. 2. first assume that p is true. 3. then also have that q p. 4. now assume that q is true. 5. then also have that q p. 6. hence q p, without assuming either p or q. Natural Deduction Proof. 1 p q 2 p 3 q p I, 2 4 q 5 q p I, 4 6 q p E, 1, 2 3, / 41
37 Negation and Truth Rules not introduction ( I) not elimination ( E) [p] p p. F F p if assuming p gives a contradiction, p is wrong so p must hold. Proof by Contradiction (PC) Truth [ p] F. p to prove p, assume p and derive a contradiction. truth, i.e. T, can always be established without assumptions. T 30 / 41
38 Example: double negation introduction (derived rule) p p Informal Proof. It is raining It is not the case that is is not raining Natural Deduction Proof. 1. Fix the assumption p. 2. additionally assume that p. 3. then F as p and p (under assn p) 4. hence p is contradictory, so p. 1 p 2 p 3 F E, 1, 2 4 p I, / 41
39 Example: contradiction elimination (derived rule) Anything follows form a contradiction F q 1 F 2 q 3 F R, 1 4 q PC, 2 3 R stands for repeat. F holds and continues to hold within the scope of the assumption q. assuming q a technical trick. 32 / 41
40 Example: double negation elimination (derived rule) p p 1 p 2 p 3 F E, 1, 2 4 p PC, / 41
41 Equivalence p q means p is true if and only if q is true We can make the definition p q (p q) (q p) which would naturally give us these rules introduction rule: p q q p elimination rules: p q p q p q p q q p 34 / 41
42 Equivalence Rules Alternatively we can get rules which don t involve the symbol I ( introduction) [p] [q] E ( elimination).. q p p q p q p p q q p Note the similarities to the I and E rules q 35 / 41
43 Which rule to use next? Guided by the form of your goal, and what you already have proved form ie, look at the connective:,,, always can consider using PC (proof by contradiction) to prove p q, I (or introduction) may not work p p q q p q p may not be necessarily true, q may not be necessarily true 36 / 41
44 To prove p q, sometimes you need to do this: 1. Using PC, assume (p q) (hoping to prove some contradiction) 2. When is (p q) true? When both p and q false! 3. From (p q) how to prove p? (next slide) 4. Having proved both p and q, prove some further contradiction Tutorial Exercise. p q p q 37 / 41
45 Notor elimination (derived rule) (p q) p 1 (p q) 2 p 3 p q I, 2 4 F E, 1, 3 5 p I, / 41
46 Proving a contrapositive rule In the same way, whenever you can prove any q then you can prove p p q 1 q 2 p 3 q your proof of q from p 4 F E, 1, 3 5 p I, / 41
47 Law of the excluded middle (derived) p p Everything must either be or not be. Russell 1 (p p) 2 p E (previous slide), 1 3 p E (previous slide), 1 4 F E, 2, 3 5 p p PC, / 41
48 Summary: Major Proof Techniques Three major styles of proof in logic and mathematics Model based computation: truth tables for propositional logic Algebraic proof: equational reasoning Deductive reasoning: rules of inference (e.g. Natural Deduction) Q. Why bother? Why not write a program that does truth tables? propositional logic is decidable: can write a program other logics are not: first order logic (next week) 41 / 41
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