Propositional natural deduction COMP2600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2016 Major proof techniques Three major styles of proof in logic and mathematics Model based computation: truth tables for propositional logic Algebraic proof: simplification rules e.g. De Morgan’s Deductive reasoning: rules of inference I Natural deduction is one example I Other examples: tableau calculi, resolution, Hilbert calculi 1 / 25 T TT T T T T T T TF T T T T T T FT T T F T T T FF F F F F T F TT T F F T T F TF T F F F T F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) 2 / 25 T TF T T T T T T FT T T F T T T FF F F F F T F TT T F F T T F TF T F F F T F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T 2 / 25 T FT T T F T T T FF F F F F T F TT T F F T T F TF T F F F T F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T 2 / 25 T FF F F F F T F TT T F F T T F TF T F F F T F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T T FT T T F T T 2 / 25 F TT T F F T T F TF T F F F T F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T T FT T T F T T T FF F F F F T 2 / 25 F TF T F F F T F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T T FT T T F T T T FF F F F F T F TT T F F T T 2 / 25 F FT T F F T T F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T T FT T T F T T T FF F F F F T F TT T F F T T F TF T F F F T 2 / 25 F FF F F F F T Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T T FT T T F T T T FF F F F F T F TT T F F T T F TF T F F F T F FT T F F T T 2 / 25 Example proof using truth tables Statement to be proved: (p ^ (q _ r)) ! ((p ^ q) _ r)) For all 8 (= 23) possibilities of p;q;r, calculate truth value of the statement p q r q _ r p ^ (q _ r) p ^ q (p ^ q) _ r (p ^ (q _ r)) ! ((p ^ q) _ r)) T TT T T T T T T TF T T T T T T FT T T F T T T FF F F F F T F TT T F F T T F TF T F F F T F FT T F F T T F FF F F F F T 2 / 25 The notion of a deductive proof A proof is a sequence of steps. Each step is either: I an axiom or an assumption; or I a statement which follows from previous steps via a valid rule of inference. Natural deduction: I for each connective, there is an introduction and an elimination rule I rules are formal, but resemble natural, i.e., human reasoning 3 / 25 Example of a natural deduction proof Statement to be proved: (p ^ (q _ r)) ! ((q ! s) _ p) 1 p ^ (q _ r) Assumption 2 p ^-E, 1 3 (q ! s) _ p _-I, 2 4 (p ^ (q _ r)) ! ((q ! s) _ p) !-I, 1–3 4 / 25 ^-E (and elimination) p ^ q p ^ q p q Conjunction rules ^-I (and introduction) p q p ^ q 5 / 25 Conjunction rules ^-I (and introduction) p q p ^ q ^-E (and elimination) p ^ q p ^ q p q 5 / 25 Example Commutativity of conjunction (derived rule) p ^ q q ^ p 1 p ^ q 2 p ^-E, 1 3 q ^-E, 1 4 q ^ p ^-I, 2, 3 6 / 25 !-E (implication elimination) p p ! q q Implication rules !-I (implication introduction) [ p ] . q p ! q This notation means that by assuming p, you can prove q. p is then discharged - no longer an assumption. 7 / 25 Implication rules !-I (implication introduction) [ p ] . q p ! q This notation means that by assuming p, you can prove q. p is then discharged - no longer an assumption. !-E (implication elimination) p p ! q q 7 / 25 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 the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 8 / 25 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 the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 1 p ! q 8 / 25 3 p 4 q !-E, 1, 3 5 r !-E, 2, 4 6 p ! r !-I, 3–5 Lines 1 and 2 are the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 1 p ! q 2 q ! r 8 / 25 4 q !-E, 1, 3 5 r !-E, 2, 4 6 p ! r !-I, 3–5 Lines 1 and 2 are the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 1 p ! q 2 q ! r 3 p 8 / 25 5 r !-E, 2, 4 6 p ! r !-I, 3–5 Lines 1 and 2 are the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 1 p ! q 2 q ! r 3 p 4 q !-E, 1, 3 8 / 25 6 p ! r !-I, 3–5 Lines 1 and 2 are the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 1 p ! q 2 q ! r 3 p 4 q !-E, 1, 3 5 r !-E, 2, 4 8 / 25 Lines 1 and 2 are the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 1 p ! q 2 q ! r 3 p 4 q !-E, 1, 3 5 r !-E, 2, 4 6 p ! r !-I, 3–5 8 / 25 Example - transitivity of implication (derived rule) p ! q q ! r We prove p ! r 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 the assumptions given, you may use these throughout the proof Line 3 is an assumption made within the proof, you may use it only within its scope (lines 3 to 5) 8 / 25 1 p ! q 2 p 3 q !-E, 1, 2 4 p ! q !-I, 2–3 This is a rather silly proof, we succeed in proving what we started with.