Warm-Up Problem. Write a Resolution Proof for. Res 1/32

Transcription

1 Warm-Up Problem Write a Resolution Proof for Res 1/32

2 A second Rule Sometimes throughout we need to also make simplifications: You can do this in line without explicitly mentioning it (just pretend you simplified after applying the resolution rule - this actually makes more sense using the set notation 2/32

3 Propositional Logic: Natural Deduction Derived Rules and Examples Carmen Bruni Lecture 8 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler, and P Van Beek 3/32

4 Learning goals Natural Deduction in Propositional logic Describe rules of inference for natural deduction Prove a conclusion from given premises using natural deduction inference rules Describe strategies for applying each inference rule when proving a conclusion formula using natural deduction Derived Rules in Natural Deduction: Describe the derived rules of inference for natural deduction Prove a conclusion from given premises using natural deduction inference rules Describe strategies for applying each inference rule when proving a conclusion formula using natural deduction 4/32

5 The Natural Deduction Proof System We will now consider a proof system called Natural Deduction It closely follows how people (mathematicians, at least) normally make formal arguments It extends easily to more-powerful forms of logic 5/32

6 A proof is syntactic First, we think about proofs in a purely syntactic way A proof starts with a set of premises, transforms the premises based on a set of inference rules (which form a proof system), and reaches a conclusion We write ND or simply if context is clear if we can find such a proof in our proof system of Natural Deduction that starts with a set of premises and ends with the conclusion 6/32

7 Revisit Soundness and Completeness We revisit Soundness and Completeness just like we did for Resolution: (Soundness) 1 Does ND imply? 2 Does every proof establish a semantic entailment? 3 If a proof of to exists, is it true that semantically entails? (Completeness) 1 Does imply ND? 2 For every semantic entailment, can I find a proof for it? 3 If semantically entails, does a proof of to exist? 7/32

8 Reflexivity / Premise If you want to write down a previous formula in the proof again, you can do it by reflexivity Name -notation inference notation Reflexivity, or Premise ND The notation on the right: Given the formulas above the line, we can infer the formula below the line The version in the centre reminds us of the role of assumptions in Natural Deduction Other rules will make more use of it Basic Rules 8/32

9 An example using reflexivity Here is a proof of ND 1 Premise 2 Premise 3 Reflexivity: 1 Alternatively, we could simply write 1 Premise and be done Basic Rules 9/32

10 For each symbol, the rules come in pairs An introduction rule adds the symbol to the formula An elimination rule removes the symbol from the formula Basic Rules 10/32

11 Rules for Conjunction Name -notation inference notation -introduction ( i) If ND and ND, then ND Name -notation inference notation -elimination ( e) If ND, then ND and ND Conjunction Rules 11/32

12 Example: Conjunction Rules Example Show that ND Conjunction Rules 12/32

13 Example: Conjunction Rules Example Show that ND 1 Premise 2 e: 1 3 e: 1 4 i: 2, 3 Conjunction Rules 12/32

14 Example: Conjunction Rules (2) Example Show that, ND Conjunction Rules 13/32

15 Example: Conjunction Rules (2) Example Show that, ND 1 Premise 2 Premise 3 e: 1 4 i: 3, 2 Conjunction Rules 13/32

16 Rules for Implication: e Name -notation inference notation -elimination ( e) (modus ponens) If ND and ND, then ND In words: If you assume is true and implies, then you may conclude Implication Rules 14/32

17 Rules for Implication: i Name -notation inference notation -introduction ( i) If ND, then ND The box denotes a sub-proof In the sub-proof, we starts by assuming that is true (a premise of the sub-proof), and we conclude that is true Nothing inside the sub-proof may come out Outside of the sub-proof, we could only use the sub-proof as a whole Implication Rules 15/32

18 Example: Rule i and sub-proofs Example Give a proof of ND To start, we write down the premises at the beginning, and the conclusion at the end 1 Premise 2 Premise??? What next? Implication Rules 16/32

19 Example: Rule i and sub-proofs Example Give a proof of ND To start, we write down the premises at the beginning, and the conclusion at the end 1 Premise 2 Premise 3 Assumption i:?? What next? The goal contains Let s try rule i Implication Rules 16/32

20 Example: Rule i and sub-proofs Example Give a proof of ND To start, we write down the premises at the beginning, and the conclusion at the end 1 Premise 2 Premise 3 Assumption 4 e: 1, 3 5 e: 2, 4 6 i:?? What next? The goal contains Let s try rule i Inside the sub-proof, we can use rule e Implication Rules 16/32

21 Example: Rule i and sub-proofs Example Give a proof of ND To start, we write down the premises at the beginning, and the conclusion at the end 1 Premise 2 Premise 3 Assumption 4 e: 1, 3 5 e: 2, 4 6 i: 3 5 What next? The goal contains Let s try rule i Inside the sub-proof, we can use rule e Done! Implication Rules 16/32

22 Rules of Disjunction: i and e Name -notation inference notation -introduction ( i) -elimination ( e) If ND, then ND and ND If, ND and, ND, then, ND e is also known as proof by cases Disjunction Rules 17/32

23 Example: Or-Introduction and -Elimination Example: Show that ND Disjunction Rules 18/32

24 Example: Or-Introduction and -Elimination Example: Show that ND 1 Premise 2 Assumption 3 Assumption 4 Reflexivity: 2 5 i: i: 5 7 Assumption 8 Assumption 9 Reflexivity: 7 10 i: i: e: 1, 2 6, 7 11 Disjunction Rules 18/32

25 Negation We shall treat negation by considering contradictions We shall use the notation to represent any contradiction It may appear in proofs as if it were a formula Do not confuse this with the from resolution! Negation 19/32

26 Negation Rules The elimination rule for double negations: Name -notation inference notation -elimination ( e) If ND, then ND If an assumption leads to a contradiction, then derive Name -notation inference notation -introduction ( i) If ND, then ND Negation 20/32

27 The Last Two Basic Rules for Contradiction Introduction (also known as -elimination ( e)): Name -notation inference notation -introduction,, ND If we have both and, then we have a contradiction Contradiction Elimination: Name -notation inference notation -elimination ( e) If ND, then ND Negation 21/32

28 Example: Negation Example Show that ND Negation 22/32

29 Example: Negation Example Show that ND 1 Premise?? Negation 22/32

30 Example: Negation Example Show that ND 1 Premise 2 Assumption 3 4?? 5 i: 2? Negation 22/32

31 Example: Negation Example Show that ND 1 Premise 2 Assumption 3 e: 1, 2 4?? 5 i: 2? Negation 22/32

32 Example: Negation Example Show that ND 1 Premise 2 Assumption 3 e: 1, 2 4 i: 2, 3 5 i: 2 4 Negation 22/32

33 A Redundant Rule The rule of -elimination is not actually needed Suppose a proof has 27 some rule 28 e: 27 We can replace these by 27 some rule 28 Assumption 29 Reflexivity: i: e: 30 Thus any proof that uses e can be modified into a proof that does not Negation 23/32

34 Example: Modus tollens The principle of modus tollens: ND Negation 24/32

35 Example: Modus tollens The principle of modus tollens: ND 1 Premise 2 Premise?? Negation 24/32

36 Example: Modus tollens The principle of modus tollens: ND 1 Premise 2 Premise 3 Assumption i:?? Negation 24/32

37 Example: Modus tollens The principle of modus tollens: ND 1 Premise 2 Premise 3 Assumption 4 e: 3, i:?? Negation 24/32

38 Example: Modus tollens The principle of modus tollens: ND 1 Premise 2 Premise 3 Assumption 4 e: 3, i: 3 5 What should go on line 5? Negation 24/32

39 Example: Modus tollens Modus tollens is sometimes taken as a derived rule : MT Negation 25/32

40 Derived Rules Whenever we have a proof of the form ND, we can consider it as a derived rule: If we use this in a proof, it can be replaced by the original proof of ND The result is a proof using only the basic rules Using derived rules does not expand the things that can be proved But they can make it easier to find a proof Negation 26/32

41 More Derived Rules Double-Negation Introduction: i Negation 27/32

42 More Derived Rules Double-Negation Introduction: i 1 Premise?? Negation 27/32

43 More Derived Rules Double-Negation Introduction: i 1 Premise 2 Assumption 3 4 i:?? Negation 27/32

44 More Derived Rules Double-Negation Introduction: i 1 Premise 2 Assumption 3 i: 1, 2 4 i:?? Negation 27/32

45 More Derived Rules Double-Negation Introduction: i 1 Premise 2 Assumption 3 i: 1, 2 4 i: 2 3 Negation 27/32

46 More Derived Rules Proof by contradiction (reductio ad absurdum): PBC Negation 28/32

47 More Derived Rules Proof by contradiction (reductio ad absurdum): PBC 1 Premise 5?? Negation 28/32

48 More Derived Rules Proof by contradiction (reductio ad absurdum): PBC 1 Premise?? 5 e:?? Negation 28/32

49 More Derived Rules Proof by contradiction (reductio ad absurdum): PBC 1 Premise 2 Assumption 3 4 i:?? 5 e: 4 Negation 28/32

50 More Derived Rules Proof by contradiction (reductio ad absurdum): PBC 1 Premise 2 Assumption 3 e: 1, 2 4 i:?? 5 e: 4 Negation 28/32

51 More Derived Rules Proof by contradiction (reductio ad absurdum): PBC 1 Premise 2 Assumption 3 e: 1, 2 4 i: e: 4 Negation 28/32

52 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM Negation 29/32

53 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 9?? Negation 29/32

54 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1?? 9 e:?? Negation 29/32

55 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption?? 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

56 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3?? 4?? 5 i:?? 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

57 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3?? 4?? 5 i:?? 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

58 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4?? 5 i:?? 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

59 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4?? 5 i:?? 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

60 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i:?? 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

61 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: 2-4 6???? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

62 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: 2-4 6?? 7 i:?? 8 i:?? 9 e:?? Negation 29/32

63 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: i: 5 7 i:?? 8 i:?? 9 e:?? Negation 29/32

64 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: i: 5 7 i: 1,6 8 i:?? 9 e:?? Negation 29/32

65 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: i: 5 7 i: 1,6 8 i: e:?? Negation 29/32

66 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: i: 5 7 i: 1,6 8 i: e: 8 Negation 29/32

67 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: i: 5 7 i: 1,6 8 i: e: 8 Negation 29/32

68 More Derived Rules The Law of Excluded Middle (tertiam non datur): LEM 1 Assumption 2 Assumption 3 i: 2 4 i: 1,3 5 i: i: 5 7 i: 1,6 8 i: e: 8 Negation 29/32

69 Strategies for natural deduction proofs 1 Work forward from the premises Can you apply an elimination rule? 2 Work backwards from the conclusion What introduction rule do you need to use at the end? 3 Stare at the formula Notice its structure Use it to guide your proof 4 If a direct proof doesn t work, try a proof by contradiction Additional Examples and Techniques 30/32

70 Further Examples of Natural Deduction Example Show that ND Write down premises and conclusion (step 1) No elimination applies (step 2) Thus try i (step 3) 1 Premise?? Additional Examples and Techniques 31/32

71 Further Examples of Natural Deduction Example Show that ND In the sub-proof, try -elimination on the assumption (step 2) 1 Premise 2 Assumption?? 9?? Additional Examples and Techniques 31/32

72 Further Examples of Natural Deduction Example Show that ND To justify lines 4 and 7: No elimination applies from the assumptions (step 2) What about -introduction for the conclusion (step 3)? 1 Premise 2 Assumption 3 Assumption 4?? 5 Assumption 6 7?? 8 e:?? 9 i: 2 8 Additional Examples and Techniques 31/32

73 Further Examples of Natural Deduction Example Show that ND It works! 1 Premise 2 Assumption 3 Assumption 4 i: 3 5 Assumption 6 e: 5, 1 7 i: 6 8 e: 2, 3 4, i: 2 8 Additional Examples and Techniques 31/32

74 Handout Try some of the problems on the handout! You may work in small groups if you wish or on your own Additional Examples and Techniques 32/32