CS 512, Spring 2016, Handout 02 Natural Deduction, and Examples of Natural Deduction, in Propositional Logic

Transcription

1 CS 512, Spring 2016, Handout 02 Natural Deduction, and Examples of Natural Deduction, in Propositional Logic Assaf Kfoury January 19, 2017 Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 1 of 41

2 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 2 of 41

3 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 3 of 41

4 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 4 of 41

5 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 5 of 41

6 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 6 of 41

7 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: IF P AND (NOT Q) THEN R Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 7 of 41

8 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: ( P Q ) R Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 8 of 41

9 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: ( P Q ) R R Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 9 of 41

10 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: ( P Q ) R R P Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 10 of 41

11 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: ( P Q ) R R P THEREFORE Q Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 11 of 41

12 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: ( P Q ) R R P Q Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 12 of 41

13 from informal/common reasoning to formal reasoning: IF the train arrives late AND there are NO taxis THEN John is late for the meeting John is NOT late for the meeting the train did arrive late THEREFORE there were taxis again symbolically: ( P Q ) R R P Q more succintly: P Q R, R, P Q Formal Proof of the Sequent Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 13 of 41

14 a sequent (also called a judgment) is an expression of the form: 1,..., n ψ where: 1. 1,..., n, ψ are well-formed formulas (also called wff s) 2. the symbol is pronounced turnstile 3. the wff s 1,..., n to the left of are called the premises (also called antecedents or hypotheses) 4. the wff ψ to the right of is called the conclusion (also called succedent) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 14 of 41

15 a sequent is said to be valid (also deducible or derivable) if there is a formal proof for it a formal proof (also called deduction or derivation) is a sequence of wff s which starts with the premises of the sequent and finishes with the conclusion of the sequent: 1 2. n. ψ premise premise premise conclusion where every wff in the deduction is obtained from the wff s preceding it using a proof rule Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 15 of 41

16 Examples of Proof Rules ψ ψ i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 16 of 41

17 Examples of Proof Rules ψ ψ ψ i e 1 Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 17 of 41

18 Examples of Proof Rules ψ ψ ψ ψ ψ i e 1 e 2 Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 18 of 41

19 Examples of Proof Rules ψ ψ ψ ψ ψ i e 1 e 2 i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 19 of 41

20 Examples of Proof Rules ψ ψ i ψ e 1 ψ ψ e 2 i e (cannot be used in intuitionistic logic) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 20 of 41

21 Examples of Proof Rules ψ ψ e (or MP for Modus Ponens) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 21 of 41

22 Examples of Proof Rules ψ ψ e (or MP for Modus Ponens) ψ ψ MT ( for Modus Tollens) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 22 of 41

23 Examples of Proof Rules ψ ψ e (or MP for Modus Ponens) ψ ψ MT ( for Modus Tollens). ψ ψ i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 23 of 41

24 Examples of Proof Rules ψ ψ e (or MP for Modus Ponens) ψ ψ MT ( for Modus Tollens). ψ ψ i open a box when you introduce an assumption (wff in rule i) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 24 of 41

25 Examples of Proof Rules ψ ψ e (or MP for Modus Ponens) ψ ψ MT ( for Modus Tollens). ψ ψ i open a box when you introduce an assumption (wff in rule i) close the box when you discharge the assumption Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 25 of 41

26 Examples of Proof Rules ψ ψ e (or MP for Modus Ponens) ψ ψ MT ( for Modus Tollens). ψ ψ i open a box when you introduce an assumption (wff in rule i) close the box when you discharge the assumption you must close every box and discharge every assumption in order to complete a formal proof Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 26 of 41

27 Proof Rules Associated with Only One and with So far, we have an elimination rule and an introduction rule for double negation, namely e and i, but not for single negation. We now compensate for this lack: e ( or LNC for Law of Non-Contradiction) where (a single symbol) stands for contradiction Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 27 of 41

28 Proof Rules Associated with Only One and with So far, we have an elimination rule and an introduction rule for double negation, namely e and i, but not for single negation. We now compensate for this lack: e ( or LNC for Law of Non-Contradiction) where (a single symbol) stands for contradiction. i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 28 of 41

29 Proof Rules Associated with Only One and with So far, we have an elimination rule and an introduction rule for double negation, namely e and i, but not for single negation. We now compensate for this lack: e ( or LNC for Law of Non-Contradiction) where (a single symbol) stands for contradiction. i e ( if you can prove, you can prove every WFF ) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 29 of 41

30 Two Derived Proof Rules The two following rules are derived rules the first from rules i, i, e, and e (see [LCS, pp 24-25]); the second from rules i, i, e, and e (see [LCS, pp 25-26]):. PBC (for Proof by Contradiction) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 30 of 41

31 Two Derived Proof Rules The two following rules are derived rules the first from rules i, i, e, and e (see [LCS, pp 24-25]); the second from rules i, i, e, and e (see [LCS, pp 25-26]):. PBC (for Proof by Contradiction) LEM (for Law of Excluded Middle) Because e is rejected in intuitionistic logic, so are PBC and LEM Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 31 of 41

32 Two Derived Proof Rules The two following rules are derived rules the first from rules i, i, e, and e (see [LCS, pp 24-25]); the second from rules i, i, e, and e (see [LCS, pp 25-26]):. PBC (for Proof by Contradiction) LEM (for Law of Excluded Middle) Because e is rejected in intuitionistic logic, so are PBC and LEM (a summary of all proof rules and some derived rules in [LCS, p. 27]) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 32 of 41

33 Examples of Natural Deductions formal proof of the sequent P Q (P Q) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 33 of 41

34 Examples of Natural Deductions formal proof of the sequent P Q (P Q) 1 P 2 Q 3 P Q i 1, 2 4 Q (P Q) i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 34 of 41

35 Examples of Natural Deductions formal proof of the sequent P (Q R) P Q R Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 35 of 41

36 Examples of Natural Deductions formal proof of the sequent P (Q R) P Q R 1 P (Q R) 2 P Q 3 P e Q R e 1, 3 5 Q e R e 4, 5 7 P Q R i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 36 of 41

37 Examples of Natural Deductions formal proof of the sequent P Q R P (Q R) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 37 of 41

38 Examples of Natural Deductions formal proof of the sequent P Q R P (Q R) 1 P Q R 2 P 3 Q 4 P Q i 2, 3 5 R e 1, 4 6 Q R i 7 P (Q R) i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 38 of 41

39 Examples of Natural Deductions formal proof of the sequent P (Q R) (P Q) (P R) Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 39 of 41

40 Examples of Natural Deductions formal proof of the sequent P (Q R) (P Q) (P R) 1 P (Q R) 2 P Q 3 P 4 Q e 2, 3 5 Q R e 1, 3 6 R e 5, 4 7 P R i 8 (P Q) (P R) i Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 40 of 41

41 Formal Proof of the Initial Sequent: Initial Sequent 1 P Q R premise 2 R premise 3 P premise 4 Q assume 5 P Q i 3, 4 6 R e 1, 5 7 e 6, 2 8 Q i 9 Q e 8 Assaf Kfoury, CS 512, Spring 2017, Handout 02 page 41 of 41