Ling 130: Formal Semantics. Spring Natural Deduction with Propositional Logic. Introducing. Natural Deduction

Transcription

1 Ling 130: Formal Semantics Rules Spring 2018

2 Outline Rules Rules

3 What is ND and what s so natural about it? A system of logical proofs in which are freely introduced but discharged under some conditions. Introduced independently and simultaneously (1934) by Gerhard Gentzen and Stanis law Jaśkowski Rules The book & slides/handouts/hw represent two styles of one ND system: there are several. Introduced originally to capture the style of reasoning used by mathematicians in their proofs.

4 Ancient antecedents Aristotle s syllogistics can be interpreted in terms of inference rules and proofs from. Rules Stoic logic includes a practical application of a ND theorem.

5 ND rules and proofs Rules There are at least two rules for each connective: an introduction rule an elimination rule The rules reflect the meanings (e.g. as represented by truth-tables) of the connectives. Parts of each ND proof You should have four parts to each line of your ND proof: line number, the formula, justification for writing down that formula, the goal for that part of the proof. # formula justification rule Goal 1 A we proved this... Goal: something else

6 Conjunction introduction Rules &-introduction, or &I To introduce a conjunction formula, you have to have already introduced both conjuncts. 1. A we proved this... Goal:A&B 2.. (other junk) 3. B and this A&B &I (1, 3) because that s the only way the conjunction would be true, when both conjuncts are true.

7 Conjunction elimination Rules &-elimination, or &E If you managed to write down a conjunction, you can go on to write down one or both conjuncts by themselves. 1. A&B we proved this somehow Goal: A, B 2. B &E(1) 3. A &E(1) because if conjunction is true, you re in the first line of the truth-table, so each conjunct is true.

8 Disjunction introduction (Monkey s uncle!) Rules Truth of one disjunct is sufficient to make the whole disjunction true. -introduction, or I To introduce a disjunction formula, you have to have one of the disjuncts. B can be any formula at all! 1. A we proved this... Goal:A B 2. A B I (1)

9 Disjunction introduction (Monkey s uncle!) Rules Truth of one disjunct is sufficient to make the whole disjunction true. -introduction, or I To introduce a disjunction formula, you have to have one of the disjuncts. Or, if you want, A can be any formula at all! 1. B we proved this... Goal:A B 2. A B I (1)

10 Conditional elimination (Modus Ponens) Rules -elimination, E or Modus Ponens, MP If you have a conditional formula, and, separately, the formula matching the antecedent of the conditional, you can write down the formula matching the consequent of the conditional. 1. A B we proved this somehow... Goal:B 2.. (other junk) 3. A and this B E(1, 3) conditional=1, so not the 2 nd line of that truth-table. antecedent=1, so not lines 3, 4 of that truth-table. So you re in the 1 st line, so consequent=1.

11 Negation elimination, E Rules -elimination, or E, or Contradiction rule You re not actually eliminating the negation: when you have both a formula and its negation, it s a contradiction. 1. A we proved this... Goal: 2.. (other junk) 3. A and this E(1, 3) / Contradiction(1, 3) Negation truth-table: no line where A = 1 and A = 1 Why would you ever want to derive a contradiction? We ll see shortly!

12 Negation elimination, E Rules -elimination, or E, or Contradiction rule You re not actually eliminating the negation: when you have both a formula and its negation, it s a contradiction. I find it clearer to do this, but you can choose: 1. A we proved this... Goal: 2.. (other junk) 3. A and this A A I (1, 3) 5. E(4) / Contradiction(4)

13 Double negation rules -elimination, or E Double negative makes a positive. Rules 1. A we got this... Goal: A 2. A E -introduction, or I Positives makes a double negative. 1. A we got this... Goal: A 2. A I

14 Rules and proofs Rules Writing proofs You can assume anything you want, as long as you know how to get rid of that assumption later (discharge it). Starting the assumption, and until the assumption is discharged, lines/formulas are enclosed in a box. During the proof, you can use your assumption as well as anything already established. But you can t use that assumption or anything that depends on that assumption later. This means that once the box is closed, all the stuff inside the box becomes unusable.

15 Rules and proofs 1. stuff this Goal: something Rules 2. A as. Box goal: blah 3. proof hard struggle 4. blah we made it! 5. B assumption discharged Box goal often differs from goals outside the box. While writing lines (2-4), we can use stuff from line 1. From line 5 on, we cannot use formulas from lines (2-4)

16 Rules and proofs Rules Typesetting note: Boxes have to include at least the actual formulas; optionally also the explanations, goals, and/or line numbers. 1. stuff this Goal: something 2. A as. Box goal: blah 3. proof hard struggle 4. blah we made it! 5. B assumption discharged

17 Conditional introduction, I Rules -introduction ( I ), or Conditional/Hypothetical proof To prove a conditional A B, show that A entails B: suppose that A and show that B follows. Goal :A B 1. A as. Box goal: B 2. proof hard struggle 3. B we made it! 4. A B I (1, 3)

18 Negation introduction, I Rules Reductio ad absurdum (proof by contradiction), RAA, or -introduction ( I ) To prove that A is true, show that A entails a contradiction, so A must be false Goal: A 2. A as. Box goal: 3. proof struggle subgoal: q, q 4. yay, A means trouble! 5. A RAA(1, 3)

19 Negation introduction, I Rules To get the contradiction, we ll need E rule: find any formula q and its negation q, together they entail Some creativity is required to come up q that works Goal: A 2. A as. Box goal: 3. proof struggle subgoal: q, q 4. yay, A means trouble! 5. A RAA(1, 3)

20 Proof by cases Rules Proof by cases, PBC, or -elimination ( E) To prove something from A B, you need to prove it by assuming A, AND prove it by assuming B. I hate the name E, since you are not eliminating anything. You are just proving that the conclusion follows in either case. You know A B is true, but you don t know why it s true: could be because A is true, could be because of B, or both. Since A B means at least one of the disjuncts is true, showing that the conclusion follows in either case proves that it follows from A B

21 Proof by cases Rules We don t know that A is true, or that B is true (just that at least one of them is) So, A and B are, not usable later, so are put in (side-by-side) boxes. 1. A B we got this somehow Goal: C 2. A as. Box goal: C 5. B as. Box goal: C st proof nd proof 4. C got it once! 7. C got it again! 8. C PBC(1, 2 4, 5 7)