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5. Soundness 41 i j A ⊥ n ¬A ¬I i–j Let v be any valuation that makes all of ∆ n true. Note that all of ∆ n are among ∆ i , with the possible exception of A itself. By hypothesis, line j is shiny. But no valuation can make ‘⊥’ true, so no valuation can make all of ∆ j true. Since the sentences ∆ i are just the sentences ∆ j , no valuation can make all of ∆ i true. Since v makes all of ∆ n true, it must therefore make A false, and so make ¬A true. So ∆ n ⊨ ¬A. ■ Lemma 5.11. TND is rule-sound. Proof. Assume that every line before line n on some TFL-proof is shiny, and that TND is used on line n. So the situation is: i j A B k ¬A l B n B TND i–j, k–l Let v be any valuation that makes all of ∆ n true. Either v makes A true, or it makes A false. We shall reason through these (sub)cases separately. Case 1: v makes A true. The sentences ∆ i are just the sentences ∆ j . By hypothesis, line j is shiny. So ∆ i ⊨ B. Furthermore, all of ∆ i are among ∆ n , with the possible exception of A. So v makes all of ∆ i true, and hence makes B true. Case 2: v makes A false. Then v makes ¬A true. Reasoning in exactly the same way as above, considering lines k and l, v makes B true. Either way, v makes B true. So ∆ n ⊨ B. ■ Lemma 5.12. →I, →E, ↔I, and ↔E are all rule-sound. Proof. I leave these as exercises. ■ This establishes that all the basic rules of our proof system are rule-sound. Finally, we show: Lemma 5.13. All of the derived rules of our proof system are rule-sound. Proof. Suppose that we used a derived rule to obtain some sentence, A, on line n of some TFL-proof, and that every earlier line is shiny. Every use of a derived rule can be replaced (at the cost of long-windedness) with multiple uses of basic rules. (This was proved in fx C§30.) That is to say, we could have used basic rules to write A on some line n + k, without introducing any further
5. Soundness 42 assumptions. So, applying Lemmas 5.3–5.12 several times (k+1 times, in fact), we can see that line n + k is shiny. Hence the derived rule is rule-sound. ■ And that’s that! We have shown that every rule – basic or otherwise – is rulesound, which is all that we required to establish Lemma 5.2, and hence the Soundness Theorem (see §5.1). But it might help to round off this chapter if I repeat my informal explanation of what we have done. A formal proof is just a sequence – of arbitrary length – of applications of rules. We have spent this section showing that any application of any rule will not lead you astray. It follows (by induction) that no formal proof will lead you astray. That is: our proof system is sound. Practice exercises A. Complete the Lemmas left as exercises in this chapter. That is, show that the following are rule-sound: • ∨I. (Hint: this is similar to the case of ∧E.) • ⊥E. (Hint: this is similar to the case of ⊥I.) • →I. (Hint: this is similar to ∨E.) • →E. • ↔I. • ↔E.
Page 1 and 2: Metatheory for truth-functional log
Page 3 and 4: Contents 1 Induction 3 1.1 Stating
Page 5 and 6: Contents 2 The corresponding notion
Page 7 and 8: 1. Induction 4 knocking the first o
Page 9 and 10: 1. Induction 6 So the induction pro
Page 11 and 12: 1. Induction 8 sentences. But thing
Page 13 and 14: 1. Induction 10 Case 1: A is atomic
Page 15 and 16: Substitution 2 In this chapter, we
Page 17 and 18: 2. Substitution 14 2.2 Interpolatio
Page 19 and 20: 2. Substitution 16 Step 3: Repeat s
Page 21 and 22: 2. Substitution 18 We shall use the
Page 23 and 24: Normal forms 3 In this chapter, I p
Page 25 and 26: 3. Normal forms 22 Case 2: S is tru
Page 27 and 28: 3. Normal forms 24 until no (bi)con
Page 29 and 30: 3. Normal forms 26 4a: B ∗ contai
Page 31 and 32: 3. Normal forms 28 (cnf2) Every occ
Page 33 and 34: Expressive adequacy 4 In chapter 3,
Page 35 and 36: 4. Expressive adequacy 32 4.2 Indiv
Page 37 and 38: 4. Expressive adequacy 34 In fact,
Page 39 and 40: 4. Expressive adequacy 36 negations
Page 41 and 42: 5. Soundness 38 Again, it is easy t
Page 43: 5. Soundness 40 m i j k l A ∨ B A
Page 47 and 48: 6. Completeness 44 6.2 An algorithm
Page 49 and 50: 6. Completeness 46 been expanded. N
Page 51 and 52: 6. Completeness 48 We have managed
Page 53 and 54: 6. Completeness 50 1 (A ∨ (¬B
Page 55 and 56: 6. Completeness 52 Step 1. Start a
Page 57 and 58: 6. Completeness 54 Now let D be any
Page 59 and 60: 6. Completeness 56 Now, when we are
Page 61 and 62: 6. Completeness 58 Finally: by Lemm
Page 63 and 64: 6. Completeness 60 Practice exercis

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