# HW3 - induction on derivations. In doing so, formulate the...

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Homework 3 PHIL 151 Due Wed. Jan 26th in class at 10 am 1. (10 points) Do exercises 1 (d), 2c and 4 on page 39. In problem 1d), the expression abbreviates the conjunction of two implications. So to prove it you need to prove each direction separately. 2. (6 points) Exercise 8 on page 40 asks for a proof by induction on ψ . Work out the inductive case for ψ = ψ 1 ψ 2 . As with 1(d), abbre- viates a conjunction of two implications. 3. (a) (4 points) Deﬁne a length function l ( D ) that returns the length of the longest branch in D . (b) (2 points) As with proof by induction on a proposition’s rank, proof by induction on a derivation’s length proceeds by complete induction. Formulate this induction prinicple. That is, formu- late the principle by which complete induction on the length of a derivation proceeds. (Hint: the analogous principle for induction on the rank of propositions is given as theorem 1.1.8 on page 13.) (c) (6 points) Prove that the principle given in 3b) implies normal

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Unformatted text preview: induction on derivations. In doing so, formulate the principle of normal induction on derivations as a conditional statement. Then prove this conditional with what you have formulated in 3b). (Hint: an analogous proof is given—as a subproof—in the proof of theorem 1.1.8. Also, the conditional statement that expresses the principle of normal induction on derivations is a complicated one. See theorem 1.1.3 for the analogous statement for induction on propositions.) 4. (6 points) Do exercise 1 on page 47. 5. (8 points) Deﬁne a consistent set Γ such that (a) Γ has only one maximally consistent set containing it. (b) There is a natural number k such that all propositions in Γ have rank less than or equal to k . 1 Your proof that the set you deﬁne satisﬁes (a) and (b) can use any of the facts proven about maximally consistent sets in section 1.5. 6. (6 points) Do exercise 6 on page 48. 2...
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HW3 - induction on derivations. In doing so, formulate the...

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