Philosophy
Question:
Completely stuck with final assignment for PHIL course. Here’s the premises and conclusions: Here are Arrow’s two assumptions or premises about weak preferences written in a more familiar language for symbolic logic than he uses. The predicates have an obvious interpretation. P1: ∀x∀y(WeakPref(x,y)∨WeakPref(y,x)) P2: ∀x∀y∀z((WeakPref(x,y)∧WeakPref(y,z))→WeakPref(x,z)) The two axioms only state rules about weak preferences work. So Arrow also provides two definitions relating strong preference and indifference to weak preference. We’ll label these as premises, just as we did with the assumptions or axioms. P3: ∀x∀y(StrongPref(x,y)↔ ¬WeakPref(y,x)) P4: ∀x∀y(Indiff(x,y)↔(WeakPref(y,x)∧WeakPref(x,y))) Arrow says that from these axioms and definitions you can prove a bunch of results or conclusions — what he calls lemmas — about preferences. In fact, there are more than this. Using the same language for symbolic logic, here are the conclusions you can get: C1: ∀xWeakPref(x,x) C2: ∀xIndiff(x,x) C3: ∀x∀y(Indiff(x,y)↔Indiff(y,x)) C4: ∀x∀y∀z((Indiff(x,y)∧Indiff(y,z))→Indiff(x,z)) C5: ∀x∀y(StrongPref(x,y)→WeakPref(x,y)) C6: ∀x∀y(StrongPref(x,y)→ ¬StrongPref(y,x)) C7: ∀x∀y∀z((StrongPref(x,y)∧StrongPref(y,z))→StrongPref(x,z)) C8: ∀x∀y(Indiff(x,y)→¬(StrongPref(y,x)∨StrongPref(x,y))) C9: ∀x∀y∀z((Indiff(x,y)∧StrongPref(y,z))→StrongPref(x,z)) C10: ∀x∀y∀z((Indiff(x,y)∧StrongPref(z,x))→StrongPref(z,y)) C11: ∀x∀y∀z((StrongPref(x,y)∧WeakPref(y,z))→StrongPref(x,z)) C12: ∀x∀y(WeakPref(x,y)∨StrongPref(y,x)) C13: ∀x∀y(StrongPref(x,y)→ ¬Indiff(x,y)) C14: ∀x∀y((StrongPref(x,y)∨Indiff(x,y))→WekPref(x,y)) C15: ∀x∀y(WeakPref(x,y)→(StrongPref(x,y)∨Indiff(x,y))) You should show using derivations that you can prove some of these lemmas from some or all of these axioms and definitions. This is exercise 9.5.1 in webCT Vista. There are other ways to set up the rules for how preferences work. These other ways have exactly the same results as Arrow’s way of doing this. Here is one of these ways. The axioms or assumptions are — instead of P1 to P4 –: P5: ∀x∀y(StrongPref(x,y)→ ¬StrongPref(y,x)) P6: ∀x∀y∀z((StrongPref(x,y)∧StrongPref(y,z))→StrongPref(x,z)) P7: ∀xIndiff(x,x) P8: ∀x∀y(Indiff(x,y)→Indiff(y,x)) P9: ∀x∀y∀z((StrongPref(x,y)∧Indiff(y,z))→StrongPref(x,z)) P10: ∀x∀y(StrongPref(x,y)∨Indiff(x,y)∨StrongPref(y,x)) And then a definition relates these rules to weak preference: P11: ∀x∀y(WeakPref(x,y)↔(StrongPref(x,y)∨Indiff(x,y))) And here are the questions: 1. make a proof or derivation of C13 from P5 through P11 2. make a proof or derivation of C6 from P5 through P11 3. make a proof or derivation of C11 from P5 through P11 4. make a proof or derivation of C6 from P1 through P4 Any help is appreciated.
