I've been stumped on this one question in particular for several days now and I'm hoping to get some help on where I'm going wrong.
Given the following premises:
- ∀x∀y(StrongPref(x,y)→ ¬StrongPref(y,x))
- ∀x∀y∀z((StrongPref(x,y)∧StrongPref(y,z))→StrongPref(x,z))
- ∀xIndiff(x,x)
- ∀x∀y(Indiff(x,y)→Indiff(y,x))
- ∀x∀y∀z((StrongPref(x,y)∧Indiff(y,z))→StrongPref(x,z))
- ∀x∀y(StrongPref(x,y)∨Indiff(x,y)∨StrongPref(y,x))
- ∀x∀y(WeakPref(x,y)↔(StrongPref(x,y)∨Indiff(x,y)))
Prove: (Not required to use all the premises)
∀x∀y∀z((Indiff(x,y) ∧ Indiff(y,z)) → Indiff(x,z))
Since we are trying to prove a conditional, my initial inclination is to assume Indiff(x,y) ∧ Indiff(y,z)
and try to prove Indiff(x,z)
. Since none of the premises explicitly tells us much information about deriving Indiff
, I figured it would be best to prove by contradition - that is, assuming ~Indiff(x,z)
and deriving a contradiction.
I begin the proof by initializing the quantifiers, using c,d,e in place of x,y,z. Since only premises (5) and (6) give us some information about Indiff
to work off of, I am thinking to initially use (6) and go through each case/combination of StrongPref(c,d) ∨ Indiff(c,d) ∨ StrongPref(d,c)
and StrongPref(d,e) ∨ Indiff(d,e) ∨ StrongPref(e,d)
in order to get c
and e
together (which seems unreasonable since that would mean doing 9 proofs but I'm getting desperate). Then, to get the contradictions, I used (5) to derive something of the form StrongPref(x,x) which we can get a contradiction from (1).
I am able to get contradictions for many of the cases, but I begin getting stuck when it comes to getting a contradiction in the case where we have both Indiff(c,d) and Indiff(d,e).
I have shown my work below (took out some of the repeated work for working contradictions to save space). I have a feeling I'm completely off since it seems unreasonable of have to do 9 subproofs, but I can't see another perspective to take. Any help would be much appreciated, thank you!
Indiff
, whereas the other one is proving something withStrongPref