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))
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
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!