So I have to prove that this set S is inconsistent:
S = {{P(x),P(f(a)), ¬Q(z)}, {P(g(x’,x)),Q(x)},{¬P(y)}}
I just have no idea where to start. The only time I learned about inconsistency is when I learned about Skolem form, but I got no idea how to prove it. Any help will be much appreciated! Thanks!
A set of statements is inconsistent if and only if it has no model, i.e. it is impossible for all statements to be true at the same time.
This is equivalent to saying that the conjunction of all those sentences is a contradiction.
It is also equivalent to saying that the contradiction is a logical consequence of that set of sentences.
All of which means:
To show a set is inconsistent using resolution: simply start with the sentences, put into clauses, and derive the empty clause!
Or: if the set is already a bunch of clauses: simply derive the empty clause! (and that on its turn shows that the original formula $F$ that the clauses came from is a contradiction (or, as a singleton set of sentences $\{ F \}$, is inconsistent ... which I assume is what they mean by $F$ being inconsistent)