I have the following task:
"Show that Pa V Pb -> Ex Px" where E stands for the existential quantifier.
I have done the following:
- Let M denote a model with domain D, and assume that M |= Pa V Pb
- It suffices then to show that M |= Ex Px
- Let s be an element in D, arbitrarily chosen
- By assumption, we know that M |= Ps (since we consider disjunction, it suffices to only include one, from Pa V Pb)
- Thus, it is the case that M |= Ex Ps
- Since s was arbitrarily chosen, it will be so that M |= Ex Px, that Ex Px is true in M.
Is this the right way to prove validity?