Predicate logic and validity

[aprilrocks92]

Hi,

I have the following task:

"Show that Pa V Pb -> Ex Px is valid" 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?

 

[HallsofIvy]

Hi,

I have the following task:

"Show that Pa V Pb -> Ex Px is valid" 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)

No, we don't. We know that either Pa or Pb. We do NOT know that "Ps" for arbitrary s.
You need, rather "if Pa then ..." and "if Pb then ..."

- 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?