Determining Semantic Properties with a Model Universe

Solution 1

Test the following arguments for validity:

1. (∃x)(Sx ⊃ Px), Sa /// (∃x)Px

Since there are two distinct propositional functions, Sx and Px, our maximum domain will be a domain of 4. In choosing a domain size to start with, we note that the premises contain a singular proposition and an existential one.  Since the first premise may refer to something other than the "a" in the second premise, we'll start with a domain of 2. (It should be clear that the argument is valid in a domain of one. If that's not clear, check it out.

domain: {a,b}

(Sa ⊃ Pa) v (Sb ⊃ Pb)
Sa                       
Pa v Pb

So is there a row of the truth table for this instantiation where the premises are true and the conclusion false?

Sa is T
Sb is F
Pa is F
Pb is F

So the argument is invalid in a domain of 2, and so invalid.