Determining Semantic Properties with a Model Universe

Solution 5

Test the following arguments for validity:

(∃x)(Rx & ~Tx), Ta, /// (x)Rx

Since there are two distinct propositional functions, Rx and Tx, our maximum domain will be a domain of 4. 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}

(Ra & ~Ta) v (Rb & ~Tb) v (Rc & ~Tc)
Ta                       
(Ra & Rb) &Rc

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

Ra is T
Rb is T
Ta is T
Tb is F
Rc is F
Tc is F

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