# Proof of the Model Universe Theorem: Proving the invalidity arguments in quantifier logic

I am studying how to prove an argument in quantifier logic is invalid. The textbook I am using by Virginia Klenk claims that you can use a Model Universe that contains a finite number of objects to substitute in to the propositional functions. The book claims: "Where n is the number of different predicate letters in an argument form, the largest domain you need to test is one with \$2^n\$ individuals." Can someone explain or refer me to a proof for why this is so?

• I am guessing that all the predicates involved are monadic (one place), otherwise this is false, as arithmetic shows. If there are n monadic predicates and each can be assigned T/F there are 2^n possible assignments, and the truth value of the formula depends only on that. With 2^n individuals in the domain every possible assignment can be made to be satisfied by an individual. Jan 7, 2022 at 3:44
• @Conifold, what example from arithmetic would show that this is not case for anything other than monadic predicates?
– Name
Jan 10, 2022 at 0:20
• "Every element has a successor" with a dyadic successor predicate from arithmetic is unsatisfiable on any finite domain. Jan 10, 2022 at 1:12