I'm doing predicate logic models and can't seem to get the right answers to any of these problems. here is the question:

 

" test the following formulas in two-element domains, using any technique you wish. Which are true in each model? which are false in some? For those that have counter-examples, present a counterexample as a table of values for F and G."

 

help with any of the following would be amazing:

 

2. (Ax)(Fx>(Gx v ~Gx))

3. (Ex) Fx v (Ex) ~Fx

 

I tried to prove that they were tautologies by making a tree for their negand, but all of the branches always remained open since in one branch there never were two formulaes that had the same variables. So I'm not sure what to do. I have an exam monday, hopefully someone can help me out!

The domain contains only two elements. This makes the problem extremely simple. If you let the domain be [latex]\{a,b\}[/latex] you can test any statement, say [latex]P(x)[/latex], by considering the statements [latex]P(a)[/latex] and [latex]P(b)[/latex]. If both of them are true, [latex](\forall x)\left(P(x)\right)[/latex] is true; if at least one of them is true, [latex](\exists x)\left(P(x)\right)[/latex] is true; and so on.

Edited April 8, 201312 yr by Nehushtan

You can rewrite them in simple sentential logic, then use a assignment table to quite trivially see that they are both tautologies. In fact, the second one is of the form of an axiom of classical logics.