Atomic Formulas in Second Order Logic

Question

I'm studying second-order logic and I would like to know if the phrase about atomic formulas in Figure 1 is correct. If addition, I would like to know what means a second-order predicate like $P^n_k$ or $R_nk$ like in the Figure 1.

Thanks!!

Answer 1

Yes, it is correct. In SOL we have second-order predicates, i.e. predicates having pradicates as arguments.

They express properties of properties:

redness is a property that is instantiated by apples;

being a color is a property that applies to redness.

In FOL we have individual variables, that we may quantify, and predicate symbols, that we interpret.

Thus, we may have the formula : $\forall x P(x)$ that reads "every object is $P$", whose meaning (truth value) will depend on the interpretation (how the predicate symbol $P$ is interpreted).

In FOL, in addition to the above, we have predicate variables, that we may quantify, and second-order predicate symbols, that we interpret.

Thus, we may have the formula : $\forall X \mathcal R(X)$ that reds "every (first-order) property is $\mathcal R$", whose meaning (truth value) will depend on the interpretation (how the second-order predicate symbol $\mathcal R$ is interpreted).

And we may have also $\forall X \forall x X(x)$, that reads "every object has every property".