Finding a guaranteed consequent of a satisfiable sentence

Question

Let A = ∀x∀y(P(x,y)). A is satisfiable. Find any B such that it isn't a tautology and is a consequent of A.

Now if I understand it correctly, then a correct and somewhat trivial solution to this would be any sentence logically equal to A. For example neg(∃x∃y(neg(P(x,y))).

Is there anything else, or is this the only simple solution?

Thank you in advance.

Answer 1

How about reflexivity - $\forall x (P(x, x))$?

Similarly, transitivity $\forall x\forall y\forall z(P(x, y)\wedge P(y, z)\implies P(x, z))$ and symmetry $\forall x\forall y(P(x, y)\implies P(y, x))$. And there are lots of others . . .

Answer 2

Other consequents are $\exists x\exists y(P(x, y))$ if you don't allow empty domain structures in your semantics (as usual). Or you could add things infinitely: $\forall x\forall y(P(x,y))\vee\varphi$ for all $\varphi$ actually.