show that a horn sentence is preserved under a direct product.

Question

show that a horn sentence is preserved under a direct product.

If $\varphi$ is a horn sentence and $\mathfrak{A}_i, i \in \text{I}$ is a model for $\varphi$ namely $\mathfrak{A}_i \vDash \varphi$ then $\Pi_{i \in I} \mathfrak{A}_i \vDash \varphi$

is this best done inductively?

Answer 1

I don't see any direct need for induction.

Hint. Call a wff $\psi$ "smooth" if, whenever you have a variable assignment in each of the $\mathfrak A_i$s that makes $\psi$ true, the pointwise product of the assignment makes $\psi$ true in the product model.

Prove that an unquantified Horn clause is always smooth.

Prove that the class of smooth wffs is closed under universal and existential quantification.

Thus (here's room for a bit of induction if you're very formal) a Horn sentence is smooth.

Clearly truth of a smooth sentence is preserved by products.