What is satisfiable by a reduct of a model is satisfiable by the original model (and vice versa)?

Question

My professor told me that any formula that is satisfiable by a reduct of a model is satisfiable by the model it is a reduct of, and vice versa (as long as the formula is interpretable on the reduct(?))This seems intuitively true to me, but I want to prove it and I don't know how. I think I should use proof by induction on formulas, but I am at a loss as to what to say besides something like "It's obviously true for atomic formulas, and it's also obviously true for more complex formulas built using the formula building operations". Any ideas?

Answer 1

See Reduct of a structure $\mathcal M$ : it is obtained omitting some of the operations and relations of that structure. The domain is not affected by the "reduction".

For the question :

that any formula that is satisfiable by a reduct of a model is satisfiable by the model it is a reduct of,

we have to consider that for an atomic $P(x)$ to be satisfiable in the "reduced" structure $\mathcal M^R$ means that there is an element $a$ in the "common" domain of the two structures such that $\mathcal M^R \vDash P(x)[a]$; but then also : $\mathcal M \vDash P(x)[a]$.

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This is the basis case for the induction; the cases for the connectives are strightforward ...

For the $\exists$ quantifier, the argument is the same : if there is some $a \in |\mathcal M^R|$ such that $\mathcal M^R \vDash \varphi(x)[a]$, then obviously $\mathcal M \vDash \varphi(x)[a]$, because the two structures have the same domain.