Is the set of wffs $ Γ = \{¬Ry | y ∈ V\} ∪ \{∃x Rx\} $ satisfiable?

Question

Let $L$ be a logic language with one unary predicate symbol $R$, no contant symbols, no function symbols. Let $V$ be a set of variables. We define the set of wffs $Γ = \{¬Ry | y ∈ V \} ∪ \{∃x Rx\}$. Is $Γ$ satisfiable? Could someone please give me some hints? Much appreciated. Thanks.

Answer 1

Yes, the set of formulas $\Gamma$ is satisfiable. To show this, you have to give an interpretation that satisfies all its formulas. Intuitively, your formulas say that the predicate $R$ should not hold for the variables in $V$, but that there should be some element for which $R$ holds.

An interpretation consists of a structure and an assignment of the free variables. In your case, the structure would be of the form $(A,R_A)$, for some set of elements $A$ and a set $R_A \subseteq A$, which interprets the predicate $R$. The assignment would be of the form $\nu: V \to A$, mapping the variables $V$ to elements of your structure.

Consider the interpretation with $A = \{0,1\}$, $R_A = \{0\}$ and $\nu(y) = 1$ for all $y \in V$.

• For every $y \in V$, the interpretation satisifies $\lnot Ry$, because $\nu(y) = 1 \notin R_A$.

• The interpretation satisfies $\exists x Rx$, becaus $R_A$ is non-empty.

The interpretation satisfies all formulas in $\Gamma$, therefore $\Gamma$ is satisfiable.