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You are given the signagure $σ = 〈 ; ; P〉$ in which P is a unary predicate symbol. Find a formula F such that $\forall x F = 0$ and $\exists x$ $F = 1$. Prove that it meets these requirements.

From what I understand, we need to assume that the domain of discourse D may be any non-empty set. Also we need to assume that the predicate P may be either true for all $x \in D$ (1), for some $x \in D$ (2) or for no $x \in D$ (3).

I tried to solve it in 3 parts. In case (1) the appropriate formula F would just be P(x). But then once I tried to do the other cases, I got stuck since if all of your values are 0 or 1, how can you change only some of them around by only using the symbols we have $(\neg, \land, \lor, \rightarrow, \leftrightarrow, \forall, \exists)$? Therefore, help would be much appreciated.

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This seems to be impossible ... When the domain contains exactly one object, $\forall x \phi$ and $\exists x \phi$ should always have the same truth-value, no matter what $\phi$ is.

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