**Definition**[Quine Transform]

The

*Quine transform*of $\mathcal{A}$ under $\pi$, written $\mathcal{A}^{\pi}$, is given by:

$\mathcal{A}^{\pi}: = (A, \pi[\vec{R}], \pi[\vec{f}])$.For example, suppose the model $\mathcal{A} = (A,R)$ is specified as follows:

$A= \{0,1, 2\}$Let $\pi: A \to A$ be the transposition that swaps $0$ to $1$. Then,

$R= \{(0,1), (0,2), (1,2) \}$.

$\pi[R] = \{(1,0), (1,2),(0,2)\}$Consequently, $R$ and $\pi[R]$ are extensionally distinct. However, $\mathcal{A}$ and $\mathcal{A}^{\pi}$ are

*isomorphic*under $\pi$. More generally, one can see that:

**Lemma**["Quine Transform Lemma"]

Let $\pi : A \to A$ be any bijection. Then: $\mathcal{A}^{\pi} \cong \mathcal{A}$.

This is all quite simple discrete mathematics. But it has interesting applications.

objection: relevance?

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