## Derivation of the Law of Extensions

[Note: We use $$\epsilon F$$ to denote the extension of the concept $$F$$.]

We want to show, for an arbitrarily chosen concept P and an arbitrarily chosen object $$c$$, that $$c\in\epsilon P\equiv Pc$$.

$$(\rightarrow)$$ Assume $$c\in\epsilon P$$ to show $$Pc)$$. Then, by the definition of $$\in$$, it follows that

$$\exists H(\epsilon P\eqclose \epsilon H \amp Hc)$$

Suppose that $$Q$$ is such a property. Then, we know

$$\epsilon P\eqclose \epsilon Q \amp Qc$$

But, by Basic Law V, the first conjunct implies $$\forall x(Px\equiv Qx)$$. So from the fact that $$Qc$$, it follows that $$Pc$$.

$$(\leftarrow)$$ Assume $$Pc$$ (to show $$c\in\epsilon P)$$. Then, by the Existence of Extensions principle, $$P$$ has an extension, namely, $$\epsilon P$$. So by the laws of identity, we know $$\epsilon P =\epsilon P$$. We may conjoin this with our assumption to conclude

$$\epsilon P\eqclose \epsilon P\amp Pc$$

Now by existential generalizing on the concept $$P$$, it follows that

$$\exists H(\epsilon P\eqclose \epsilon H \amp Hc)$$

Thus, by the definition of $$\in$$, it follows that $$c\in\epsilon P$$.