TITLE: Equivalence and Presupposition
QUESTION [7 upvotes]: This is a very basic question about the relationship between presupposition and implication. The question is as follows: Let $A$, $B$ and $C$ be three propositions and let $A\Leftrightarrow B$, do we have ($\star$)? $$\textrm{$A$ presupposes $C$ iff $B$ presupposes $C$}\quad\quad\quad(\star)$$ In other words, if two propositions are logically equivalent, must they carry the same presuppositions?

REPLY [2 votes]: Yes. The definition of presupposition is
$$A \text{ presupposes } C \text{ iff } A \vDash C \text{ and } \neg A \vDash C \text{ and } \nvDash C$$
That is, $A$ presupposes $C$ iff $C$ follows both from $A$ and its negation but $C$ is not tautological.
If $A \Leftrightarrow B$, then $\neg A \Leftrightarrow \neg B$. So
$\begin{align*}
& A \text{ presupposes } C \\
\text{ iff } & A \vDash C \text{ and } \neg A \vDash C \\
\text{ iff } & B \vDash C \text{ and } \neg B \vDash C \\
\text{ iff } & B \text{ presupposes } C
\end{align*}$
So $\textrm{$A$ presupposes $C$ iff $B$ presupposes $C$}$.