# Equivalence and Presupposition

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?

## 1 Answer

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 $$\textrm{A presupposes C iff B presupposes C}$$.

• Thanks! But I don't know if this generalizes. Consider this simple example: Mary is happy iff she receives the first prize from the competition. Let $A$ stands for "Mary is happy'' and $B$ for "she receives the first prize from the competition". Clearly, $B$ presupposes that there is such a competition. But this is not presupposed by $A$, right? – Ximei Jan 19 at 17:36
• What you have in your example is a biconditional ($\leftrightarrow$), not logical equivalence ($\Leftrightarrow$). These are two different things. Logical equivalence means that $A$ and $B$ have the same truth values in all conveivable situations. But what is said in the competition example is just that the two states of affairs coincide in this particular situation. This doesn't, however, make the $A$ and $B$ sentences equivalent, i.e. identical in meaning: We can, in principle, easily think of a situation where Mary is unhappy despite having won the competition. – lemontree Jan 19 at 17:41
• So the transitivity is not applicable here because your example is not one of logical equivalence. Indeed, as you notice, it fails here; the argument doesn't work for biconditional claims. – lemontree Jan 19 at 17:43
• Ah, okay. I see the difference. Thanks a lot! – Ximei Jan 19 at 17:45