# Why does the Deduction Theorem use Union?

We have an initial set of premises $S$.

We are given or observe or assume sentence(s) $A$ is/are true.

We can then prove $B$.

Formally, $S \cup \left\{A\right\} \vdash B$.

Shouldn't it be an intersection? Only when we have both $S$ and ${A}$ can we prove $B$. Not when we have $S$ or ${A}$, which is what my mental picture of what Union is.

• You are "using" $\lor$ in the wrong way ... The set $S \cup \{ A \}$ is not a set that is "equal" to $S$ or to $A$ but is the set of those objects $x$ that belong to $S$ or belong to $\{ A \}$ : $x \in S \cup \{ A \}$ iff $x \in S \lor x \in \{ A \}$ i.e. $x \in S \cup \{ A \}$ iff $x \in S \lor x = A$. – Mauro ALLEGRANZA Jun 28 '15 at 19:16

When you say you ate soup and salad, you don't mean that you ate something that is both soup and salad. You mean you ate a soup, and you also ate a salad. Your lunch, then, consisted of the union $$\{\text{Soup}\} \cup \{\text{Salad}\},$$ not the intersection.
That is the sense that is being used here. When you have a set with both $S$ and $A$, you have $$S\cup \{A\}$$ which contains $S$ and also includes $A$.
Also note that the intersection is clearly wrong here, since $S\cap \{A\}$ is equal to either $\emptyset$ or $\{A\}$, so doesn't contain anything from $S$ other than $A$.