Deduction Theorem
Theorem
Let $\mathscr H$ be instance 1 of a Hilbert proof system.
Then the deduction rule:
- $\dfrac{U,\mathbf A \vdash \mathbf B}{U \vdash \mathbf A \implies \mathbf B}$
is a derived rule for $\mathscr H$.
Proof
For any proof of $U, \mathbf A \vdash \mathbf B$, we indicate how to transform it into a proof of $U \vdash \mathbf A \implies \mathbf B$ without using the deduction rule.
This is done by applying the Second Principle of Mathematical Induction to the length $n$ of the proof of $U,\mathbf A \vdash \mathbf B$.
If $n = 1$, then one of the following must occur:
In the first case, obviously $U \vdash \mathbf B$.
By Axiom 1, $U \vdash \mathbf B \implies \paren {\mathbf A \implies \mathbf B}$.
By Modus Ponens, $U \vdash \mathbf A \implies \mathbf B$.
In the second case, $U \vdash \mathbf A \implies \mathbf A$ by the Law of Identity.
Finally, in the third case, we have $U \vdash \mathbf B$.
As in the first case, we conclude $U \vdash \mathbf A \implies \mathbf B$.
If $n > 1$, the only other option for arriving at $U, \mathbf A \vdash \mathbf B$ is through Modus Ponens.
That is to say, two earlier lines of the proof contain:
- $U, \mathbf A \vdash \mathbf C$
- $U, \mathbf A \vdash \mathbf C \implies \mathbf B$
for some WFF $\mathbf C$.
But then these sequents have shorter proofs.
Hence, they satisfy the induction hypothesis.
Thus, we may infer:
- $U \vdash \mathbf A \implies \mathbf C$
- $U \vdash \mathbf A \implies \paren {\mathbf C \implies \mathbf B}$
This allows us to give the following proof of $U \vdash \mathbf A \implies \mathbf B$:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | $U \vdash \mathbf A \implies \mathbf C$ | Hypothesis | ||||
| 2 | $U \vdash \mathbf A \implies \paren {\mathbf C \implies \mathbf B}$ | Hypothesis | ||||
| 3 | $U \vdash \paren {\mathbf A \implies \paren {\mathbf C \implies \mathbf B} } \implies \paren {\paren {\mathbf A \implies \mathbf C} \implies \paren {\mathbf A \implies \mathbf B} }$ | Axiom 2 | ||||
| 4 | $U \vdash \paren {\mathbf A \implies \mathbf C} \implies \paren {\mathbf A \implies \mathbf B}$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 2 , 3 | |||
| 5 | $U \vdash \mathbf A \implies \mathbf B$ | Modus Ponendo Ponens: $\implies \mathcal E$ | 1 , 4 |
The result follows by the Second Principle of Mathematical Induction.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): deduction theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): deduction theorem