# Should multiple premises of a natural deduction inference rule always have the same context?

Consider the conjunction introduction and implication elimination rules of natural deduction:

$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I) \qquad \frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(single)}$$

and note that the context $\Gamma$ of both premises of $(\to E)$ and $(\land I)$ must be the same.

Because this need not be the case in general, why not to write those rules like this instead:

$$\frac{\Gamma\vdash\alpha \quad \Delta\vdash\beta}{ \Gamma,\Delta\vdash \alpha \land \beta} (\land I') \qquad \frac{ \Gamma \vdash \alpha \to \beta \quad \Delta \vdash \alpha} {\Gamma, \Delta\vdash\beta} (\to E') \qquad \text{(multiple)}$$

i.e. with the rules stated like this one might allow premises with distinct contexts.

Questions:

1. Should multiple premises of a natural deduction inference rule always have the same context?

2. In spite of their generality, why most (if not all) textbook or canonical presentations of the inference rules of the natural deduction refrain from using $\text{(multiple)}$-like rules? Because they are less didactical?

3. Aren't $\text{(multiple)}$-like rules valid as well in the natural deduction?

Thanks!

## 2 Answers

Your generalized rules are equivalent to the original rules as long as you have the weakening rule (which every reasonable system would have):

$\dfrac{Γ \vdash φ}{Γ,Δ \vdash φ}$

• If these rules are taught as part of an introductory course in logic, then yes pedagogically it is more sensible to use the original rules, which are obviously valid, as compared to the generalizations, which are not so obviously valid. – user21820 Jun 20 '16 at 6:11

Multiple like rules work out that we increase the amount assumed in order to get to the derived formula. Generally speaking, it comes as better to derive something from fewer premises than from more premises.

Really, the original form comes as bad enough, as often enough you don't need every premise in gamma, you only need a few of those premises to derive what lies on the right side of $\vdash$.