# Contexts in Natural Deduction

This is my first post. I have a basic question about the use of context in natural deduction. If $$A$$ is true in an empty context, written as

$$\vdash A$$

then, by monotonicity, in any context $$\Gamma$$, $$A$$ is true as well, written as

$$\Gamma\vdash A.$$

However, the interpretation of $$\Gamma\vdash A$$ is usually that $$A$$ is true under the assumptions in $$\Gamma$$.

My question is, if there is any way to signal that $$A$$ is true and independent from any assumption in $$\Gamma$$ in the sequent $$\Gamma\vdash A$$? In other words, how is it possible to keep the interpretation of $$\vdash A$$ while adding a context $$\Gamma$$ (by monotonicity) in front of the sequent? Thanks!

• As you said, to say that $A$ is derivable independently from every context $\Gamma$ we have $\vdash A$. Jul 7, 2020 at 14:28
• You can try with a contrived: $\Gamma \supseteq \emptyset \vdash A$, but I've never seen it... Jul 7, 2020 at 14:31
• @MauroALLEGRANZA Thanks! I've never seen it either. It looks strange... Jul 7, 2020 at 14:57

The interpretation of Γ⊢𝐴 is usually that 𝐴 is true under the assumptions in Γ.

Not quite: It is "under at most the assumptions in $$\Gamma$$". The definition of $$\vdash$$ reads

$$\Gamma \vdash A$$ iff there is a derivation $$\mathcal{D}$$ with end formula $$A$$ and $$\text{Hyp}(\mathcal{D}) \subseteq \Gamma$$ (where $$\text{Hyp}(\mathcal{D})$$ is the set of open assumptions of the derivation).

Note the $$\subseteq$$ rather than $$=$$: It is nowhere required that all or even any of the assumptions in $$\Gamma$$ actually be used in the derivation of $$A$$. The possibility that there are formulas in $$\Gamma$$ on which $$A$$ is not dependent, or conversely, the ability to add arbitrary premises to the left-hand side of the turnstile, is already built into the definition of derivability.

To express that an empty context suffices, write just that: $$\vdash A$$.

And to express that all premises in a given context are necessary for the derivation and can not be reduced further, write that: "$$\Gamma \vdash A$$, and there exists no $$\Delta \subsetneq \Gamma$$ such that $$\Delta \vdash A$$".

• Thanks! Yes, I agree, it's sufficient for the hypotheses of $A$ be a subset of those contained in $\Gamma$. But your suggestion didn't answer my question. I agree with you on the way to define the smallest context. But what I am wondering is that how we can signal the smallest context (in my question, it's an empty context) while using a larger one in front of the sequent. Do you have any idea about this? Jul 7, 2020 at 15:46
• I don't know of any more compact notation than the one in my last sentence. Jul 7, 2020 at 15:47