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I am learning the basics of Natural Deduction and I meet a problem about the role of context.

In Natural Deduction, if we have $\vdash A\rightarrow B$, we also have $A\vdash B$; if we have $A\vdash B$, we obtain $\vdash A\rightarrow B$.

My question is: if we have $\Gamma\vdash A$ where $\Gamma$ denotes a context, do we have $\vdash\Gamma\rightarrow A$; and if we have $\vdash\Gamma\rightarrow A$, do we have $\Gamma\vdash A$? I ask this because I didn't see any occurrence of contexts written as $\Gamma,\Delta$ after $\vdash$, is it wrong to write in that way?

Thank you in advance!

Natalie Clarius
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Fred
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    If $\Gamma$ is a finite set of formulas, then yes, we can instead simply consider the conjunction of them as a single formula $\gamma:=\bigwedge \Gamma$, so that $\Gamma\vdash A$ if and only if $\vdash \gamma\to A$. – Berci Jun 09 '20 at 19:17
  • @Berci Thanks Berci! – Fred Jun 09 '20 at 22:03
  • @Review I disagree with the vote as duplicate. The linked question asks about the conceptual difference between the notions $\vdash$ vs. $\to$; this one presupposes this knowledge and asks about the applicability and notation of the deduction theorem. – Natalie Clarius Jun 11 '20 at 11:28

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Yes, we do, in a way, have the generalization $\Gamma \vdash A\ \Longleftrightarrow\ \vdash "\Gamma \to A"$, and yes, it is wrong to write it that way, because $\Gamma$ is not a formula, and only formulas may occur as the antecedent to an object-language implication $\to$.

The solution is that when importing the set of premises into the conclusion, the premises become conjoined: The generalized deduction theorem says that

$$\Gamma \vdash A \ \Longleftrightarrow \ \vdash (\gamma_1 \land \ldots \land \gamma_n) \to A$$

(where $\gamma_i \in \Gamma$).

For more convenient notation, one may introduce a symbol for generalized conjunction:

$$\bigwedge \Gamma = \gamma_1 \land \ldots \land \gamma_n$$

and thus write

$$\Gamma \vdash A \ \Longleftrightarrow \ \vdash \bigwedge \Gamma \to A.$$

However, as user21820 correctly points out, this is only applicable if the set $\Gamma$ is finite, since there is no such thing as infinitely conjoined formulas.

Natalie Clarius
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  • Thanks! I would like to know more about that. Do you have any references for this kind of notation? Because I didn't see it in my textbooks and I tried to search for similar notations but I didn't find anything. – Fred Jun 09 '20 at 22:04
  • Out of the textbooks I had lying around, I found $\bigwedge$ (which suggests there are other notions introduced as well) in W. Rautenberg A concise introduction to mathematical logic and D. van Dalen Logic and structure. I haven't read the first one in detail, but the second one I found a decent textbook. However, be aware that when it comes to notation, different textbooks may use different symbols. – Natalie Clarius Jun 09 '20 at 22:31
  • Hi, thanks a lot! I find it in van Dalen's classical textbook. Just an afterthought: from $\Gamma\vdash A$, we have $\vdash\bigwedge\Gamma\rightarrow A$, it seems that we can also have $\Gamma\vdash\bigwedge\Gamma\rightarrow A$ (by adding $\Gamma$ as a context), right? – Fred Jun 10 '20 at 18:03
  • Yes; to clarify, t the reason we can do this is the following: We have $\Gamma \vdash A \ \Longrightarrow \ \Gamma, \Delta \vdash A$ (and hence also $\Longrightarrow \ \Gamma \vdash \bigwedge \Delta \to A$) for any set of propositions $\Delta$ (and thereby also for $\Delta = \Gamma$). This property of classical logic is known as monotonicity or weakening. – Natalie Clarius Jun 10 '20 at 18:16
  • You find some more references here: https://math.stackexchange.com/q/1577114/344246 – Natalie Clarius Jun 11 '20 at 12:37
  • Hi! Please emphasize that the conjunction of an axiom set as used in your post is meaningful only if that set is finite, otherwise we simply cannot apply the deduction theorem. Thanks! =) – user21820 Mar 11 '21 at 09:09
  • @user21820 Of course; thanks for pointing it out. – Natalie Clarius Mar 11 '21 at 09:22
  • Thank you very much! =) – user21820 Mar 11 '21 at 09:26