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I was reading the Wiki article on Hilbert systems and came across this passage:

A characteristic feature of the many variants of Hilbert systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules.

What does this mean exactly?

Does a "context" in a Hilbert system refer to the set of assumptions/non-logical axioms we are building our proofs from in the first place?

For example if we have some proof starting off with $\Delta \vdash \ldots \,$ then we can basically use $\Delta \vdash$ in front of every line of our proof. Is this our "context" and is this what they mean by "it never changes"? And in ND we are able to change contexts?

What does this refer to exactly?

Taroccoesbrocco
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user525966
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1 Answers1

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The Wikipedia article refers to the fact that in natural deduction and sequent calculus there are inference rules that can change the set of assumptions (called context there) in a derivation. For instance, in both natural deduction and sequent calculus there is the following inference rule for introducing implication on the right of $\vdash$ (called $\to_\text{intro}$ in natural deduction, and $\to_\text{right}$ in sequent calculus):

\begin{align} \dfrac{\Gamma, A \vdash B}{\Gamma \vdash A \to B} \end{align}

Indeed, in the premise of such an inference rule the assumptions are $\Gamma, A$ (i.e. the formulas in $\Gamma \cup \{A\}$); while in the conclusion of such an inference rule the assumptions are just the formulas in $\Gamma$, the formula $A$ has been discharged from the assumptions.

It is true that in Hilbert systems the deduction theorem mimics the inference rule above (if $\Gamma, A \vdash B$ is derivable then $\Gamma \vdash A \to B$ is derivable), but it is a methatheorem about the system (it claims that if there is a derivation $\Gamma, A \vdash B$ in a Hilbert system, then there is another derivation $\Gamma \vdash A \to B$ in such a Hilbert system, with a completely different structure from the former), it is not an inference rule in the system, as is the case in natural deduction and sequent calculus.

Taroccoesbrocco
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  • What if we start out with a set of assumptions in one context, a set in another context, and then we want to combine the assumptions together to make a new set/system (i.e. combining different areas of mathematics)? Is this a meta theorem in all logic systems? – user525966 Sep 22 '18 at 17:46
  • @user525966 - Please, be more precise. What is the metatheorem that you want to prove, exactly? In which system? – Taroccoesbrocco Sep 22 '18 at 17:50
  • I don't have a specific example, just wondering. For instance if we have some set of assumptions for natural numbers or something and then another set for some other area of mathematics and we want to be able to use them together (random example) – user525966 Sep 22 '18 at 18:00
  • @user525966 - Putting together different areas of mathematics is not so obvious: first, maybe you should extend your language, moreover you can get a set of axioms that is contradictory or that has only trivial (and hence not interesting) models. For instance, try to put together Peano axioms with the usual axioms for $\mathbb{R}$: then, the induction principle holds no longer for all the elements in your domain, but just for the elements in $\mathbb{N} \subsetneq \mathbb{R}$. – Taroccoesbrocco Sep 22 '18 at 18:06
  • @user525966 - Maybe your question deserves another post, with a more precise explanation of your troubles or of what do you want to get. – Taroccoesbrocco Sep 22 '18 at 18:09
  • I don't have anything in mind, was more just curious how that concept works – user525966 Sep 22 '18 at 18:17