Use of contexts in formal proofs

Question

I have the idea that a formal system is defined by a set of inference rules that can be used to extend a finite set of theorems starting from the axioms. If I look at the wikipedia page or if I ask on stackexchange I see that people use a more relaxed definition of inference which requires the use of some kind of context or nested proof. For example to prove $P\implies Q$ one creates a context where $P$ is assumed to be true, then operates in this context until is able to prove $Q$. Then one can deduce $P\implies Q$ outside the context.

In the wikipedia page this is explained saying that $P\implies Q$ can be derived from $P \vdash Q$. I cannot fully understand how this can be turned into a formal definition. If I take it literally it requires to formally define $P \vdash Q$. I think that it is of fundamental importance to know if the inference rules are computable (so that a computer can make a formal proof verification). So it is very important to explain which is the algorithm to check if $P\vdash Q$. I think that in general there is not such an algorithm (in view of Turing's Theorem) hence I think that $P\vdash Q$ should not be used in defining the inference rules of a formal system. Am I wrong?

I can instead understand how the use of context could be formalized. However the introduction of contexts makes inference rules much more complicated since every theorem must be associated to the context where it was proven. So I can imagine that $P\implies Q$ can be added to the set of theorems in a context if there is a sub-context where $P$ is taken as an axiom and $Q$ is a theorem. This explains the algorithmic bit that was missing in the previous approach.

Now, finally, the point of my question. Why don't we simply prepend each theorem in the context with the assumptions of the context? In the case where the context has the proposition $P$ added as an axiom we could simply replace each theorem $T$ with $P\implies T$ and avoid the use of contexts. The only problem being that the formal inference rules should address this: for example we cannot simply say that $Q\land Q$ can be derived from $Q$. We should say that also $P\implies (Q\land Q)$ can be derived by $P\implies Q$ since $P\implies Q$ can be viewed as the theorem $Q$ in the context where $P$ holds.

Clearly the use of contexts seems easier to understand and is what usually mathematicians do in everyday life. But I would say that using contexts in the formal definitions is hiding some of the formal difficulties (among which the restrictions on the usage of free and bounded variables). So I wonder if there is some discussion of these issues and in case if there is some exposition of propositional logic, predicate logic, or set theory which is context free.

Answer 1

I wish to set down some remarks that might help shape and delineate thoughts about the notion of context in logic.

Two senses of context can be identified in the discourse of logic:

• A broad one which can be referred to as an antecedent given context of deducibility after Nuel Belnap, construed as the "assumptions about the nature of deducibility" we have beforehand. This is a leading topic in philosophical logic; one can follow it up via Belnap's seminal paper "Tonk, Plonk and Plink," and
• A narrow one, briefly to say, denoting all what stand on the left-hand side of $\vdash$ with respect to those on the right-hand side, viewed as interrelated by structural rules that were set forth by Gerhard Gentzen.

Within a total setting constituted by the context in the broad sense working in the background and the ones in the narrow sense working in the foreground, we have articulated logical systems (incorporating instantiation/generalisation, introduction/elimination rules, etc.) that have fared quite well over the decades.

A particularly significant area of study on the notion of context is dropping any structural rule, thus yielding substructural logics (hence the prefix 'sub-' in the name as in the word 'substandard'). As Greg Restall says in his SEoP article "Substructural Logics,"

[d]ifferent logical systems differ in the rules governing particular logical concepts. Intuitionistic logic famously differs from classical logic in its treatment of negation. The structural rules are an essential part of any explanation of the behaviour of individual logical concepts, because we use the ambient context of logical consequence to characterise the behaviour of those concepts.

So, for example, by dropping contraction and weakening, we get linear logic, and by dropping only contraction, we get affine logic.

Overall, it seems, we have to do more in the way to adequately characterise context as a generic concept, before turning to the questions concerning operational stipulations.

Answer 2

What Context means in a deduction?

It is the collection of the axioms of the theory that are implicitly assumed for the proof, like e.g. axioms for set theory in real analysis.

Wrt your linked question, in that case the "context" is the derivation: outside it, free variables have no meaning. Specifically, the proof start from axiom ∀x(x=x) and the following step instantiates it to (a=a).

This is correct whatever a is: either or variable or a constant, because the axiom states that reflexivity holds ALWAYS. This means that e.g. in the case of arithmetic we are licensed to derive (0=0) and so on.

Nitpicking issue: what you call "predicates" are "open formulas" i.e. formulas with free occurrences of variables. –