We need to make sure that this actually corresponds to (or is analogous to) the "context of statements in English Language" because if this doesn't turn out to be the case then how can we say "mathematical logic doesn't ignore context"?
The answer to this question is that "context" in logic can be precisely defined syntactically, and the syntax is intended to mimic the idea of subordination that we find in natural reasoning and language. So yes, logic is intended to syntactically represent contexts in the same way as natural language. But no, it cannot actually capture the contexts of natural language in an absolute sense, because there must always be a layer of interpretation that converts the symbolic contexts and statements in any logic system to the semantic meanings that (we believe) we understand.
For a simple example, we can say:
Let L denote the statement that natural language is useful.
The logic system can never perform that mental connection between "natural language is useful" and the actual meaning it conveys to most speakers of English. Moreover, different people can interpret it differently and it is opaque to the logic system. However, we purposely ignore that issue so that we can manipulate statements logically according to inference rules before we go back and interpret our conclusions. If the audience agrees with the soundness of the inference rules according to their interpretations, then it is enough to force them to accept conclusions that are validly deduced from accepted premises. Their interpretations may be totally different from ours, and we may even have no way to tell. That aspect can never captured by logic or mathematics.
How is it true that Natural Deduction systems doesn't ignore context and in what sense?
Natural deduction is merely a paradigm, not a system by itself. Its main defining feature is that it utilizes syntactically defined contexts to allow and facilitate symbolic reasoning inside and outside them. In particular we often have introduction and elimination rules to govern contexts.
For instance, the modal logic S4 has a necessity operator "☐" that can be governed by the following context rules loosely described as follows.
If you have deduced:
☐:
...
P.
You can deduce:
☐P.
And vice versa.
We also permit classical reasoning in any context, which together with the ability to create a new context under a "☐", will give rise to the distribution axiom as defined in this SEP article. If we further add the necessitation inference rule, we automatically get both it and (4), which shows the close relation between the two in the framework of natural deduction.
As an aside, a brief discussion with Not_Here revealed that there are some differences in our perspectives, so I'll point them out. Firstly, we both agree that mathematics (and in particular natural deduction) can easily deal with and analyze the same statement in different contexts. However, he/she also claims that model theory is the answer to semantics, and links to SEP's article on model theory. I consider that it presents an incomplete and misleading picture, from a logician's perspective.
The reason is that ultimately any form of reasoning has to be based on some meta-system. Without a precisely defined meta-system, there is no way to validate or invalidate arguments. But the only known way to precisely define a meta-system, as of today, is through syntax. One common way is to specify inference rules of the form "If you have deduced statements of the form ..., then you can deduce the statement ...", and then specify that the meta-system deduces no other statements besides those generated by the inference rules. However, in the interest of making my claim precise, this type of formal system is too restrictive. To achieve the most generality, we define a formal system S to be a system that has a proof verifier V, which is a program (in some fixed Turing-complete language) that, given as input a pair (P,X) (in some fixed encoding), decides (always outputs "yes" or "no") whether P is a valid proof of the statement X or not. While I use the terms "proof" and "statement", they are merely to help the intuition and are not actually part of the definition. In other words, we actually define that a finite string P is a proof over S of a finite string X iff V(P,X) is true, in which case we say X is a theorem of S.
Now we can get down to the contended claim. Model theory, as the term is used by modern logicians today, refers to a branch of mathematics, and modern mathematics is usually based on a particular formal system called ZFC set theory. But ZFC set theory is just one of many incompatible possible foundations for mathematics. So before one can claim that anything can capture real-world meaning correctly, one would first have to justify that it is based on a formal system that is truth-preserving for statements about the real-world. In the case of modern model theory, one would have to justify that ZFC set theory has real-world meaning in this very sense, and this would require that one can give a real-world interpretation of any sentence over ZFC, and that one can show the axiom schemas of ZFC to be truth-preserving (here I'm already granting that classical logic is sound). This is a very tall order, and no logician has ever done this, and remember that there are many incompatible candidates for a foundation of mathematics, so no two of them can be both compatible with the real-world.
Of course, "model theory" as used by philosophers may mean much less than "model theory" used by logicians. That is perfectly normal. But the first consequence of the issue of having to precisely specify the meta-system is that one should not just refer to model theory as if there is only one such notion. In fact, there is a fine spectrum of meta-systems and corresponding philosophical justifications or lack thereof, and I give a brief account and some links in this post.
It should be clear after looking at the spectrum of possible meta-systems that things are not so easily justified as absolute as one might think. For instance, one cannot ever consider "the natural numbers" to be absolute. All useful meta-systems will have a structure that satisfies the axioms of PA, but it is peculiar to each meta-system. No meta-system is able to refer to the so-called 'real' natural numbers in the real-world, even if they exist. The reason is simple and can be justified as follows (in a suitable meta-system).
Take any (consistent) useful formal system S that can construct any collection of 'natural numbers' that is first-order definable over PA. Then there is an arithmetical sentence Con(S) such that:
S does not prove that S is consistent.
S proves that N satisfies Con(S) iff S is consistent.
Now S cannot prove its own consistency otherwise (1) would make S inconsistent. So S cannot prove that N satisfies Con(S). Also:
( S + S is inconsistent ) is consistent.
( S + S is inconsistent ) proves that N satisfies PA.
Of course, we reject ( S + S is inconsistent ) as being actually useful, but why? Simply because (here in the meta-system) we can prove that ( S + S is inconsistent ) must have the wrong notion of N, namely that it is different from the 'real' N (that the meta-system knows). But this shows that PA is utterly insufficient to capture the notion of natural numbers. Historically, Godel also did this to prove the existence of a nonstandard model of PA.
But this has serious consequences. We are unable to pin down the natural numbers, but we need to specify a meta-system that has that very collection (that one might perhaps believe has some platonic existence). No extension of PA will be enough, and so our meta-system may very well be referring to some strange model of PA that is not really the same as the natural numbers of the real world (if such a structure exists). Even the meta-system itself knows this possibility (as shown above)!
This is why ultimately it is a philosophical question whether some given formal system has real world meaning, which then has vast implications for model theory that is carried out in that formal system as the meta-system of choice. Too weak, and one would not be able to deduce basic things about logic, such as described in the linked post. Just one 'wrong' inference rule, and the meta-system would necessarily have a collection of 'natural numbers' that it believes is standard but is actually not. If we believe that there is a collection of representations in physical media that obeys PA, when suitably interpreted using algorithms, (which explains why HTTPS and other known algorithms work,) then we necessarily believe that this model of PA is standard by definition of the physical representations, and hence we cannot accept any meta-system that does not have a standard model of PA. In logic terms, we only accept a meta-system that has an ω-model. The problem is, as you may expect by now, we cannot define "ω-model" except relative to an existing model of PA...
(Actually there is ω-logic, that can actually pin down the natural numbers and the entire first-order theory, but naturally of course ω-logic has no effective deductive system.)
