Suppose we have the formula . We can then enter the context c and infer the formula . Conversely, if we have the formula we can infer by exiting the context c. We don't always want to be explicit about the sequence of all the contexts that were entered, but the logic needs to be such that the system always exits into the context it was in before entering. The enter and exit operations can be thought of as the push and pop operations on a stack. In the logic presented in  the sequence of contexts that has been entered is always explicitly stated.
We can regard as analogous to , and the operation of entering c as analogous to assuming c in a system of natural deduction as invented by Gentzen and described in many logic texts. Indeed a context is a generalization of a collection of assumptions, but there are important differences. For example, contexts contain linguistic assumptions as well as declarative and a context may correspond to an infinite and only partially known collection of assumptions. Moreover, because relations among contexts are expressed as sentences in the language, allows inferences within the language that could only be done at the meta-level of the usual natural deduction systems.
There are various ways of handling the reasoning step of entering a context. The way most analogous to the usual natural deduction systems is to have an operation . Having done this, one could then write any p for which one already had . However, it seems more convenient in an interactive theorem proving to use the style of Jussi Ketonen's EKL interactive theorem prover . In the style of that system, if one had , one could immediately write p, and the system would keep track of the dependence on c. To avoid ambiguity as to where an occurrence of came from, one might have to refer to a line number in the derivation. Having obtained p by entering c and then inferring some sentence q, one can leave c and get . In natural deduction, this would be called discharging the assumption c.
Human natural language risks ambiguity by not always specifying such assumptions, relying on the hearer or reader to guess what contexts makes sense. The hearer employs a principle of charity and chooses an interpretation that assumes the speaker is making sense. In AI usage we probably don't usually want computers to make assertions that depend on principles of charity for their interpretation.
We are presently doubtful that the reasoning we will want our programs to do on their own will correspond closely to using an interactive theorem prover. Therefore, it isn't clear whether the above ideas for implementing entering and leaving contexts will be what we want.
Sentences of the form can themselves be true in contexts, e.g. we can have . In this draft, we will ignore the fact that if we want to stay in first order logic, we should reify assertions and write something like , where is a term rather than a wff. Actually the same problem arises for p itself; the occurrence of p in might have to be syntactically distinct from the occurrence of p standing by itself. Alternatively to reifying assertions we could use a modal logic; this approach is investigated in [52, 9, 4, 59].