Relations among Contexts

There are many useful relations among contexts and also context valued functions. Here are some.

1. $specialize\hbox{-}time(t,c)$ is a context related to $c$ in which the time is specialized to have the value $t$. We may have the relation

$$\begin{eqnarray*} \begin{array}{l} c0: \quad ist(specialize\hbox{-}time(t,c),at(... ...\ \ \equiv ist(c,at\hbox{-}time(t,at(jmc,Stanford))). \end{array}\end{eqnarray*}$$

Here $at\hbox{-}time(t,p)$ is the assertion that the proposition $p$ holds at time $t$. We call this a lifting relation. It is convenient to write $at\hbox{-}time(t,foo(x,y,z))$ rather than $foo(x,y,z,t)$, because this lets us drop $t$ in certain contexts. Many expressions are also better represented using modifiers expressed by functions rather than by using predicates and functions with many arguments. Actions give immediate examples, e.g. $slowly(on\hbox{-}foot(go))$ rather than $go(on\hbox{-}foot,slowly)$.

Instead of using the function $specialize\hbox{-}time$, it may be convenient to use a predicate $specializes\hbox{-}time$ and an axiom

$$\begin{eqnarray*} \begin{array}{l} c0:\quad specializes\hbox{-}time(t,c1,c2)\lan... ...p,c1)\\ \ \ \ \ \supset ist(c2,at\hbox{-}time(t,p)). \end{array}\end{eqnarray*}$$

This would permit different contexts $c1$ all of which specialize $c2$ to a particular time.

There are also relations concerned with specializing places and with specializing speakers and hearers. Such relations permit lifting sentences containing pronouns to contexts not presuming specific places and persons.

2. If $q$ is a proposition and $c$ is a context, then $assuming(p,c)$ is another context like $c$ in which $p$ is assumed, where ``assumed'' is taken in the natural deduction sense of section .

3. There is a general relation $specializes$ between contexts. We say $specializes(c1,c2)$ when $c2$ involves no more assumptions than $c1$ and every proposition meaningful in $c1$ is translatable into one meaningful in $c2$. We have nonmonotonic relations

$$specializes(c1,c2) \land \lnot ab1(p,c1,c2) \land ist(c1,p) \supset ist(c2,p).$$

and

$$specializes(c1,c2) \land \lnot ab2(p,c1,c2) \land ist(c2,p) \supset ist(c1,p).$$

This gives nonmonotonic inheritance of $ist$ in both from the subcontext to the supercontext and vice versa. More useful is the case when the sentences must change when lifted. See below for an example.

4. A major set of relations that need to be expressed are those between the context of a particular conversation and a subsequent written report about the situation in which the conversation took place. References to persons and objects are decontextualized in the report, and sentences like those given above can be used to express their relations.

5. Consider a wire with a signal on it which may have the value 0 or 1. We can associate a context with this wire that depends on time. Call it $c_{wire117}(t)$. Suppose at time 331, the value of this signal is 0. We can write this

$$ist(c_{wire117}(331),signal = 0).$$

Suppose the meaning of the signal is that the door of the microwave oven is open or closed according to whether the signal on $wire117$ is 0 or 1. We can then write the lifting relation

$$\forall\ t(ist(c_{wire117}(t),signal = 0) \equiv door\hbox{-}open(t).$$

The idea is that we can introduce contexts associated with particular parts of a circuit or other system, each with its special language, and lift sentences from this context to sentences meaningful for the system as a whole.