Whenever we write an axiom, a critic can say that the axiom is true only in a certain context. With a little ingenuity the critic can usually devise a more general contex in which the precise form of the axiom doesn't hold. Looking at human reasoning as reflected in language emphasizes this point. Consider axiomatizing ``on'' so as to draw appropriate consequences from the information expressed in the sentence, ``The book is on the table''. The critic may propose to haggle about the precise meaning of ``on'' inventing difficulties about what can be between the book and the table or about how much gravity there has to be in a spacecraft in order to use the word ``on'' and whether centrifugal force counts. Thus we encounter Socratic puzzles over what the concepts mean in complete generality and encounter examples that never arise in life. There simply isn't a most general context.
Conversely, if we axiomatize at a fairly high level of generality, the axioms are often longer than is convenient in special situations. Thus humans find it useful to say, ``The book is on the table'' omitting reference to time and precise identifications of what book and what table. This problem of how general to be arises whether the general common sense knowledge is expressed in logic, in program or in some other formalism. (Some people propose that the knowledge is internally expressed in the form of examples only, but strong mechanisms using analogy and similarity permit their more general use. I wish them good fortune in formulating precise proposals about what these mechansims are).
A possible way out involves formalizing the notion of context and combining it with the circumscription method of nonmonotonic reasoning. We add a context parameter to the functions and predicates in our axioms. Each axiom makes its assertion about a certain context. Further axioms tell us that facts are inherited by more restricted context unless exceptions are asserted. Each assertions is also nonmonotonically assumed to apply in any particular more general context, but there again are exceptions. For example, the rules about birds flying implicitly assume that there is an atmosphere to fly in. In a more general context this might not be assumed. It remains to determine how inheritance to more general contexts differs from inheritance to more specific contexts.
Suppose that whenever a sentence p is present in the memory of a computer, we consider it as in a particular context and as an abbreviation for the sentence holds(p,C) where C is the name of a context. Some contexts are very specific, so that Watson is a doctor in the context of Sherlock Holmes stories and a baritone psychologist in a tragic opera about the history of psychology.
There is a relation meaning that context c2 is more general than context c1. We allow sentences like so that even statements relating contexts can have contexts. The theory would not provide for any ``most general context'' any more than Zermelo-Frankel set theory provides for a most general set.
A logical system using contexts might provide operations of entering and leaving a context yielding what we might call ultra-natural deduction allowing a sequence of reasoning like
This resembles the usual logical natural deduction systems, but for reasons beyond the scope of this lecture, it is probably not correct to regard contexts as equivalent to sets of assumptions -- not even infinite sets of assumptions.
All this is unpleasantly vague, but it's a lot more than could be said in 1971.
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