We shall begin by discussing how to express such facts as ``Pat knows the combination of the safe'', although the idea of treating a concept as an object has application beyond the discussion of knowledge.
We shall use the symbol safe1 for the safe, and combination(s) is our notation for the combination of an arbitrary safe s. We aren't much interested in the domain of combinations, and we shall take them to be strings of digits with dashes in the right place, and, since a combination is a string, we will write it in quotes. Thus we can write
as a formalization of the English ``The combination of the safe is 45-25-17''. Let us suppose that the combination of safe2 is, co-incidentally, also 45-25-17, so we can also write
Now we want to translate ``Pat knows the combination of the safe''. If we were to express it as
the inference rule that allows replacing a term by an equal term in first order logic would let us conclude knows(pat,combination(safe2)), which mightn't be true.
This problem was already recognized in 1879 by Frege, the founder of modern predicate logic, who distinguished between direct and indirect occurrences of expressions and would consider the occurrence of combination(safe1) in (8) to be indirect and not subject to replacement of equals by equals. The modern way of stating the problem is to call Pat knows a referentially opaque operator.
The way out of this difficulty currently most popular is to treat Pat knows as a modal operator. This involves changing the logic so that replacement of an expression by an equal expression is not allowed in opaque contexts. Knowledge is not the only operator that admits modal treatment. There is also belief, wanting, and logical or physical necessity. For AI purposes, we would need all the above modal operators and many more in the same system. This would make the semantic discussion of the resulting modal logic extremely complex. For this reason, and because we want functions from material objects to concepts of them, we have followed a different path--introducing concepts as individual objects. This has not been popular in philosophy, although I suppose no-one would doubt that it could be done.
Our approach is to introduce the symbol Safe1 as a name for the concept of safe1 and the function Combination which takes a concept of a safe into a concept of its combination. The second operand of the function knows is now required to be a concept, and we can write
to assert that Pat knows the combination of safe1. The previous trouble is avoided so long as we can assert
which is quite reasonable, since we do not consider the concept of the combination of safe1 to be the same as the concept of the combination of safe2, even if the combinations themselves are the same.
and say that safe1 is the denotation of Safe1. We can say that Pegasus doesn't exist by writing
still admitting Pegasus as a perfectly good concept. If we only admit concepts with denotations (or admit partial functions into our system), we can regard denotation as a function from concepts to objects--including other concepts. We can then write
The functions combination and Combination are related in a way that we may call extensional, namely
and we can also write this relation in terms of Combination alone as
or, in terms of the denotation predicate,
It is precisely this property of extensionality that the above-mentioned knows predicate lacks in its second argument; it is extensional in its first argument.
Suppose we now want to say ``Pat knows that Mike knows the combination of safe1''. We cannot use knows(mike,Combination(Safe1)) as an operand of another knows function for two reasons. First, the value of knows(person,Concept) is a truth value, and there are only two truth values, so we would either have Pat knowing all true statements or none. Second, English treats knowledge of propositions differently from the way it treats knowledge of the value of a term. To know a proposition is to know that it is true, whereas the analog of knowing a combination would be knowing whether the proposition is true.
We solve the first problem by introducing a new knowledge function
Knows(Mike,Combination(Safe1)) is not a truth value but a proposition, and there can be distinct true propositions. We now need a predicatetrue(proposition), so we can assert
which is equivalent to our old-style assertion
We now write
to assert that Pat knows whether Mike knows the combination of safe1. We define
which forms the proposition that a person knows a proposition from the truth of the proposition and that he knows whether the proposition holds. Note that it is necessary to have new connectives to combine propositions and that an equality sign rather than an equivalence sign is used. As far as our first order logic is concerned, (11) is an assertion of the equality of two terms. These matters are discussed thoroughly in (McCarthy 1979b).
While a concept denotes at most one object, the same object can be denoted by many concepts. Nevertheless, there are often useful functions from objects to concepts that denote them. Numbers may conveniently be regarded has having standard concepts, and an object may have a distinguished concept relative to a particular person. (McCarthy 1977b) illustrates the use of functions from objects to concepts in formalizing such chestnuts as Russell's, ``I thought your yacht was longer than it is''.
The most immediate AI problem that requires concepts for its successful formalism may be the relation between knowledge and ability. We would like to connect Mike's ability to open safe1 with his knowledge of the combination. The proper formalization of the notion of can that involves knowledge rather than just physical possibility hasn't been done yet. Moore (1977) discusses the relation between knowledge and action from a similar point of view, and (McCarthy 1977b) contains some ideas about this.
There are obviously some esthetic disadvantages to a theory that has both mike and Mike. Moreover, natural language doesn't make such distinctions in its vocabulary, but in rather roundabout ways when necessary. Perhaps we could manage with just Mike (the concept), since the denotation function will be available for referring to mike (the person himself). It makes some sentences longer, and we have to use an equivalence relation which we may call eqdenot and say ``Mike eqdenot Brother(Mary)'' rather than write ``mike = brother(mary)'', reserving the equality sign for equal concepts. Since many AI programs don't make much use of replacement of equals by equals, their notation may admit either interpretation, i.e., the formulas may stand for either objects or concepts. The biggest objection is that the semantics of reasoning about objects is more complicated if one refers to them only via concepts.
I believe that circumscription will turn out to be the key to inferring non-knowledge. Unfortunately, an adequate formalism has not yet been developed, so we can only give some ideas of why establishing non-knowledge is important for AI and how circumscription can contribute to it.
If the robot can reason that it cannot open safe1, because it doesn't know the combination, it can decide that its next task is to find the combination. However, if it has merely failed to determine the combination by reasoning, more thinking might solve the problem. If it can safely conclude that the combination cannot be determined by reasoning, it can look for the information externally.
As another example, suppose someone asks you whether the President is standing, sitting or lying down at the moment you read the paper. Normally you will answer that you don't know and will not respond to a suggestion that you think harder. You conclude that no matter how hard you think, the information isn't to be found. If you really want to know, you must look for an external source of information. How do you know you can't solve the problem? The intuitive answer is that any answer is consistent with your other knowledge. However, you certainly don't construct a model of all your beliefs to establish this. Since you undoubtedly have some contradictory beliefs somewhere, you can't construct the required models anyway.
The process has two steps. The first is deciding what knowledge is relevant. This is a conjectural process, so its outcome is not guaranteed to be correct. It might be carried out by some kind of keyword retrieval from property lists, but there should be a less arbitrary method.
The second process uses the set of ``relevant'' sentences found by the first process and constructs models or circumscription predicates that allow for both outcomes if what is to be shown unknown is a proposition. If what is to be shown unknown has many possible values like a safe combination, then something more sophisticated is necessary. A parameter called the value of the combination is introduced, and a ``model'' or circumscription predicate is found in which this parameter occurs free. We used quotes, because a one parameter family of models is found rather than a single model.
We conclude with just one example of a circumscription schema dealing with knowledge. It is formalization of the assertion that all Mike knows is a consequence of propositions P0 and Q0.