It has precursors, but Russell's paradox of 1901 shows that the obvious set theory, as proposed by Frege has to be modified in unpleasant ways. Frege's basic idea is to let us define the set of all objects having a given property, in more modern notation

giving the set of all *x* with the property . Thus the
set of all red dogs is denoted by , or if
the set of dogs is denoted *dogs* and the set of red objects as
*reds*, we can also write . This
notation for forming sets is very convenient and is much used in
mathematics. The principle is called *comprehension*.

Bertrand Russell in his 1901 letter to Gottlob Frege pointed out that forming the set

i.e. the set of all sets that are not members of themselves, leads promptly to a contradiction. We get .

There are many ways of restricting set theory to avoid the
contradiction. The most commonly chosen is that of Zermelo, whose
set theory Z allowed only writing , where
*A* is a previously defined set. This turned out to be not quite
enough to represent mathematics and Fraenkel introduce a further axiom
schema of *replacement* giving a system now called ZF.

ZF is less convenient than Frege's inconsistent system because of the
need to find the set *A*, and the unrestricted comprehension schema is
often used when it is clear that the needed *A* could be found.

A more direct inconvenience for giving robots consciousness is the paradox discovered by Richard Montague [Montague, 1963] concerning a set of desirable axioms for knowledge of sentences.

We might denote by *knows*(*person*,*sentence*) the assertion that
*person* knows *sentence* and consider this as holding at some time
*t* in in some situation *s*. However, Montague's paradox arises even
when there is only one knower, and we write *Kp* for the knower
knowing the sentence *p*. Montague's paradoxes arise under the
assumption that the language of the sentences *p* is rich enough for
``elementary syntax'', i.e. allows quantifiers and operations on
sentences or on Gödel numbers standing for sentences.

The axioms are

and

Intuitively these axioms state that if you know something, it's true, if you know something, you know you know it, and you can do modus ponens. Added to this are schemas saying that you know some sentences of elementary logic.

From these, Montague constructed a version of the paradox of the liar. Hence they must be weakened, and there are many weakenings that restore consistency. Montague preferred to leave out elementary syntax, thus getting a form of modal logic.

I think it might be better to weaken (18) by introducing a
hierarchy of *introspective knowledge operators* on the idea that
knowing that you know something is knowledge at an introspective
level.

Suppose that we regard knowledge as a function of time or of the situation. We can slither out of Montague's paradox by changing the axiom to say that if you knew something in the past, you now know that you knew it. This spoils Montague's recursive construction of the paradox.

None of this has yet been worked out for an AI system.

Mon Jul 15 13:06:22 PDT 2002