The alleged paradox begins with a sceptical inquiry about my right to
claim that my past usage of '+' (i.e., my past usage of the plus sign)
was used to denote the function plus rather than the function quus. The
definition of quus is: x quus y = x + y, if x, y < 57; otherwise, x
quus y = 5. (Kripke uses an encircled plus sign to represent the quus
sign. I can't reproduce that sign here so I'll just use 'quus').
Basically, the problem is that on all of the problems that I have done
so far, the plus and quus functions demand the same answers. So,
whether I know it or not, my past responses to computations have been
in accord with plus just as much as they have been in accord with quus.
Thus, there seems to be no reason at all to prefer the claim that I've
been plussing to the claim that I've been quussing, given my
hypothetical past history. After all, the two functions are identical
over the cases that I am allowed to appeal to so far.
Given this setup, Kripke boldly asks: "Who is to say that [quus] is not
the function I previously meant by '+'?" The sceptic claims that no one
can legitimately claim that quus is not the function I previously meant
by '+', given the hypothetical situation he proposes, because no one
can find a fact that shows that I meant plus rather than quus. More
specifically, the sceptic challenges those of us who disagree with him
to produce a fact that shows that I meant plus rather than quus. Also,
the sceptic requires that the fact in question "must, in some sense,
show how I am justified in giving the answer '125' to '68 + 57'"
(rather than '5'). (K, p. 11). (I have always seen the second form of
the challenge, as Kripke calls it, viz., the requirement that the fact
show how I am justified in answering '125' rather than '5', as simply a
way of clarifying the task at hand. After all, challenging someone to
produce a fact that I meant plus rather than quus is not the most
obvious challenge in the world, for a variety of reasons. Since it's
clear that if I such a fact could be produced, it would justify my
saying '125' rather than '5', the second form of the challenge does not
impose a new task on us. Rather, it clarifies the task to be
performed).