The
Problem of Non-Self-Exemplification
Here’s a famous paradox of properties. Observe that
some properties exemplify themselves.
For example, lacking red exemplifies lacking red. But most properties don’t
exemplify themselves. For example, being
red does not exemplify being red.
Consider, then, the property of lacking
exemplifying oneself. Call this property, non-E. Does non-E exemplify
non-E? Well, if it does, then it would lack exemplifying itself, which means
that non-E would not exemplify non-E. On the other hand, if non-E lacks non-E,
then non-E lacks lacking exemplifying oneself, which
entails exemplifying oneself. Therefore, non-E exemplifies non-E if and only if
non-E lacks non-E, which is absurd. It follows that there is no such thing as
non-E.
This is a paradox because it seems true of many
different things that they do not exemplify themselves. They seem to have non-E
in common. So it looks like non-E should exist, yet non-E cannot exist without
absurdity.
Let’s take a step back and consider the slightly
simpler property, E, which is the property of exemplifying oneself. Ask: does E exemplify E? Suppose not. Then, E
lacks E. That is, E lacks exemplifying oneself, which entails that E lacks E.
That might sounds a little fishy maybe, but I see nothing absurd here. The same
goes for the proposal that E exemplifies E. E exemplifies E entails that E
exemplifies exemplifies oneself, which entails that E
exemplifies E. This isn’t an absurd result, but there does seem to be something
fishy going on. It’s fishy that we have no way to see whether E exemplifies E
or whether E lacks E. This is strange because E’s definition is supposed to
give us direct insight into E, and if we have direct insight into a thing, I
think we should be able to tell whether that thing exemplifies itself or not. I
think the problem is that we don’t
have direct insight into E because we don’t have a clear definition of E. And
the reason we don’t have a clear definition of E is because the definition
contains the term, ‘oneself’. But what does ‘oneself’
mean?
The term, ‘oneself’ is a context sensitive term. Its
meaning is always borrowed from its
context. It’s like the term, ‘it’. In the sentence, “John picked up the ball
and then threw it,” Here, ‘it’ refers to the ball, and
the meaning of ‘it’ is the same as the meaning of ‘the ball’. But in the
sentence, “Sally picked up the vase and then threw it”,
‘it’ refers to the vase. It would be meaningless
to say that Sally and John both throw it
unless there is a context to tell us what ‘it’ means. The term, ‘oneself’ works
the same way: it borrows its meaning from the context. But when we say that a
bunch of things all exemplify oneself,
we haven’t specified what ‘oneself’ means. There is an illusion that we know what ‘oneself’ means
there, but I suspect that this illusion arises from the fact that one and the
same word is being used in each of
the example cases. We think there is a common property because we use a common
word, i.e. ‘oneself’ in our description of each thing. But in reality, there
isn’t a common property being ascribed because the word, ‘oneself’, means
something different in each case. So, I’ve come to think that the expression
‘exemplifies oneself’ is meaningless without context because ‘oneself’ is
meaningless without context.
At this point, we might consider the property of being such that the predicate ‘exemplifies
oneself’ correctly applies to it. After all, isn’t it true that
‘exemplifies oneself’ does correctly apply to many things? But this is just to
make the same mistake again, only this time with the term ‘it’. What do we mean
when we say without context that the
predicate ‘exemplifies oneself’ applies to it. What does ‘it’ mean here? I
suspect it is meaningless.
I’ve discussed this with Peter van Inwagen, and his response was to point out a connection
between ‘it’ and variables in quantificational logic: e.g. there is an x, such
that it exemplifies itself = there is an x, such that x exemplifies x. So as
long as ‘it’ is bound by quantificational operators, we can understand it. But
my reply is that such expressions in
quantificational logic are meaningless for the same reason: the variables do
not mean anything unless they can be interpreted. I realize this is a bold
claim, but I believe it is the sober truth. Before you completely write me off
here, I should say that although I think that some quantificational sentences are strictly speaking meaningless, I do think that the vast majority of
quantificational sentences successfully convey the propositions we wish to
convey. That is, I think that we normally understand the intended meaning of a sentence in quantificational logic but only because we’ve learned to
instinctively interpret the variables in a way that allows us to grasp the
intended proposition: for example, ‘for all x, if x is A, then x is B’ brings
our minds to the proposition that A implies B, where the meanings of ‘A’,
‘implies’, and ‘B’ are grasped directly (though the fact that those terms have
those meanings may not itself be grasped directly); and ‘there is an x such
that x is A’ brings our minds to the proposition that A is exemplified.
If I’m right (and I might not be), then not only is
there no such thing as non-self-exemplification, but there is also no such
thing as bearing R to oneself, for any
R. Interestingly, this means there is no such thing as being self-identical.